measures of central tendency final

7/28/2019 Measures of Central Tendency Final

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Quantitative Aptitude 2.1 Measures of Central Tendency

CHAPTER-2

MEASURES OF CENTRAL TENDENCY

Introduction:After classifying and tabulating the data, the statistician is usually interested in analyzing

the data. This is done with the help of various numerical measures which describe theinherent characteristics of a frequency distribution.A statistical measure summarizes the data and brings out the important characteristics ofthe data in such a way that a clear and accurate picture emerges. We will discuss certainstatistical measures which will be helpful in analyzing the data.

Meaning:The word measures means methods and the word central tendency means averagevalue of any statistical series.The combined term Measures of central tendency meansthe methods of finding out the central value or average value of a statistical series or anyother series of quantitative information.An average is a single figure that is computed from a given series to give complete ideaabout the entire series. Its value lies between the maximum and the minimum value of aseries and represents all the items belonging to the series.

Need for the average:1. It is not possible to remember each and every fact relating to a field of enquiry as

human mind is not capable of storing all facts.2. It is not desirable to remember or take note of all the facts relating to a study because

it may lead to confusion in the mind of a person if all the facts are presented before him.3. It is the average value which is quite fit to give a clear picture or efficacy about the

field under study for guidance and necessary conclusion.

Definition:1. Average is a value which is typical or representative of a set of data.2. Average is an attempt to find one single figure to describe whole of figures.3. An average is a single number describing as number some features of a set of data.4. The average is sometimes described as number which is typical of the whole

group.

Characteristics of an average:1. It is a single figure expressed in some quantitative form.2. It lies between the extreme values of a series.3. It is a typical value that represents all the values in a series.4. It is capable of giving a central idea about the series it represents.5. It is determined by some method or procedure.Objectives of an average:The main objectives of an average are:1. To determine one single value that may be used to describe the characteristics of the

entire series.2. To facilitate comparison at a particular point of time or over a period of time.


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3. To facilitate statistical inference: An average obtained from a sample is used inestimating the average of the population.

4. To facilitate quick understandingof complex data.5. To help the decision-making process: The averages help the managers in decision-

making. The managers are often interested in knowing normal output of a plant,

representative sales volume, overall productivity index, price index, etc., these all arethe connotations of an average.

Characteristics of an Ideal average:1. It should be rigidly defined so that there is no confusion with regard to its meaning

connotation.

2. It should be easy to understand.3. It should be simple to compute.4. Its definition should be in the form of mathematical formula.5. It should be based on all the items in the data.

6. Any single item or a group of items should not unduly influence it.7. It should be capable of further algebraic treatment.8. It should be capable of being used in further statistical computation.9. It should have sampling stability.

Advantages of an average:1. It simplifies the complexity of data.2. It facilitates comparative study.3. It is easy to remember.Disadvantages of an average:1. It may give rise to a figure which may not belongto the series.2. It may give rise to a result which is practically impossible and absurd.3. It may lead to dangerous results.4. It may give different results.5. It does not reveal the entire story of a phenomenon.6. It does not have sympathy for individual item.Different Types of Average:There are three types of averages;

A.Mathematical Averages:

1. Arithmetic Average or mean2. Geometric Mean3. Harmonic Mean

B. Positional Averages:1. Median2. Mode

C. Positional Averages:1. Moving average2. Progressive average


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A.Mathematical Averages:1.Arithmetic Mean:

Meaning of Arithmetic Mean:The arithmetic mean is a measure of central tendency and is popularly known as mean.Arithmetic mean is obtained by dividing the sum of the values of all items of a series by

the number of items of that series. Normally, arithmetic mean is denoted by X which isread as X bar. It can be computed for unclassified or ungrouped data or individual seriesas well as classified or grouped data or discrete or continuous series.

1.Arithmetic MeanA. Simple arithmetic Mean.B. Weighted arithmetic Mean.C. Combined arithmetic Mean.A.Simple Arithmetic Mean:The simple arithmetic mean can be computed in two ways;

1) Direct method.2) Short cut method.

Computation of simple Arithmetic Mean:Individual series:Direct Method Steps involved are:1. Identify the variable and denote it by x.2. Sum up the observations i.e. x.3. Count the number of observations. i.e. n.4. Apply the following formula.

nxx

Where, x =Arithmetic Mean, x =sum of the variable x, and n=the total number of

observations.

Short cut MethodSteps involved are:1. Identify the variable and denote it by x.2. Select an assumed mean or provision mean A from the values of variables or any

value.3. Take the deviations of the values of the variablex from the assumed mean,

i.e., (d=x-A). The deviations are denoted byd4. Sum up the deviations d i.e. d 5. Divide the sum of deviations by the number of observations and add the resulting

figure to the assumed mean or apply the following formula.

n

dAX

Where,

X = Arithmetic meanA = Assumed meand = Sum of the deviations of individual values from assumed mean.

N = f=Number of observations.


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Discrete SeriesDiscrete series means where frequencies are given but without class intervals.Direct Method Steps involved are:1. Identify the variable (x) and frequency (f) from the data given.2.Multiply each variable with respective frequency. i.e. fXx3.Apply the following formula.

N

fxX

Where, f = frequency,x = the value of the variable and N = the sum of frequency or f

Short Cut Method Steps involved are:1. Identify the variable (x) and frequency (f) from the data given.2.Select an assumed mean or provision mean A from the values of variables or any value.3.Take the deviations of the values of the variable x from the assumed mean, i.e., (d=x-A).

The deviations are denoted by d

4.Multiply each frequency with respective deviation (fXd) and their summation (fXd).5.Apply the following formula.

f

fdAX

Where, A = Assumed Mean, dx = (x A), f = frequency And f or N = total number of

items.

Continuous Series:Continuous series means where frequencies are given along with class intervals:Direct Method Steps involved are:

Here we take the mid point of every class interval as x multiple these values withrespective frequencies.1.Convert the class series into exclusive series if given in the open-ended or cumulative

series. It may be noted that the mean can be easily calculated if the problem is given ininclusive series or mid-value series.

2.Find out the mid-value of each class and denote them byx.3.Find N; the total of the frequencies (N = f )4.Multiply each mid-value with the respective frequency and find out (fx).5.Apply the following formula

N

fx

f

fx

X

.

Where, f is frequency; x the mid point of interval and N = f .

Short Cut Method Steps involved are:1.Convert the class series into exclusive series if given in the open-ended or cumulative


2. Find out the mid-value of each class and denote them byx.3.Select an assumed mean or provision mean A from the values of variables or any value.4.Take the deviations of the values of the variable X from the assumed mean, i.e., (d=x-A).

The deviations are denoted by d


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5.Multiply each frequency with respective deviation (fxd) and find their summation (fxd).6.Apply the following formula.

N

fdAX

Where, A = Assumed Mean dx= A); f = frequency And or N = total number of

items.

B.Weighted Arithmetic MeanThe term weight stands for the relative importance of the different items of the series.Weighted arithmetic mean refers to the arithmetic mean calculated after assigning weightsto different values of variable. It is suitable where the relative importance of different itemsof variable is not the same. It is especially useful in problems relating to:a. Construction of index numbers.b. Standardised birth and death rates.In case actual weights are available, actual weights should be used. In case actual weights

are not available, arbitrary weights may be used.The relationship between simple arithmetic mean and weighted arithmetic mean issummarized below:1. AM=WAM if equal weights are assigned to all the items of a series.2. AM>WAM if smaller weight is assigned to the higher values and greater weight is

assigned to the lower values.3. AM


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3. The formula of arithmetic mean can be extended to compute the combined average oftwo or more related series.

4. If the given items ofx can be changed to items on Y = a + bx, then Y = a + b.5. If each of the values of a variable x is increased or decreased by some constant C, the

arithmetic mean also increases or decreases by C. Similarly, when the values of a

variable xare multiplied by a constant, say k, the arithmetic mean is also multiplied bythe same quantity k. when the values of variable x are divided by a constant say d,the arithmetic mean is also divided by the same quantity d.

