measure of dispersion in statistics

Measure of Dispersion

We are Group 4

Tasnim Ansari Hridi (ID-09)

Md. Mehedi Hassan Bappy (ID-21)

Debanik Chakraborty (ID-25)

Syed Ishtiak Uddin Ahmed (ID-31)

Devasish Kaiser (ID-49)

Definition of Measure of Dispersion

In statistics, dispersion (also called

variability, scatter, or spread) is the extent

to which a distribution is stretched or

squeezed. Common examples

of measures of statistical dispersion are the

variance, standard deviation, and

interquartile range.

Example

Centre: Same

Variation: Different

Year 2000: Close Dispersion

Year 2015: Wide Dispersion

Better Quality Data: Data of Year 2000

Why Measure of Dispersion

Serve as a basis for the

control of the variabilityTo compare the variability

of two or more series

Facilitate the use of other

statistical measures.

Reliable

Determine the reliability of

an average

Why Measure of Dispersion

Characteristics of an Ideal


Must be based on all observations of the data.

It should be rigidly defined

It should be easy to understand and calculate.

Must be least affected by the sampling

fluctuation.

Must be easily subjected to further mathematical

operations

Characteristics of an Ideal


It should not be unduly affected by the extreme

values.

Types of Measures of

Dispersion

Classification of

Measures of dispersion

in Statistics

Measures of

Dispersion

Algebraic

Absolute Relative

Graphical

Algebraic Measure of Dispersion

× Mathematical way to calculate the

measure of dispersion.

Example: Calculation of Standard Deviation

or Co-efficient of Variance by using numbers

and formulas.

Characteristics of Algebraic


• Mathematical Way

• Algebraic Variables are used

• Numerical Figures are used here

• Formulas & Equations are used

Graphical Measure of Dispersion

× The way to calculate the measure of

dispersion by figures and graphs.

Example: Calculation of Dispersion among

the heights of the students of a class from

the average height using a graph.

Characteristics of Graphical


• It is a visual way of measuring dispersion

• Graphs, figures are used

• Sometimes, it cannot give the actual result

• It helps the reader to have an idea about the

dispersion practically at a glance

Absolute Measure of Dispersion

Absolute Measure of Dispersion gives an idea about the

amount of dispersion/ spread in a set of observations. These

quantities measures the dispersion in the same units as the

units of original data. Absolute measures cannot be used to

compare the variation of two or more series/ data set.

Classification of

Algebraic Measure of

Dispersion

Absolute Measure of

Dispersion

Absolute Measure of Dispersion gives an idea about the

amount of dispersion/ spread in a set of observations. These

quantities measures the dispersion in the same units as the

units of original data. Absolute measures cannot be used to

compare the variation of two or more series/ data set.

Relative Measure of Dispersion

These measures are a sort of ratio and are called coefficients.

Each absolute measure of dispersion can be converted into

its relative measure.

It can be used to compare two or more data sets

Difference Between Absolute and Relative Measure of

Dispersion

3

This is calculated from original dataThese measure are calculated absolute

measures

2

It is not expressed in terms of percentage It is expressed in terms of percentage

1

It has the variable unit It has no unit

Absolute Measure Relative Measure

6

There is no change in variables and with the absolute measures.

There is changes in variables with relative measures.

5

These measure cannot be used to compare the variation of two or more series

These measure can be used to compare the variation of two or more series.

4

No use of ratio Use of ratio

Absolute Measure Relative Measures

Absolute measures of Dispersion

Classification of Absolute measure

Mean Deviation

Quartile Deviation Standard Deviation

Range

“Range

Range

The difference between the maximum and

minimum observations in the data set.

R= H-L

5, 10 , 15 , 20, 7, 9, 12 , 17 , 13 , 6 , 10 , 11

, 17 , 16

Range = H- L

= 20- 5 = 15

Merits and Demerits of Range

Gives a quick answer

Cannot be calculated in open ended

distributions

Affected by sampling fluctuations

Changes from one sample to the

next in population

Gives a rough answer and is not

based on all observationSimple and easy to

understand

“Mean deviation

Mean deviation

The average of the absolute values of

deviation from the mean(median or mode) is

called mean deviation.

