07 central tendency

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Research Methodology Dr. Nimit Chowdhary, Professor

© Dr. Nimit Chowdhary Research Methodology 2 Saturday, April 14, 2012

Mean

Median

Mode

Trimmed mean

Arithmetic meanGeometric Mean

Harmonic Mean

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Measures of central tendency refer to the summary measures used to describe the most "typical" value in a set of values.

There are hundred of colleges in India

Each college has different tuition

Average college tuition is Rs. 2000

Saturday, April 14, 2012 © Dr. Nimit Chowdhary Research Methodology Workshop p. 3

Central tendency

Mean Median

Compute mean and median

Explain pros and cons of each measure

Describe how they change after:

Adding a constant to each value in the data set

Multiplying each value in the data set by a

constant


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Computing mean is a two step process

1. Find the sum of all values in a group of values

2. Divide the sum by the number of values in the

group


valuesofNumber valuesall of Sum

Mean

POPULATION MEAN

= Sum the following

X = Sum all X values

N = Number of X values

= Population mean

SAMPLE MEAN


= Sum the following

x = Sum all x values

n = Number of x values

= sample meanx

NX n

xx

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BOWLING SCORE

Game Score

1 100

2 170

3 130

4 140


1354/540/4

540140130170100

/

NXN

X

NX

Mean bowling score

The median is the midpoint in a set of data

List scores from smallest to largest

With an odd number of scores, the median is

the middle score

With an even number of scores, the median is

the sum of the middle two scores divided by 2Median= Sum of middle two scores / 2


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BOWLING SCORE

Game Score1 1002 1703 1304 1405 160

List the scores in ascending order100 130 140 160 170

With an odd number of scores, identify the middle score100 130 140 160 170

BOWLING SCORE

Game Score1 1002 1703 1304 140

List the scores in ascending order100 130 140 170

With an even number of scores, identify the two middle scores100 130 140 170

The median is the sum of the middle scores divided by two

1352

140130

Median

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MEAN

The mean is better if a large set of scores does not have an outlier

MEDIAN

The median is better if a small set of scores has an outlier


Rs. 60,000/- Rs. 70,000/- Rs. 80,000/- Rs. 90,000/- Rs. 10,000,000/-

Mean: = X/ N= (60,000 + 70,000 + 80,000 +

90,000 + 10,000,000)= Rs. 2,060,000

Median: Rs. 80,000/-

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Marks obtained by 30 students of class X of a certain school in a mathematics text consisting of 100 marks are as follows:


Marks obtained

Number of students

Marks obtained

Number of students

10 1 70 4

20 1 72 1

36 3 80 1

40 4 88 2

50 3 92 3

56 2 95 1

60 4

Marks obtained (xi) No. of students (fi) fixi

10 1 10

20 1 20

36 3 108

40 4 160

50 3 150

56 2 112

60 4 240

70 4 280

72 1 72

80 1 80

88 2 176

92 3 276

95 1 95

Total fi=30 fixi = 1779

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3.5930

1779

i

ii

fxf

Mean

Class interval No. of students(fi)

Class mark (xi) fixi

10-25 2 17.5 35.0

25-40 3 32.5 97.5

40-55 7 47.5 332.5

55-70 6 62.5 375.0

70-85 6 77.5 465.0

85-100 6 92.5 555.0

fi= 30 fixi =1860.0


6230

1860

i

ii

fxf

Mean

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Suppose we have to find the median of the following data which gives the marks, out of 50, obtained by 100 students in a test.


Marks obtained 20 29 28 33 42 38 43 25

No. of students 6 28 24 15 2 4 1 20

No. of data points is even, 100 The median is the middle-most value, that is,

the value between 50th and 51st points Median will therefore be the average of 50th

and 51st point


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Marks obtained No. of students Cumulative frequency

20 6 6

25 20 26

28 24 50

29 28 78

33 15 93

38 4 97

42 2 99

43 1 100


n/2 th value is 50th, that is 28 (n/2 + 1)th value is 51st , that is 29 So the median is


5.282

2928

Median

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hf

cfn

lMedian

2

Where,

l = Lower limit of median class

n = Number of observations

cf = Cumulative frequency of class preceding median class

f = Frequency of median class

h = Class size

A survey of heights (in cm) of 51 girls of class X of a school was conducted and the following data was obtained


Height in cms No. of girls

Less than 140 4

Less than 145 11

Less than 150 29

Less than 155 40

Less than 160 46

Less than 165 51

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Create a frequency table


Class intervals Frequency Cumm. frequency

Below 140 4 4

140-145 7 11

145-150 18 29

150-155 11 40

155-160 6 46

160-165 5 51

n= 51, therefore n/2 =25.5; this means 145-150 is the median class

l= 145 cf= 11 f= 18, and h= 5


hf

cfn

lMedian

2

518

115.25145

Median

03.14918

5.72145

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Two important measures of central tendency-mean and median

Grouped and ungrouped data How to calculate mean How to calculate median When to use mean and when to use median


07 central tendency

Technology