6. The arithmetic mean of the first n natural numbers can be calculated by applying theformula:

Merits of Arithmetic Mean:1. It is easy to understand.2. It is simple to calculate.3. It is based on all the items of the series. In its computation no item is ignored.4. It is rigidly defined by a mathematical formula so that the same answer is derived by

every one who computes it.5. It is capable of further algebraic treatment so that its utility is enhanced. The formula of

arithmetic mean can be extended to compute the combined average of two or morerelated series.

6. It has sampling stability. It is least affected by sampling fluctuations.7. Its computation does not require arrangement of items since it is not based on position

in the series.8. It is characterized as a centre of gravity a point of balance, balancing the value on

either side of it.

9. Arithmetic average can be calculated if we know the number of items and aggregate.10. It provides a good basis for comparison.11. The mean is a more stable measurer of central tendency (Ideal average)

Limitations of Arithmetic Mean:1. Since it includes all the items, its value may be distorted by extreme values.2. It cannot be calculated if any item of the series is missing.3. The average may not coincide with any of the actual items in a series.4. In cases where the items cannot be represented quantitatively, like intelligence, honesty

and character but can be ranked, the arithmetic average is not an appropriate measure

of central tendency.5. It cannot be located by observation or the graphic method.6. It gives greater importance to bigger items of a series and lesser importance to smaller

items.7. It cannot be calculated in a distribution with open-ended class series and cumulative

series without converting the class series into exclusive series.8. It fails to provide a characteristic value or a representative value where the distribution

of the series is not normal and gives a U shaped curve rather than a bell shaped ( curve.


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2. Geometric Mean:A. Simple Geometric Mean.B. Weighted Geometric Mean.C. Combined Geometric Mean.A.Simple Geometric Mean.Meaning of geometric mean:

Geometric mean is the nthroot of the product of n number of values. It means if there aretwo values then Geometric mean is square root of the product of two values. Symbolically,

G.M. = n nXXXX ........321

Where, X1, X2. refer to the values of various items of the seriesn = Total no. of items of the series.

Computation of Geometric Mean:Practical steps involved in the computation of G.M. in case of individual series

1.Take the logarithms of each item of variable, and enter in the column headed as log Xand obtain their total i.e., log X.

2. Calculate G.M. As follows:G.M. = Antilog

n

Xlog

Where, n = Total no. of items

Calculation of Geometric Mean Discrete SeriesLet X1, X2, X3.Xn be the given values of a variable and f1, f2, f3 fn, are their respectivefrequencies. Then the geometric mean is given by the formula.

Log g =n

nn

ffffXfXfXf

......log......loglog

321

2211

Log g =

n

i

iXf1

log

or g = Antilog

n

i

iXf1

log

Steps:1. Take log values of all the items of a given series.2. Multiply each log value to its respective frequencies3. Add the values and divide by the total number of frequencies4. Take the value of antilog from the antilog table and the result would be the geometric

mean.

Calculation of Geometric Mean Continuous SeriesIn case of continuous series, the following steps are observed.1. Convert the class series into exclusive series if given in the open-ended or cumulative


2. Take the mid-value of each of the class group and denote it by X.3. Take log value of these mid-values i.e. Log x.4. Multiply log values with their respective frequencies i.e. f x log X.


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Note: If any three values of the four used in the above formula are known, the fourth valuecan be found out.Tutorial Note: In case annual rate of decrease in variable (e.g. depreciation on fixed assets)is to be calculated, the Compound Interest Formula will be modified as follows:Pn = Po (1-r) n

Properties of Geometric mean:1. The logarithm of G for a set of observationsx1,x2,x3, x4,.xn is the arithmetic mean of the

logarithm of the observation, i.e. Log G=

n

nxx klog.......loglog 21 .

2. If all the observations of a variable assumes the same constant value k, i.e. x1=x2=x3=x4.xn=k, then the geometric mean of these observations is also k.

3. For any series of positive values the geometric mean is smaller than the arithmeticmean.

4. If any value is zero, the value of geometric mean is infinity and thus inappropriate.5. It cannot be calculated if number of negative values is odd.6. Since geometric mean or G=

nXXXMG .....

21. , the geometric mean raised to the

nth power equals to the product of values.7. The geometric mean of two variables is the product of their G.M. i.e., if Z=xy, then G.M.

of Z= (G.M. ofx) x (G.M. of y).8. The geometric mean of the ratio of two variables is the ratio of the G.M. of these two

variables i.e., if Z=x/y. then G.M. of Z= (G.M. ofx) (G.M. of y).

Merits of Geometric mean

1.

It is based on all observations2. It is rigidly defined3. It is useful in averaging ratios and percentages and in determining rates of increase and

decrease.4. It is capable of algebraic treatment. Its formula can be extended to calculate combined

G.M. as follows:

12.M.G = Antilog

21

2211 ..log..log

NN

MGNMGN

5. It gives less weight to large items and more to small ones than does the arithmeticaverage. It is because of this reason that geometric mean is never larger than the

arithmetic mean, on occasions it may turn out to be same as the arithmetic mean, butusually it is smaller.

6. It is useful in studying economic and social data.Limitations of Geometric Mean1. It is difficult to understand.2. It is difficult to compute. Non-mathematical persons cannot do calculations.3. The Geometric mean cannot be computed if any item in the series is negative or zero.4. It has restricted application.5. It is biased for small values as it gives more weight to small values.6. At times it gives a value which may not be found in the series and even may be absurd.


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3. Harmonic Mean:A. SimpleHarmonic Mean.B. WeightedHarmonic Mean.C. Combined Harmonic Mean.A. Simple Harmonic Mean.Meaning of Harmonic Mean:Harmonic mean of various items of a series is the reciprocal of the arithmetic mean of theirreciprocals.

Individual series:Steps to calculate H.M1. Identify the variable and denote it by X.2. Calculate the reciprocal of values of variable i.e. 1/x and find its summation.3. Apply the following formula.

nXXXX

nMH1

...111

..

321

Where, X1, X2..Xn refer to the value of various items of the series.n = Total no. of items of the series.

Note: To simplify the calculations, reciprocals of the various items are taken from themathematical tables.

Discrete series:Steps to calculate H.M

1. Identify the variable(x) and frequency (f) from the data given.2. Divide each frequency with the respective value of the variable i.e. (f/x) and find theirsummation i.e. (f/x).

3. Apply the following formula.

nx

f

x

f

x

f

x

f

NMH

...

..

321

Where, N= f.

Continuous Series:Steps to calculate

1. Convert the class series into exclusive series if given in the open-ended or cumulativeseries. It may be noted that the mean can be easily calculated if the problem is given ininclusive series or mid-value series.

2. Take the mid-value of each of the class group and denote it by X.3. Divide each frequency with the respective value of the variable i.e. (f/x) and find their

summation i.e. (f/x).4. Apply the following formula.

n

x

f

x

f

x

f

x

f

NMH

...

..

321

Where, N= f.


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B. Weighted Harmonic MeanSteps to calculate:

1. Identify the variable(X) and weights (W) from the data given.2. Divide each weight with the respective value of the variable i.e. (w/x) and find their

summation i.e. (w/x).

3. Apply the following formula: xWW

Wx

Wx

Wx

WMH

n

n

w/1

....11

..

2

2

1

1

C. Combined Harmonic MeanSteps to calculate:1.Identify the number of observations in each group and denote them n1, n2, n3, n4nm.2.Identify the respective harmonic means H.M1, H.M2, H.M3, H.M4 H.Mm.3.Apply the formula:

2

2

1

1

21..

H

n

H

nnnMH

Properties of Harmonic Mean1. If all the observations of a variable assumes the same constant value k, i.e. x1=x2=x3=

x4.xn=k, then the geometric mean of these observations is also k.2. Harmonic mean can be computed from a series with any number of negative values.3. Harmonic mean cannot be computed from a series from a series if any of its value is

zero.

4. For any series in which all the values are not equal nor any value is zero, the value ofthe harmonic is less than the geometric mean and arithmetic mean.

5. If there are two groups with n1 and n2 observations and H1 and H2 as respective H.Msthen the combined H.M. is given by

2

2

1

1

21..