=𝒇 | 𝒙−𝒙 |

𝑵

Merits of Mean deviation

Simplifies calculations

Can be calculated by mean, median

and mode

Is not affected by extreme measures

Used to make healthy

comparisons

Demerits of Mean Deviation

Not reliable

Mathematically illogical to assume all

negatives as positives

Not suitable for comparing

series

“Quartile Deviation

Quartile Deviation

The half distance

between 75th

percentile i.e. 3rd

quartile (Q1) and 25th

percentile i.e. 1st

quartile (Q3) is

Quartile deviation or

Interquartile range.

Q.D = Q3 – Q1

𝟐

Has better result than

range mode.

Is not affected by

extreme items

Merits of Quartile Deviation

Demerits of Quartile Deviation

It is completely dependent on the central items.

All the items of the frequency distribution are not given equal importance in finding the values of Q1 and Q3

Because it does not take into account all the items of the series, considered to be inaccurate.

“Standard Deviation

Standard Deviation

Standard deviation is calculated as the

square root of average of squared

deviations taken from actual mean.

It is also called root mean square

deviation.

= √ 𝒙− 𝒙

𝟐

𝒏

68.2%

95.4%

99.7%

Merits of standard deviation

It takes into account all the items and is capable of future algebraic treatment and statistical analysis.

It is possible to calculate standard deviation for two or more series

This measure is most suitable for making comparisons among two or more series about variability.

Demerits of Standard Deviation

It is difficult to compute. It assigns more

weights to extreme items and less

weights to items that are nearer to

mean.

Classifications ofRelative Measures of

Dispersion

Chart of classification

Relative Measure

Coefficient of Range

Coefficient of Quartile

Deviation

Coefficient of Mean

Deviation

Coefficient of Variation

Coefficient of

Range


The measure of the distribution based on range

is the coefficient of range also known as range

coefficient of dispersion.

Formula:

Coefficient of Range= 𝑅𝑎𝑛𝑔𝑒

𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑉𝑎𝑙𝑢𝑒+𝐿𝑜𝑤𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒× 100

“Coefficient of

Quartile Deviation

Coefficient of Quartile Deviation

A relative measure of dispersion based on the

quartile deviation is called the coefficient of

quartile deviation.

Formula:

Coefficient of Quartile Deviation = 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

𝑀𝑒𝑑𝑖𝑎𝑛× 100

= Q3 – Q1

Q3 + Q1

× 100 [By Simplification]

Merits & Demerits of Coefficient of Quartile

Deviation

Merits

1. Easily understood

2. Not much Mathematical

Difficulties

3. Better Result than


Sampling fluctuation

Ignorance of last 25% of data sets.

Values being irregular

Demerits

Coefficient of

Mean Deviation

Coefficient of Mean Deviation

The relative measure of dispersion we get by dividing

Mean Deviation by Mean or Median, is called Coefficient

of Mean Deviation.

Formula:

Coefficient of MD= 𝑀𝑒𝑎𝑛 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

𝑀𝑒𝑑𝑖𝑎𝑛 𝑜𝑟 𝑀𝑒𝑎𝑛× 100

Merits & Demerits of Coefficient of Mean

Deviation

Merits

1. Better Result than Range

& Quartile Coefficient.

2. Least sampling fluctuation.

3. Rigidly defined.

Fractional Average.

Cannot be used for

sociological studies

Less reliable than


Demerits

Coefficient of

Variation


Coefficient of Variation is a measure of spread

that describes the amount of variability relative to the mean.

Formula:

Coefficient of Variation= 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

𝑀𝑒𝑎𝑛× 100

Merits & Demerits of Coefficient of Variation

Merits

1. Best one

2. Most appropriate one

3. Based on Mean and

Standard Deviation

4. COV is dimensionless or non-

unitized

It is impossible to calculate if

Mean is 0

It is difficult to calculate if

the values are both positive

and negative numbers & if

the mean is close to 0.

