chapter2 measure of central tendency

8/3/2019 Chapter2 Measure of Central Tendency

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Measures of Central Tendency

Measures of Location

Descriptive Statistics

Measures of Symmetry

Measures of Peakdness


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The central tendency is measured by averages.

These describe the point about which the

various observed values cluster.

In mathematics, an average, or central

tendency of a data set refers to a measure ofthe "middle" or "expected" value of the data

set.


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Arithmetic Mean

Geometric Mean

Harmonic Mean

Median

Mode


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Choosing a measure of central tendency the level of measurement of the variable

concerned (nominal, ordinal, interval or ratio);

the shape of the frequency distribution;

what is to be done with the figure obtained.

Measure of central tendency

The mean is really suitable only for ratio andinterval data. For ordinal variables, where thedata can be ranked but one cannot validly talk

of `equal differences' between values, themedian, which is based on ranking, may beused. Where it is not even possible to rank thedata, as in the case of a nominal variable, themode may be the only measure available.


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Summary

1. The purpose of central tendency is to determine the single valuethat best represents the entire distribution of scores. The three

standard measures of central tendency are the mode, the median,

and the mean.

2. The mean is the arithmetic average. It is computed by summing all

the scores and then dividing by the number of scores. Conceptually,

the mean is obtained by dividing the total (IX) equally among the

number of individuals (N or n). Although the calculation is the same

for a population or a sample mean, a population mean is identified

by the symbol and a sample mean is identified by X.

3. Changing any score in the distribution will cause the mean to be

changed. When a constant value is added to (or subtracted from)every score in a distribution, the same constant value is added to

(or subtracted from) the mean. If every score is multiplied by a

constant, the mean will be multiplied by the same constant. In

nearly all circumstances, the mean is the best representative value

and is the preferred measure of central tendency.


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Summary1. The median is the value that divides a distribution exactly in half.

The median is the preferred measure of central tendency when

a distribution has a few extreme scores that displace the value

of the mean. The median also is used when there are

undetermined (infinite) scores that make it impossible to

compute a mean.

2. The mode is the most frequently occurring score in a .

frequency distribution graph. For data measured on a nominal

scale, the mode is the appropriate measure of central tendency.

It is possible for a distribution to have more than one mode.

3. For symmetrical distributions, the mean will equal the median.If there is only one mode, then it will have the same value, too.

4. For skewed distributions, the mode will be located toward the

side where the scores pile up, and the mean will be pulled

toward the extreme scores in the tail. The median will be

located between these two values.


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Arithmetic Mean

The arithmetic mean is the sum of a set of

observations, positive, negative or zero,

divided by the number of observations. If we

have n real numbers

,.......,,,, 321 nxxxx

their arithmetic mean, denoted by , can be

expressed as:

n

xxxxx n

.............321

n

x

x

n

i

i 1

x


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Arithmetic Mean of Group Data

if are the mid-values and

are the corresponding

kzzzz .,,.........,, 321

kffff ,........,,, 321

,

the number of classes, then the mean is

i

ii

f

zfz


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Geometric Mean

Geometric mean is defined as the positive root of theproduct of observations. Symbolically,

n/1

It is also often used for a set of numbers whose values aremeant to be multiplied together or are exponential in nature,

such as data on the growth of the human population orinterest rates of a financial investment.

Find geometric mean of rate of growth: 34, 27, 45, 55, 22, 34

n321


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Geometric mean of Group data

If the n non-zero and positive variate-values

occur times, respectively,

then the eometric mean of the set of

nxxx ,........,,

21 nfff ,.......,, 21

observations is defined by:

Nn

i

f

iN

f

n

ffin xxxxG

1

1

1

2121

n

i

ifN1

Where


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Geometric Mean (Revised Eqn.)

321 fff

Ungroup Data Group Data

2 n

n

ii

xLogN

AntiLogG

1

1

n

iiixLogf

NAntiLogG

1

1

321 n


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Harmonic Mean

Harmonic mean (formerly sometimes called thesubcontrary mean) is one of several kinds of

average.

Typically, it is appropriate for situations when theaverage of rates is desired. The harmonic mean isthe number of variables divided by the sum of thereciprocals of the variables. Useful for ratios suchas speed (=distance/time) etc.


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Harmonic Mean Group Data

The harmonic mean H of the positive real

numbers x1,x2, ..., xn is defined to be

n

i i

i

x

f

n

H

1

n

i ix

n

H

1

1

Ungroup Data Group Data


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Exercise-1: Find the Arithmetic ,

Geometric and Harmonic Mean

Class Frequency

(f)

x fx f Log x f / x

20-29 3 24.5 73.5 4.17 8.17

30-39 5 34.5 172.5 7.69 6.9

40-49 20 44.5 890 32.97 2.23

50-59 10 54.5 545 17.37 5.45

60-69 5 64.5 322.5 9.05 12.9

Sum N=43 2003.5 71.24 35.64


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Weighted Mean

The Weighted mean of the positive real numbers x1,x2,

..., xn with their weight w1,w2, ..., wn is defined to be

n

i

i

n

i

ii

w

xw

x

1

1


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Median

The implication of this definition is that amedian is the middle value of the

observations such that the number of o servat ons a ove t s equa to t e num erof observations below it.