H

n

H

n

nnMH

Uses of Harmonic Mean:1. For the rates and ratios involving speed, time and distance, harmonic mean is used to

find out the average speed at which journey has been performed.

2.

For the rates and ratios involving price and quantity (both amount of money spent andthe units per rupee are given), harmonic mean is used. To find out the average price atwhich an article had been sold or purchased.

3. When it is desired to assign greater weight to smaller values and smaller weight tolarger values of a variate, its use is recommended.

4. In a given data set if there are a few large values, the reciprocals will turn down theeffect of large numbers. In such cases harmonic mean is to be used.

Merits:1. It is rigidly defined.2. It is based on the all the observations of the series.3. It is suitable in case of series having wide dispersion.


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4. It is suitable for further mathematical treatment.5. It gives less weight to large items and more weight to small items.6. It is the most suitable average for measuring the time, speed, etc.,Demerits:

1.

It is difficult to calculate and not easy to understand.2. All the values must be available for computation.3. It is not popular.4. It is usually a value which does not exist in series.5. It cannot be computed when one or more items are zero6. It gives largest weight to smallest items. Hence, it is not useful for analyzing the

economic data.

Relationship among the Averages

(a)where values of all items are identical(b)where values of all items are not identical A.M. = G.M. = H.M.A.M. > G.M. > H.M.

B. Positional Averages:1. Median

Meaning of Median:Median is the central value of the variable that divides the series into two equal parts insuch a way that half of the items lie above this value and the remaining half lie below thisvalue. Median is called a positional average because it is based on the position of a givenobservation in a series arranged in an ascending or descending order and the position ofthe median is such that an equal number of items lie on either side of it. It concentrates onthe middle or centre of a distribution. According to L.R. Connor, Median is that value of

the variable which divides the group into two equal parts, one part comprising all thevalues greater, and the other all values less than the Median. Median is usually denotedby Med. or Md. Median can be computed for both ungrouped data (Individual Series) andgrouped data (or Discrete/continuous Series).

Calculation of MedianIndividual Series

To find the value of Median in this case the values are arranged in ascending or descendingorder first; and then the middle most value is taken as Median.

The terms are arranged in ascending or descending order and then

2

1nth term is taken as

Steps to Calculate1. Arrange the terms in ascending or descending order2. Count the number of terms. Put = n3. Calculate

2

1nth value,

2

1nth value is Median.

Discrete SeriesSteps to Calculate1. Arrange the data in ascending or descending order2. find cumulative frequencies


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3. find the value of the middle item by using the formula; Size of

2

1Nth item

4. By inspecting cumulative frequencies, Find out that cumulative frequency in thecumulative frequency column which is equal to

2

1Nth or next greater than th

value.5. Locate the value of the variable corresponding to that cumulative frequency. This is the

value of median.

Continuous series:Steps to Calculate:

1. Convert the class series into exclusive series if they are given in inclusive series, mid-value series or cumulative series. Median can be calculated if class series are given inexclusive series or open-ended series.

2. Arrange the data in ascending order.3. Find the cumulative frequencies.4. Find the Median class by ascertaining the class where the cumulative frequency is equal

to or greater than N/2.

5. Apply the formula: M = L + cf

fCN o

.1

N1 = N/2

C.fo = cumulative frequency preceding the cumulative frequency of median class.F = frequency of the median class.C = class interval of the median class.

Mathematical property of median

1. If the two variablesxand y are related by Y= a+bx, where a and b are the constants.Also the value of median ofx is given, then Ymd =a+bxmd.

2. The sum of the absolute deviations (i.e., deviations ignoring signs) from the medianis minimum i.e.,

dMX is the minimum.

Merits of median1. It is easy to understand.2. It is quite rigidly defined.3. It does not require all the observations of the data for its determination.4. Median is useful in case of open-ended series such as income distribution since median

is based on the position and not on the values of items.5. Median is easier to compute than mean in case of unequal class-intervals.6. It is not affected by extreme values.7. It is most suitable average for dealing with qualitative data i.e., where ranks are given.8. It can be determined graphically.9. It is capable of being expressed in qualitative form as it is not computed but located.10.It gives a value, which very much exists in the series, and is a round figure in most of

the cases.11.It minimizes the total absolute deviations.

2

1N


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Limitations of median1. It requires the arrangement of data in ascending/descending order.2. It is not rigidly defined and as such its value cannot be computed but located.3. It is not based on all items of the series.4. It is not capable of algebraic treatment. Its formula cannot be extended to calculate

combined Median of two or more related groups.5. It is affected more by sampling fluctuations than the value of mean.6. The computation formula of a median is in a way an interpolation under the

assumption that the items in the median class are uniformly distributed, which is notvery true, i.e. In case of continuous series, the median is estimated, but not calculated.

7. Typical representative of the observations cannot be computed if the distribution ofitem is irregular. For example 1, 2, 3, 100 and 500, the median is 3.

8. It ignores the extreme items.9. Where the number of items is large, the prerequisite process. i.e. arraying the items is a

difficult process.

Partition valuesPartition values are the positional measures which divide the series into equal parts say 4equal parts or 10 equal parts or 100 equal parts. The most popular partition values used areQuartiles, Octiles, Deciles and Percentiles.

QuartilesMeaning of QuartilesThe values of a variate that divide the series or the distribution into 4 equal parts areknown as Quartiles. Since three points are required to divide the data into 4 equal parts, we

have three quartiles Q1, Q2, Q3.Range

Q1 Q2 Q3Median = Q2The first quartile (Q1), known as a lower quartile, is the value of a variate below whichthere are 25% of the observations and above which there are 75% of the observations.The second quartile (Q2) known as middle quartile or median is the value of a variatewhich divides the distribution into two equal parts. it means, there are 50% observations

above it and 50% below it.The third quartile (Q3), known as an upper quartile, is the value of a variate below whichthere are 75% observations and above which there are 25% observations.It is clear that Q1 < Q2 < Q3.

Computation of QuartilesThe procedure for computing quartiles is the same as the median as explained earlier.I. In case of Individual and discrete Series (after arranging the size of item in

ascending/descending order)

Q1 = Size of

4

1Nth item, Q2 = Size of

4

12 Nth item, and Q3 = size of

4

13 Nth item


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II. In case of Continuous Series (i.e., frequencies with Class Intervals)c

f

fcN

LQcf

fcN

LQcf

fcN

LQ

..

4

3

,

..4

2

,

..4

321

Where, L = Lower limit of quartile class

c.f = Cumulative frequency preceding the quartile classf = Simple frequency in the quartile classc= class-interval of quartile class

OctilesMeaning of OctilesThe value of a variate which divides a given series or the distribution into 8 equal parts areknown as octiles. Each octile contains 12.5% of the total number of observations. Sinceseven points are required to divide the data into 8 equal parts, we have 7 octiles in thedistribution.O1 to O7.Computation of Octiles

I. In case of Individual and Discrete Series (after arranging the size of items inascending/descending order)

8

)1(

NjofsizeOj

th item where, j = 1 to 7

(e.g. O4 = size of8

)1(4 Nth item

II. In case of continuous series (i.e., frequencies with class-intervals)i

f

fcjN

LOj

..

8 Where, j = 1 to 7

Where c.f. is the cumulative frequency preceding the j th octile class, the other symbols haveusual meaning.

DecilesMeaning of DecilesThe values of a variate that divide the series or the distribution into 10 equal parts arecalled Deciles. Each part contains 10% of total observations. Since nine points are requiredto divide the data into 10 equal parts, we have 9 deciles, D 1 to D9. They are called firstdecile, second decile, etc., The 5th decile (D5) is the median.Computation of Deciles

I. In case of Individual and Discrete Series (after arranging the size of items inascending/descending order)

Dj = Share of

10

1Nj th item. Where j = 1 to 9

II. In case of continuous series (i.e., frequencies with Class Intervals)i

f

fcjN

LDj

.10 Where j = 1 to 9

Where c.f. is the cumulative frequency preceding the j th decile class, the other symbols have

usual meaning.