Demerits

Practical Uses of Coefficient of Variance

INVESTMENT ANALYSIS

STOCK MARKET

RISK EVALUATION

COMBINED STANDARD DEVIATION OF SEVERAL GROUPS

PERFORMANCES OF TWO PLAYERS

INDUSTRIES & FACTORIES

MathematicalApplication

Coefficient of range

Let 1,2,4,6,7 is a set of values of a distribution. Here, Highest Value, XH=7 & Lowest Value, XL=1 So, Range, R= 7-1 = 6Now, Coefficient of Range = 𝐑

XH + XL × 100

= 𝟔

𝟕+𝟏× 100 =75%

Coefficient of Quartile

deviation Let the number of students in 5 classes are 110, 150, 180, 190, 240 is a set of values. Here, Q1= size of 𝐍+𝟏

𝟒th item = 130

And, Q3 = size of 𝟑(𝐍+𝟏)𝟒

th item = 215

So, Coefficient of Quartile Deviation =Q3 – Q1Q3 + Q1

× 100

= 215−130215+130 × 100= 24.64 %

Coefficient of Mean Deviation

Let the ages of 5 boys in a class is 12, 14, 14, 15, 18.So their Mean, 𝐱 = 𝟏𝟐+𝟏𝟒+𝟏𝟒+𝟏𝟓+𝟏𝟖

𝟓= 14.6

Mean Deviation, MD = | 𝒙 − 𝒙 |

𝑵

=|12−14.6| + |14−14.6| + |14− 14.6|+ |15−14.6| + |18−14.6|𝟓

= 1.52

Now, the Coefficient of MD= 𝐌𝐃

𝐱× 𝟏𝟎𝟎 = 𝟏.𝟓𝟐

𝟏𝟒.𝟔× 𝟏𝟎𝟎 = 10.41%

Coefficient of

Variation Suppose the returns on an investment for 4 years is Tk.1000, Tk.3000, Tk.4500 & Tk.5000.

So, Mean, 𝐱 = 3375 Standard Deviation, SD = 1796.99

So,Coefficient of Variation, CV= 𝐒𝐃

𝐱× 100

= 𝟏𝟕𝟗𝟔.𝟗𝟗𝟑𝟑𝟕𝟓

× 100 = 53.24%

The daily sale of sugar in a certain grocery shop is given below : Monday Tuesday Wednesday Thursday Friday Saturday 75 kg 120 kg 12 kg 50 kg 70.5 kg 140.5 kg respectively.

“

No of Days sale of sugarMonday 60Tuesday 120Wednesday 10Thursday 50Friday 70Saturday 140

𝜮 𝒐𝒇 𝑫𝒂𝒚𝒔 = 𝟔 𝜮𝒙 = 𝟒𝟓𝟎

Mean, 𝑥 = 𝑥

𝑛=

4𝟓𝟎

6= 7𝟓

“

x 𝒙𝟐

60 3600120 1440010 10050 250070 4900

140 19600𝜮𝒙 = 𝟒𝟓𝟎 𝜮𝒙𝟐= 45100

Standard deviation: 𝝈 =𝜮𝒙𝟐

𝒏−

𝜮𝒙

𝒏

𝟐=

𝟒𝟓𝟏𝟎𝟎

𝟔−

𝟒𝟓𝟎

𝟔

𝟐=

𝟕𝟓𝟏𝟔. 𝟔𝟔 − 𝟓𝟔𝟐𝟓 = 𝟒𝟑. 𝟒𝟗

Quartile Deviation

The marks of 7 students in Mathematics result are given below :

70, 85, 92,68, 75, 96, 84Find out-

• First Quartile Deviation• Third Quartile Deviation

Quartile deviation

× First quartile

𝐐𝟏 = 𝐬𝐢𝐳𝐞 𝐨𝐟𝐧 + 𝟏

𝟒

𝐭𝐡

𝐢𝐭𝐞𝐦

= size of 𝟕+𝟏

𝟒

𝐭𝐡𝐢𝐭𝐞𝐦

= size of 2nd item.= 70

×Third Quartile

𝑸𝟑 = 𝒔𝒊𝒛𝒆 𝒐𝒇𝟑 𝒏 + 𝟏 𝒕𝒉

𝟒𝒊𝒕𝒆𝒎

= size of 𝟑 𝟕+𝟏 𝒕𝒉

𝟒𝒊𝒕𝒆𝒎

= size of 6th item=92

Arranging the data in ascending order we get,68,70,75,84,85,92,96

“Thank you

measure of dispersion in statistics

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