)1(2

1

n

e XM

122

2

1nne XXM

If n is oddIf n is Even


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Median of Group Data

F

n

f

hLM

o

oe2

L0 = Lower class boundary of the median

class

h = Width of the median class f0 = Frequency of the median class

F = Cumulative frequency of the pre-

median class


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Steps to find Median of group data

1. Compute the less than type cumulative frequencies.

2. Determine N/2 , one-half of the total number of cases.

3. Locate the median class for which the cumulative frequency is

more than N/2 .

4. Determine the lower limit of the median class. This is L0.

5. Sum the frequencies of all classes prior to the median class.

This is F.

6. Determine the frequency of the median class. This is f0.

7. Determine the class width of the median class. This is h.


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Example-3:Find Median

Age in years Number of births Cumulative number of

births

14.5-19.5 677 677

19.5-24.5 1908 2585

24.5-29.5 1737 4332

29.5-34.5 1040 5362

34.5-39.5 294 5656

39.5-44.5 91 5747

44.5-49.5 16 5763

All ages 5763 -


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Example-2: Find Mean, Median and

Mode of Ungroup Data

The weekly pocket money for 9 first year pupils was

found to be:

3 , 12 , 4 , 6 , 1 , 4 , 2 , 5 , 8

Mean

5

Mode

4

Median

4


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Mode of Group Data

hLM21

110

1

1 = difference of frequency between

modal class and class before it

2 = difference of frequency between

modal class and class after

H = class interval


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Steps of Finding Mode

Find the modal class which has highest frequency

L0 = Lower class boundary of modal class

=

1 = difference of frequency of modal

class and class before modal class

2 = difference of frequency of modal class and

class after modal class


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Example -4: Find Mode

Slope Angle

()

Midpoint (x) Frequency (f) Midpoint x

frequency (fx)

0-4 2 6 12

-

10-14 12 7 84

15-19 17 5 85

20-24 22 0 0

Total n = 30 (fx) = 265


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Consider the Measurements and Frequency Table

87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,

73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,

84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,

85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68

Class Class Midpoint Total Frequency

64.5 - 69.5 67 6 0.100

69.5 74.5 72 11 0. 183

74.5 79.5 77 20 0.333

79.5 84.5 82 13 0.217

84.5 89.5 87 9 0.150

89.5 94.5 92 1 0.0167


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For the 60 temperature readings in this population we obtain:

87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,

73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,

84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,85 86 81 72 69 65 71 73 78 81 77 74 77 72 68

= (87+85+ 79 +.+72+68)/60 = 4751/60 = 79.183


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A third measure of central tendency is the median

The median of a population of size N is found by

1. Arranging the individual measurements in ascending order, and

2. If N is odd, selecting the value in the middle of this list as the median (there

will be the same number of values above and below the median)

3. If N is even find the values at position N/2 and N/2 + 1 in this list (call them

xN/2 an xN/2+1 an e me an e g ven y e ormu a me an = xN/2 +

xN/2+1)/2 or be the value halfway between these two measurements.

Note! When N is even the median will usually not be an actual value in the

population


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We now find the median of the population of temperature readings

87, 85, 79, 75, 81, 88, 92, 86, 77, 72, 75, 77, 81, 80, 77,

73, 69, 71, 76, 79, 83, 81, 78, 75, 68, 67, 71, 73, 78, 75,

84, 81, 79, 82, 87, 89, 85, 81, 79, 77, 81, 78, 74, 76, 82,

85, 86, 81, 72, 69, 65, 71, 73, 78, 81, 77, 74, 77, 72, 68

Arrange these 60 measurements in ascending order

65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75,

75, 76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81,81, 81, 81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92

Since N/2 = 30 and both the 30th and 31st values in the list are the same, we obtain

median = 78


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One further parameter of a population that may give some indication of central

tendency of the data is the mode

Define: mode = most frequently occurring value in the

population

From the previous data we see:

65, 67, 68, 68, 69, 69, 71, 71, 71, 72, 72, 72, 73, 73, 73, 74, 74, 75, 75, 75, 75,

76, 76, 77, 77, 77, 77, 77, 77, 78, 78, 78, 78, 79, 79, 79, 79, 80, 81, 81, 81, 81,

81, 81, 81, 81, 82, 82, 83, 84, 85, 85, 85, 86, 86, 87, 87, 88, 89, 92

That the value 81 occurs 8 times mode = 81

Note! If two different values were to occur most frequently, the distribution would be

bimodal. A distribution may be multi-modal.


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From the table we obtain

Class Class Midpoint (x) Total (f) Frequency f*x

64.5 - 69.5 67 6 0.100 402

69.5 74.5 72 11 0. 183 792

74.5 79.5 77 20 0.333 1540

79.5 84.5 82 13 0.217 1066

84.5 89.5 87 9 0.150 783

89.5 94.5 92 1 0.0167 92

60 4675

= i(fi * xi) / i fi = 4675/60 = 77.917

The small discrepancy between these two values for the mean is due to the

way the data is accumulated into classes. The mean of the raw data is more

accurate, the mean of the tabulated data is often more convenient to obtain.


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Numerical DataProperties & Measures

Numerical Data

Properties

Central

MeanMean

MedianMedian

ModeMode

Tendency

RangeRange

VarianceVariance

Standard DeviationStandard Deviation

ar a on

SkewSkew

ape

chapter2 measure of central tendency

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