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PercentilesMeaning of PercentilesThe value of a variate which divides a given series or distribution into 100 equal parts areknown as percentiles. Each percentile contains 1% of the total number of observations.Since ninety nine points are required to divide the data into 100 equal parts, we have 99

percentiles P1 to P99. The percentile Pj is that value of the variate upto which lie exactly j% ofthe total number of observations. For ExampleP10= Value of a variate upto which lies exactly 10% of observations. This is same as D 1.P25= Value of a variate upto which lies exactly 25% of the total number of observations. This

is same as Q1.P50= Value of a variate upto which lies exactly 50% of the total number of observations. This

is the same as D5 or Q2, or median.

Computation of PercentilesI. In case of Individual and Discrete Series after arrangement of the size of items in

ascending/descending order)

Pj = size of100

)1( Njth item where j = 1 to 99

II. In case of Continuous Series (i.e. frequencies with Class Intervals)Pj = i

f

fcjN

L

.100 where j = 1 to 99

Where, c.f. is the cumulative frequency preceding the j th percentile class. The other symbolshave usual meaning.

ModeMeaning of mode:Mode is often said to be that value in a series which occurs most frequently or which hasthe greatest frequency. But it is not exactly true for every frequency distribution. Rather it isthat value around which the items tend to concentrate most heavily. It is also called themost typical or fashionable value of a distribution because it is the value which has thegreatest frequency density in its immediate neighbourhood. It is usually denoted by M0.It maybe noted that a distribution may have one mode or two modes or several modes.

Unimodal A distribution is said to be unimodal if it has only one mode.

Bimodal A distribution is said to be bimodal if it has two modes.Multimodal A distribution is said to be multimodal if it has more than two modes.Mode may be calculated for ungrouped data or individual series and grouped data ordiscrete and continuous series.

Calculation of Mode:

Individual Series:The terms are arranged in any order, Ascending or Descending, if each term of the series isoccurring once, then there is no mode, otherwise the value that occurs Maximum numbertimes is known as Mode.


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Method to calculate Mode1. Arrange the terms in ascending or descending order2. Note the term occurring maximum number of times if any.3. This term is mode.Note: If all terms occur once or some terms occur equal number of times, we cant find

Mode by this method)

Discrete SeriesThe mode can be determined by Inspection Method or grouping method.Under Inspection method Mode is that value of the variable which has the highestfrequency.But this method is applicable if the following conditions are fulfilled:1. There must be one item which has maximum frequency.2. The maximum frequency should be located at most in the middle or centre of the series.3. The frequencies should follow a particular pattern, i.e., they should gradually rise and

reach maximum level and thereafter decline.4. The difference between highest frequency and second highest frequency should belarge.

However, some times it becomes impossible to locate Mode by inspection as concentrationof frequencies is not in a unique manner or fashion as desired for this method. In such casesgrouping method should be used to find the modal value. And for such a distribution wehave to prepare the (1) grouping Table and (2) Analysis Table

1. Grouping Table:It has Six Steps as given below:1. Take the Frequency column as column number 12. Frequencies are added in groups of two(s) and leave the incomplete group.3. Leaving first frequency, add frequencies in groups of two(s) & leave the incomplete

group.4. Add the frequencies in groups of three(s) & leave the incomplete group.5. Leaving first frequency, add the frequencies in the groups of three(s) and leave the

incomplete group.6. Leaving first two frequencies, add frequencies in groups of three(s) and leave the

incomplete group.In each case, take maximum total and put it in a circle or a box to distinguish it from others.

2. Analysis Table:It has following steps:1. Note highest value in each column.2. Note the variable(s) in each column corresponding to that total.3. Check if that total is of individual items or of group of many items (2 or 3).4. It the total consists of two or more frequencies, all such variables have to be marked

with the signs as or .5. Count or marks in each column.6. Variable with maximum or marks denotes mode.


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Continuous Series:In the case of Continuous Series, we go only one step ahead of the method for discreteseries. We get the value of Mode by Interpolation as is the case with Median.Steps involved:1. Convert the class series into exclusive series if it is given in inclusive series, mid-value

series or cumulative series. Mode can be calculated the series is exclusive or open-endedseries.

2. Arrange the data in the ascending order.3. Convert the class intervals into equal class intervals if they are unequal.4. Find out the modal class either by inspection method or grouping method.5. Apply the formula: c

fff

ffLMode

201

01

2

Where, L = is the lower limit of Modal Interval

1f is the frequency corresponding to Modal Interval

0

fis the frequency preceding Modal Interval

2f is the frequency succeeding Modal Interval

C is the length of Modal Interval.We can put this formula in following shape also:

Mode cDD

DL

21

1

Here; 011 ffD

212 ffD

Here only the positive values are takenMode can also be calculated taking upper Limit.

cfff

ffLZMode

201

21

2)(

Or cDD

DLZMode

21

1)(

Here: L is the upper limitOther values are the same as in above given formula.Note:

1. If first class is the modal class then 0f will be zero.2. Similarly if last class is modal class, then 2f is zero.3. If Modal Value lies in any other interval than with highest frequency, the following

method is suggested to calculate method

iff

fLMode

20

2

Merits of mode1. It is easy to understand as well as easy to calculate. In certain cases, it can be found out

by inspection

2. It is usually an actual value as it occurs at the most highest frequency in the series.


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3. The mode is a measure which actually indicates what many people incorrectly believethe arithmetic mean indicates. In certain situations mode is the only suitable average,e.g., modal size of garments, modal size of shoes, modal wages, etc.,

4. It is not affected by extreme values.5. Its value can be determined in open-end distribution without ascertaining the class-

limits.6. It can be used to describe qualitative phenomenon. For e.g. in market research to

determine the consumer preferences for different types of products, modal preferencecan be used.

7. Its value can be determined graphically.8. It indicates the point of maximum concentration in case of highly skewed or non-normal

distributions.

Limitations of Mode1. In case of bi-modal/multimodal series, mode cannot be determined.2. It is not capable of further algebraic treatment. For example, combined mode of two ormore series cannot be calculated.3. It is not based on all the items of series.4. It is not rigidly defined measure because different formulae give somewhat different

answers.5. Its value is affected significantly by the size of the class-intervals.6. It is difficult to compute when there are both positive and negative items in a series and

when there is one or more items are zero.7. It is stable only when the sample is large.8. Mode is influenced by magnitude of the class-intervals9. It may be give weight to extreme items.Properties of mode1. If the two variables xand y are related by Y= a+bx, where a and b are real constants

variates and Also the value of mode of x is given, then the mode of Y is given by theequation Ymo =a+bxmo.

Relationship between Mean, Median and ModeEmpirical relationship between Mean, Median and Mode was discovered and propoundedby Prof. Karl Pearson. Through his experiments he found that in a moderately skeweddistribution the distance between Mean and the median is about one-third the distance

between the mean and the mode. Thus, Karl Pearson has expressed the relationship asfollows:

Mode = 3 Median 2 MeanMedian= (Mode + 2 Mean)/3Mean = (3 Median Mode)/2

MOVING AVERAGESMeaning: Moving average is an arithmetic average of data arising over a period of timeand is calculated by replacing the first item in the average by the newly arising item. Eachmoving average is based on values covering a fixed time span which is called period of

moving averages.


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UsefulnessThe successive averaging process does a smoothing operation in the time series data, i.e., itirons out fluctuations of uniform period and intensity. They can be completely eliminatedby choosing the period of moving average that coincides with the period of the cycle.

LimitationThe moving averages cannot be computed for all the given years.

Progressive AverageMeaning Progressive average is an arithmetic mean of the progressive total (i.e.,cumulative total).

ComputationProgressive totals of variable are calculated and then divided by the progressive (i.e.,cumulative) number of items. No previous figure is left as is done in the case of movingaverage. The progressive average for the first year would remain the same; the progressive

average for the second year is equal to2ba ; for the third year

3cba , for the fourth year

4

dcba , and so on.

UsefulnessProgressive averages are useful in order to find out how a business/economy is growing.Such averages can be obtained for all the years for which the data are given.

Choice of a suitable averageNo single average can be considered as best for all circumstances. The suitability of a

particular average depends on the following factors:1. Purpose: The choice should be made according to the purpose that an average isdesigned to serve. Following guidelines are suggested in this regard:

Purpose Suitable average(a)To give equal importance to all the items of series (a)Arithmetic Mean(b)To locate the position of an item in relation to other

items(b)Median and other partition

values

(c)To find out the most common or most fashionableitem

(c)Mode(d)To give more importance to small items than big

items

(d)Geometric Mean(e)To give greatest importance to small items (e)Harmonic Mean2. Nature and form of Data: The choice should be made according to the nature and form

of data available. Following guidelines are suggested in this regard:

(a)For open-end distributions particularly when thedistribution is found to be j-shaped or reverse j-shaped. For example, price distribution andIncome Distribution

(a)Median

(b)To describe qualitative data. For example, to studythe consumer preferences for different products

(b)Mode


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(c)To compute average rates of increase/decrease,average ratios, averages percentages

(c)Geometric Mean(d)To compare the value of a variable with another

variable which is constant. For examples varyingspeed with constant distance, varying quantities

bought/sold per rupee.

(d)Harmonic Mean

(e)In other cases (e)Arithmetic MeanNote: Arithmetic mean should not be used in the following cases:1. In case of highly skewed distributions.2. In case of open-end distribution.3. When there are extreme items i.e., very large and very small items.4. To compute average ratios and rates of change.5. When the distribution is unevenly spread, concentration being small or large at

irregular points.

3. Amenability to further Algebraic Treatment: If an average is to be used for furtheralgebraic treatment, arithmetic mean is considered to be the best as it is very widelyused.

4.Special Purposes: For calculating trend in time-series analysis, the moving averagewould be the most suitable average.


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MEASURES OF DISPERSIONMeaning of Dispersion:The word dispersion means deviation, difference or spread over of certain values from theircentral value. In relation to a statistical series, dispersion refers to the deviation of thevarious items of a series from their central value or difference between any two extremevalues of the series. Further, the word measure means a method of measuring orascertaining certain values. Thus, the phrase measures of dispersion means the variouspossible methods of measuring the dispersion or deviations of the different values froma designated value of the series. The designated value may be average value or any otherextreme value. Dispersion is also known as Scatter, Spread or Variation, It may be notedthat the measures of dispersion measure only the degree (i.e., the amount of variation)but not the direction of the variation. The measures of dispersion are also called averagesof the second order because these measures give an average of the differences of variousitems from an average.

Characteristics of dispersion:1. It consists of different methods through which variation can be measured in

quantitative manner.2. It deals with a statistical series.3. It indicates the degree or extent to which various items of a series deviate from its

central value.4. It supplements the measures of central tendency in revealing the characteristics of a

frequency distribution.5. It speaks of the reliability or otherwise of the average value of a series.Significance of measuring DispersionMeasures of dispersion are calculated to serve the following purposes:1. To determine the reliability of an average Measuring variability determines the

reliability of an average by pointing out to what extent the average is representative ofthe entire data.

2. To facilitate comparison Measures of dispersion facilitate comparisons of two or moredistributions with regard to their variability.

3. To facilitate control Measures of dispersion determine the nature and cause ofvariation in order to control the variation itself.

4. To facilitate the use of other statistical measures Measuring variability facilitates theuse of other statistical measures such as correlation, regression, statistical inference, etc.,

Properties of a good measure of DispersionSince a measure of dispersion is the average of the deviations of items from an average, itshould also posses all the qualities of a good measure of an average. According to Yule andKendall, the qualities of a good measure of dispersion are as follows: -1. Simple to Understand It should be simple to understand2. Easy to Calculate It should be easy to calculate.3. Rigidly Defined - It should be rigidly defined. For the same data, all the methods

should produce the same answer. Different methods of computation leading to different

answers are not proper.


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4. Based on all Items It should be based on all items. When it is based on all items, it willproduce a more representative value.

5. Amenable to further Algebraic Treatment It should be amenable to further algebrictreatment. This means combining groups, calculation of missing values, adjustment forwrong entries, etc., which are possible without the knowledge of actual values of all

items. Such treatment should be possible with a good measure of dispersion also.6. Sampling Stability It should have sampling stability. It means that the average

difference between the values obtained from the sample and the corresponding valuesfrom the population should be the least. If it is so far a measure of dispersion, it is thebest measure.

7. Not Unduly Affected by Extreme Items It should not be unduly affected by theextreme items. Extreme items, many times, are not true representatives of the data. Sotheir presence should not affect the calculation to a large extent.

Measures of Dispersion

Measures of Dispersion may be either absolute or relative.1.Absolute Measure of Dispersion Absolute measure is the measure of dispersion which

is expressed in the same statistical unit in which the original data are given such asKilograms, Tonnes, Kilometers, and Rupees etc. These measures are suitable forcomparing the variability in two distributions having variables expressed in the sameunits and of the same average size. These measures are not suitable for comparing thevariability in two distributions having variables expressed in different units. Followingare the absolute measures of dispersion:

2. Relative Measures of Dispersion Relative measure of dispersion is the ratio of ameasure of absolute dispersion to an appropriate average or the selected items of thedata. It may be noted that the same average base should be used as has been used whilecomputing absolute dispersion. Relative measures of dispersion are also called asCoefficient if Dispersion because relative measure is a pure number that isindependent of the unit of measurement. Following are the relative measures ofdispersion

Absolute Measures of Dispersion

Based on Selected Items Based on All Items

1. Range2. Inter-Quartile Range 1. Mean Deviation2. Standard Deviation


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1. RangeRange is defined as the difference between the value of largest item and the value ofsmallest item included in the distribution. Measures of range may be absolute or relative.

Individual seriesSteps to calculate:1. Ascertain largest value and smallest value.2. Calculate Range by taking out the difference between largest value and smallest value:

Range = Largest value Smallest value.3. Calculate coefficient of Range by dividing the difference between largest value and

smallest value by the aggregate of largest value and smallest value.

Coefficient of Range = L-S/L+S.

Discrete seriesSteps to calculate:1. Identify the variable(x) and frequency (f) from the data given.2. Ascertain largest and smallest value from the values of variable.3. Calculate Range by taking out the difference between largest value and smallest value:



Coefficient of Range = L-S/L+S.

Tutorial Note: Frequencies are not considered at all for computing Range and itsCoefficient.

ContinuousseriesSteps to calculate:1. Convert the class series into exclusive series if given in any other series.2. Ascertain largest and smallest value from the values of variable i.e. Ascertain the upper

limit of highest class and lower limit of lowest class.3. Calculate Range by taking out the difference between largest value and smallest value:



Relative Measures of Dispersion

Based on Selected Items Based on All Items

1. Coefficient of Range2. Coefficient of quartile 1. Coefficient of Mean Deviation2. Coefficient of Standard Deviation

or Coefficient of Variation


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Coefficient of Range = L-S/L+S.Tutorial Note: Frequencies are not considered at all for computing Range and itsCoefficient.

Interpretation of Range

If the averages of the two distributions are almost same, the distribution with smaller rangeis said to have less dispersion and the distribution with larger range is said to have moredispersion.

Merits of RangeThe merits/advantages of range are as follows:1. It is easy to understand.2. It is easy to calculate3. It does not require any special knowledge.4. It takes minimum time to calculate the value of range.Limitations of rangeThe demerits/disadvantages/limitations of range are as follows:1. It does not take into consideration all items of the distribution2. Only two extreme values are taken into consideration3. It is affected by extreme values4. It is not possible to find out the range in open-end frequency distribution5. It does not present very accurate picture of the series.6. It is not suitable for mathematical treatment7. It does not indicate the direction of the variability.Uses of Range:1. Range is used in industries for the statistical quality control of the manufactured

product by the construction of Control Chart. If the range increases beyond a certainpoint, the product may be examined to find out the reasons for variations.

2. Range is useful in studying the variations in the prices of stock, shares and othercommodities that are sensitive to price changes from one period to one period.

3. The meteorological department uses the range for weather forecasts since public isinterested to know the limits within which the temperature is Likely to vary on aparticular day.

Properties of range

1. If all the observations assumed by a variable are constant, say k, then the range is zero.2. Range remains unaffected due to a change of origin but is affected in the same ratio due

to a change in scale, if for any two constants a and b, the two variables x and y arerelated by Y= a+bx, then the range of y is given by Ry=|b|x Rx.

Quartile Deviation:Meaning Of Quartile Deviation or Semi-Inter Quartile Range:Quartile Deviation (or Semi-Inter Quartile Range) is half of the difference between upperquartile (Q3) and lower quartile (Q1). Quartile deviation indicates the average amount bywhich the two quartiles differ from the median. In symmetrical distribution the two

quartiles (Q1

and Q3

) are equidistant from the median [i.e., Median Q1

= Q3

- median]. As


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a result Median Q.D. covers exactly 50% of the observations of the distribution.Measures of Quartile Deviation may be absolute or relative.

Individual series, Discrete series and Continuous series.Steps to calculate:

1.

Calculate lower quartile (Q1)2. Calculate Upper Quartile (Q3)3. Calculate Quartile deviation and its coefficient by applying the formula given below.Measure of Quartile Deviation Formula

1. Absolute Measure of QuartileDeviation

Q.D. =2

13 QQ

Q.D. = Quartile DeviationQ3 = Third Quartile or Upper QuartileQ1 = First Quartile or Lower Quartile

2. Relative Measure of QuartileDeviation

Coefficient of Q.D. =13

13

QQ

QQ

Tutorial Note: In case of symmetrical distribution, Median = (Q3 + Q1)/2

Merits of Quartile Deviation:1. It is easy to understand2. It is easy to calculate3. It can be used in Open-end frequency distribution4. It is lest affected by extreme items.Limitations of Quartile Deviation:

1. This method of dispersion is not based on all the items of the series. It ignores the first25% of the items and the last 25% of the items.

2. It is a positional average; hence not amenable to further mathematical treatment.3. It is very much affected by sampling fluctuations4. It is not a proper measure of dispersion since it shows the distance on a scale and not

the scatter around an average. It is a positional average and is not measured from anaverage.

5. It gives only a rough measurement.Properties of Quartile Deviation

1. Quartile Deviation remains unaffected due to a change of origin but is affected in thesame ratio due to a change in scale, if for any two constants a and b, the two variables xand y are related by Y= a+bx, then the Quartile Deviation of y is given byQ.D of y =|b|x Q.D ofx.

Mean DeviationMeaning of mean deviationMean Deviation is the arithmetic mean of the absolute deviations (i.e., deviations treated aspositive regardless of their actual signs) of all items of the distribution from a measure ofcentral tendency, which could be mean or median or sometimes even mode. It is alsoknown as first moment of dispersion.


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Individual seriesSteps to calculate:1. Calculate Mean, Median or mode of the series.2. Take deviations |D| from either mean, median or mode ignoring +(-) signs. |D| is

difference between value of the variable (x) and mean, median or mode.

3. Calculate the total of these deviations i.e. |D|.4. Calculate Mean deviation by dividing the sum of deviations by total no. of observations.5. Calculate Coefficient of Mean Deviation by applying formula given below.Discrete seriesSteps to calculate:1. Calculate Mean, Median or mode of the series.2. Take deviations |D| from either mean, median or mode ignoring +(-) signs. |D| is

difference between value of the variable (x) and mean, median or mode.3. Calculate the total of these deviations i.e. |D|.4. Multiply each deviation by its corresponding frequency and obtain the total of theseproducts i.e. f|D|.5. Calculate mean deviation by dividing the total of deviations by total no. of observations.6. Calculate Coefficient of Mean Deviation by applying formula given below.Continuous seriesSteps to calculate:1.Convert the class series into exclusive series if given in any other series.2.Calculate mid-values and denote them by x.3.Calculate Mean, Median or mode of the series.4.Take deviations |D| from either mean, median or mode ignoring +(-) signs. |D| isdifference between value of the variable or mid-values (x) and mean, median or mode.5.Calculate the total of these deviations i.e. |D|.6.Multiply each deviation by its corresponding frequency and obtain the total of these

products i.e. f|D|.7.Calculate mean deviation by dividing the total of deviations by total no. of observations.8.Calculate Coefficient of Mean Deviation by applying formula given below.

Measures of Mean Deviations Formula

1. Absolute Measure of Mean Deviations(i)

For Individual Observations M.D. =

ND

Where, M.D. = Mean Deviation

D = Deviations of items from

Mean or MedianIgnoring signs

N = No. of Observations

(ii) For Discrete Series (i.e., whenfrequencies are given) M.D. = N

Df

M.D. = Mean Deviation

D = Deviations of Items from


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Mean or Medianignoring signs

N = No. of Observations

where, Df =Product of Deviations

and respective frequencyN = No. of Total Frequencies

(iii) For Continuous Series (i.e.,when both classes andfrequencies are given)

M.D. =N

Df

Where M.D. = Mean Deviation

D = Deviation of Mid-Points of

Classes from Mean orMedian

Df = Product of Deviations and

respective frequencyN = No. of Total Frequencies

2. Relative Measure of MeanDeviation

(i) Coefficient of M.D. about Mean(ii) Coefficient of M.D. about

Median(iii) Coefficient of M.D. about Mode

=Mean

MeanaboutM.D.

=Median

MedianaboutM.D.

=Mode

ModeaboutM.D.

Interpretation of mean Deviation

Nature of Distribution Range of Mean Mean Deviation1. Normal Distribution2. Moderately Skewed Distribution

Approximately 57.7% of the items expectedto fall in this rangeApproximately 57.5% of the items expectedto fall in this range

Amount of mean Deviation Interpretation

I. Small Mean Deviation

II. High Mean Deviation

Distribution is highly uniform since morethan 50% of the items fall within smallrange around the meanDistribution is not uniform since more than

50% of the items do not fall within smallrange around the mean

Merits of Mean deviation:1. It is simple to understand.2. It is easy to calculate as compared to standard deviation.3. M.D. is a calculated value.4. It is not much affected by the fluctuations of sampling5. It is based on all items of the series and gives weight according to their size.6. It is less affected by extreme values than the standard deviation.7. It is rigidly defined.8. It is flexible, because it can be calculated from any measurer of central tendency.


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9. It is a better measure for comparison.10.It is useful for small samples when no detailed analysis is required.Demerits of mean Deviation:1. It lacks algebraic properties such as (+) and (-) signs which are not taken into

consideration. All the deviations are treated as positive.2. Algebraic positive and negative signs are completely ignored. In mean deviationmeasure +5 and 5 have the same meaning. It is mathematically unsound and illogical.

3. It is not a very accurate measure of dispersion.4. It is not suitable for further mathematical calculation.5. It is rarely used it is not as popular as standard deviation.6. It is much affected by sampling fluctuations.7. It is not capable of further algebraic treatment.8. It may not give accurate result when the degree of variability in a series is very high.Uses of mean deviation:It will help to understand the standard deviation. It is useful in marketing problems. It isuseful while using small samples. It is used in statistical analysis of economic business andsocial phenomena. It is useful in calculating the distribution of wealth in a community or anation. It is useful in forecasting business cycles.

Properties of mean deviation1. If all the observations assumed by a variable are constant, say k, then the mean deviation

is zero.2.Mean Deviation about median is the least. In other words, the sum of the deviations

taken from median is the minimum.

3. Mean Deviation remains unaffected due to a change of origin but is affected in the sameratio due to a change in scale, if for any two constants a and b, the two variables x and yare related by Y= a+bx, then the Mean Deviation of y is given by M.D. of y=|b|x M.D ofx.

Standard deviationMeaning of standard deviation or root mean square deviationStandard Deviation is the square root of the arithmetic mean of the squares of deviations ofall items of the distribution from arithmetic mean.

Measures of standard deviationMeasures of Standard Deviation may be absolute or relative

Individual seriesA.When the deviations are taken from the actual mean.Steps involved.1. Calculate the Actual Mean (X) of the series.2. Take deviations from the actual mean by taking out the difference (x-x).3. Square these deviations and obtain the total. (x-x).4. Calculate Standard Devotion () by applying the formula given below.5. Calculate Coefficient of Variation (C.V) by dividing the standard deviation by actual

mean.


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B.When actual values are taken:Steps involved.1. Identify the variable and denote it by x.2. Square the values and find summation of x and x2.3. Calculate Standard Devotion () by applying the formula given below.4. Calculate Coefficient of Variation (C.V) by dividing Standard Deviation by Actual

Mean.

C.When deviations are taken from the assumed mean.Steps involved.5. Identify the variable and denote it by x.6. Take any value of the variable as Assumed Mean i.e. A.7. Take deviations from the Assumed Mean by taking out the difference (X-A) between the value

of the variable (X) and the assumed mean (A) and denote the deviations byd and obtain thetotal of these deviations i.e. d.

8.

Square these deviations and obtain the total.d

.9. Calculate Standard Devotion () by applying the formula given below.10.Calculate Coefficient of Variation (C.V) by dividing Standard Deviation by Actual

Mean.

Discrete seriesA.When the deviations are taken from the actual mean.Steps involved.1. Calculate the Actual Mean (x) of the series.2. Calculate Deviations from the actual mean by taking out the difference (x-x) between the

value of item (x) and the actual mean (x).

3. Square these deviations (x-x).4. Multiply the squared deviation of each row with the respective frequency and obtain

the total of these products i.e. f(x-x).5. Calculate the Standard Deviation () by applying the formula given below.6. Calculate Coefficient of Variation (C.V.) by dividing standard deviation by actual mean.B.When actual values are taken:Steps involved.1. Identify the variable and denote it by x.2. Square the values of x.3. Multiply each value of x with the respective frequency and each value of x2 withrespective frequency.4. Find the summation of fx and fx2.5. Calculate Standard Devotion () by applying the formula given below.6. Calculate Coefficient of Variation (C.V) by dividing Standard Deviation by Actual

Mean.

C.When deviations are taken from the assumed mean.Steps involved.1. Identify the variable and denote it by x.2. Take any value of the variable as Assumed Mean i.e. A.


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3. Take deviations from the Assumed Mean by taking out the difference (x-A) between the valueof the variable (x) and the assumed mean (A) and denote the deviations byd and obtain thetotal of these deviations i.e. d.

4. Square these deviations.5. Multiply the deviation (d) of each row by the respective frequency and obtain the total

of these products i.e. fd.6. Multiply the squared deviation (d) of each row the respective frequency and obtain the

total of these products i.e. fd2.7. Calculate Standard Devotion () by applying the formula given below.8. Calculate Coefficient of Variation (C.V) by dividing Standard Deviation by Actual

Mean.

Continuous seriesA.When the deviations are taken from the actual mean.Steps involved.

1.

Convert the class series into exclusive series if given in the open-ended or cumulativeseries. It may be noted that the mean can be easily calculated if the problem is given ininclusive series or mid-value series.

2. Take the mid-value of each of the class group and denote it by x.3. Calculate the Actual Mean (x) of the series.4. Calculate Deviations from the actual mean by taking out the difference (x-x) between the

value of item (x) and the actual mean (x).7. Square these deviations (x-x).8. Multiply the squared deviation of each row with the respective frequency and obtain

the total of these products i.e. f(x-x).

9.

Calculate the Standard Deviation () by applying the formula given below.10. Calculate Coefficient of Variation (C.V.) by dividing standard deviation by actual mean.B.When actual values are taken:Steps involved.1. Convert the class series into exclusive series if given in the open-ended or cumulative


2. Take the mid-value of each of the class group and denote it by x.3. Square the values of x.4.

Multiply each value of x with the respective frequency and each value of x

2

withrespective frequency.5. Find the summation of fx and fx2.6. Calculate Standard Devotion () by applying the formula given below.7. Calculate Coefficient of Variation (C.V) by dividing Standard Deviation by Actual

Mean.

C.When deviations are taken from the assumed mean.Steps involved.1. Convert the class series into exclusive series if given in the open-ended or cumulative

series. It may be noted that the mean can be easily calculated if the problem is given in

inclusive series or mid-value series.


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2. Take the mid-value of each of the class group and denote it by x.3. Identify the variable and denote it by x.4. Take any value of the variable as Assumed Mean i.e. A.5. Take deviations from the Assumed Mean by taking out the difference (x-A) between the value

of the variable (x) and the assumed mean (A) and denote the deviations byd and obtain the

total of these deviations i.e. d.6. Square these deviations.7. Multiply the deviation (d) of each row by the respective frequency and obtain the total

of these products i.e. fd.8. Multiply the squared deviation (d) of each row the respective frequency and obtain the

total of these products i.e. fd2.9. Calculate Standard Devotion () by applying the formula given below.10.Calculate Coefficient of Variation (C.V) by dividing Standard Deviation by Actual

Mean.

Steps involved in the calculation of combined standard deviation.1. Calculate mean and standard deviation of each group.2. Calculate combined mean of given groups as follows.3. Calculate deviations of each group from combined mean and ignore signs.4. Calculate combined standard deviation by applying formula given below.

Measure of Standard Deviation Formula

1. Absolute Measure of Mean Deviation(i) For Individual Observations

(a) Actual Mean Method

N

x2

Where, = standard Deviation

x = (X - X ) i.e., deviationstaken from Actual Mean

x2 = Squares of DeviationsN = Total No. of observations

(b) Assumed Mean Method 22

N

d

N

d

Where, = Standard Deviations,

d = (X-A) i.e., deviations takenfrom Assumed Meand2 = Squares of DeviationsN = Total No. of observations

(iii) For Discrete Series(a) Actual Mean Method

N

fx2

Where, = standard Deviation

x = (X - X ) i.e., Deviationstaken from Actual Mean

x2 = Squares of Deviations


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fx2 = Product of square ofdeviation and respectivefrequency

N = Total No. of frequencies

(b) Assumed Mean Method 22

N

fd

N

fd

Where, = Standard Deviationd = (X A) i.e., deviations taken

fromAssumed Mean

d2 = Squares of Deviationsfd2= Product of square of

deviation and respective

frequencyN = Total No. of frequencies

(c) Step Deviation Method(When a common factor is taken fromthe given data)

CN

fd

N

fd

22

Where, = Standard Deviation

d =C

AXi.e., deviations taken from

Assumed Mean and divided byCommon factor (c)

d2

= Squares of Deviationsfd2 = Product of square ofdeviation and respectivefrequency

N = total No. of frequencies

(iii) Continuous Series (i.e., when both theclass-intervals and frequencies are given)

(a) Actual Mean MethodN

fx2

Where, = Standard Deviation

x = (m - X ) i.e., deviations taken

from Actual Meanx2 = Squares of Deviations

fx2 = Product of square of deviationand

respective frequencyN = Total No. of frequencies

III S.D. of any two values = (Largest Value Smallest value)/2or = Range/2


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Similarity between Mean deviation and standard deviationBoth the Mean Deviation and Standard Deviation measures of dispersion are based on eachand every item of the distribution.

Properties of standard deviation

The main properties of standard deviation are as follows:1. Independent of change of origin:Standard Deviation is independent of change of origin. In other words, the value ofstandard deviation remains the same if each of the observations in a series is increased ordecreased by a constant value. Thus, if Y = X + K where K is a constant quantity, thenstandard deviation of Y is equal to standard deviation of X.

2. Not Independent of change of scale:Standard Deviation is not independent of change of scale. In other words, the value ofstandard deviation is also affected if each of the observations in a series is multiplied or

divided by a constant value. Thus, if Y = Y x K or Y = Y/K where K is a constant quantity,then standard deviation of Y is not equal to standard deviation of X.

3. If all the observations assumed by a variable are constant, say k, then the standarddeviation is zero.

4. If for any two constants a and b, the two variablesx and y are related by Y= a+bx, thenthe Standard Deviation of y is given by S.D. of y=|b|x S.D ofx.

5. Fixed Relationship among Measures of Dispersion:In a normal distribution there is fixed relationship between Quartile Deviation, Mean

Deviation and Standard Deviation as follows;Quartile Deviation (Q.D) = 2/3 , Mean Deviation (M.D.) = 4/5 Thus, standard deviation is never less than Quartile Deviation and Mean Deviation.

6. Fixed Area Relationship for a symmetrical distribution:In a symmetrical distribution there is fixed area relationship as follows:Mean 1 covers 68.27% of the items,Mean 2 covers 95.45% of the items,Mean 3 covers 99.73% of the items,

7. Minimum sum of squares:The sum of squares of deviations of items in the series from their arithmetic mean isminimum. In other words, the sum of squares of the deviations of items of any series from avalue other than the arithmetic mean will always be greater than standard deviation.

8. Standard Deviation of n natural numbers:The standard deviation of n natural numbers can be computed by using the followingformula:

Standard Deviation of n natural numbers = 112

1 2 N

9. Combined Standard Deviation 12 - Combined Standard Deviation 12 of twogroups can be computed by using the following formula:


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21

2

22

2

11

2

22

2

1112

NN

dNdNNN

Where, 12 = Combined Standard deviation of two groups (1, 2)

1= Standard Deviation of First group (1)

2 = Standard Deviation of Second group (2)

1N = No. of Items in First Group

2N = No. of Items in Second Group

1d = 121 XX

1222 XXd

1X Actual Mean of First Group

2X Actual Mean of Second Group

12X Actual Mean of Two Groups

Distinction between mean deviation and standard DeviationMean Deviation differs from Standard Deviation in the following respects:

Basis of Distinction Mean Deviation Standard deviation

1. Algebraic Signs Actual Signs are ignoredand all deviations are takenas positive

Actual signs are notignored

2. Use of Measure ofCentral Tendency

Mean Deviations can becomputed from arithmeticmean, median or mode

Standard Deviation iscomputed from arithmeticmean since the sum ofsquares of the deviation of

items from arithmetic meanis minimum

3. Formulan

dfDM

)(..

n

fxDS

2

).(.

Merits of standard deviation:1. It is rigidly defined and its value is always definite and based on all the observations

and the actual signs of deviations are used.2. As it is based on arithmetic mean, it has all the merits of arithmetic mean.3.

It is the most important and widely used measures of dispersion.4. It is possible for further algebraic treatment.

5. It is less affected by the fluctuations of sampling, and hence stable.6. Squaring the deviations make all of them positive; as such there is no need to ignore the

signs (as in mean deviation)7. It is the basis for measuring the coefficient of correlation sampling and statistical

inferences.8. The standard deviation provides the unit of measurement for the normal distribution.9. It can be used to calculate the combined standard deviation of two or more groups.10. The coefficient of variation is considered to be the most appropriate method for the

purpose of comparing the variability of two or more distributions, and this is based on menand standard deviation.


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Limitations of Standard Deviation1. It cannot be used for comparing the variability of two or more series of observations

given in different units. A Coefficient of standard deviation is to be calculated for thispurpose.

2. It is difficult to compute as compared to other measures.3. It is very much affected by the extreme values. The process of squaring deviations gives

undue importance to large deviations from arithmetic mean which is obtained onlyfrom extreme items and it gives less importance to items which are nearer to mean.

4. The standard deviation cannot be computed for a distribution with open-end classes.5. As it is an absolute measure of variability, it cannot be used for the purpose of

comparison.6. It has not found favour with the economists and businessmen.Interpretation of coefficient of variation% of Coefficient of Variation Interpretation

I. Lower Group is moreconsistent/uniform/stable/homogeneous

II. Higher Group is lessconsistent/uniform/stable/homogeneous

VarianceMeaning of VarianceVariance is the arithmetic mean of the squares of deviations of all items of the distributionsform arithmetic mean. In other words, variance is the square of the standard deviation orstandard deviation is the square root of the variance.

Thus, Variance = 2

Or, = Variance

Interpretation of Variance

Value of Variance 2 InterpretationSmaller the value of 2 Lesser the variability or greater the

uniformity in the population

Larger the value of 2 Greater the variability or lesser theuniformity in the population

Lorenz CurveLorenz curve is a graphic method of studying dispersion. Basically, Lorenz curve is acumulative percentage curve in which the percentage of items is combined with thepercentage of other things as wealth, income, sales, profits etc., the steps involved in thecomposition of Lorenz curve are given below:

Practical steps involved in the construction of Lorenz curveStep 1 Calculate cumulative values of items of each of the given variables and convert

each of the cumulative values into percentage taking grand total as 100.

Step 2 Calculate cumulative frequencies of items of each of the variables and convert


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each of the cumulative values into percentage taking grand total as 100.Step 3 Plot the percentages of cumulative frequencies on X-axis (scale 0 to 100)Step 4 Pot the percentages of cumulative values of the variables on Y-axis (Scale 0 to 100)Step 5 Draw a diagonal line joining 0 on X-axis with 100 on Y-axis. This is known as line

of equal distribution. Any point on this diagonal shows the same percentage on X

as on Y.Step 6 Plot the various points corresponding to X and Y and joint them. The line so

obtained, unless all items exactly equal, will always form a curve below the line ofequal distribution. The area between line of equal distribution and the plottedcurve gives the extent of inequality in the items. If curves of various distributionsare shown on the same Lorenz presentation, the curve that is farthest from thediagonal line represents greatest inequality.

LIST OF FORMULAEMeasure Type of Series Absolute Measure Relative Measure

. Range (i) Individual/Discrete(ii) Continuous

R = L S

R = UL - LS Coefficient of Range =SL

SL

Coefficient of Range =

SL

SL

UU

UU

. Inter-Quartile RangeInter-Quartile Range= 13 QQ

. Percentile RangePercentile Range = 1090 PP Coefficient of Percentile Range =

1090

1090

PP

PP

. Quartile Deviation Quartile Deviation orSemi-Inter Quartile Range =

2

13 QQ

Coefficient of Quartile Deviation =

13

13

QQ

QQ

. Mean Deviation (i) Individual Series

(ii) Discrete Series

(iii) Continuous Series

N

DDM

..

N

DfDM

..

N

DfDM ..

Coefficient of M.D. about

Mean/median/Mode

=ModeMedianMean //

n/ModeMean/MediaaboutM.D.

. Standard Deviation (i) Individual Series(a) Actual Mean

Method

(b) Assumed MeanMethod

N

x2

22

N

d

N

d

Coefficient of Variation

= 100MeanArithemtic

DeviationStandard

LIST OF FORMULAE

Measure Type of Series Absolute Measure Relative Measure

. Range (i) Individual/Discrete(ii) Continuous R = L SR = UL - LS Coefficient of Range =

Coefficient of Range =

. Inter-QuArtile RangeInter-Quartile Range=

SL

SL

SL

SL

UU

UU

13 QQ


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. Percentile RangePercentile Range =

Coefficient of Percentile Range =

0. Quartile Deviation Quartile Deviation orSemi-Inter Quartile Range = Coefficient of Quartile Deviation =

1. Mean Deviation (i) Individual Series

(ii) Discrete Series

(iii) Continuous Series

Coefficient of M.D. about

Mean/median/Mode

=

2. Standard Deviation (i) Individual Series(a) Actual Mean

Method


Coefficient of Variation

=

(ii) Discrete Series(a) Actual Mean

Method


(c) Step-DeviationMethod

N

fx2

22

N

fd

N

fd

cN

fd

N

fd

22

(iii) Continuous Series(a) Actual Mean

Method


(c) Step-DeviationMethod

N

fx2

22

N

fd

N

fd

cN

fd

N

fd

22

3. VarianceVariance =

2

4. combined StandardDeviation

21

2

22

2

11

2

22

2

11

12nn

dndnnn

5. Standard Deviation ofany two Values

2

ItemSmallest-Itemlargest

1090 PP

1090

1090

PP

PP

2

13 QQ 13

13

QQ

QQ

N

DDM

..

N

DfDM

..

N

DfDM ..

ModeMedianMean //

n/ModeMean/MediaaboutM.D.

N

x2

22

N

d

N

d

100MeanArithemtic

DeviationStandard


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measures of central tendency final

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