# Module On Measures Of Central Tendency

Central Tendency

Measures of Central

Tendency

There are many ways of describing of a

given set of data. A good number of descriptive

measures exist in statistics whose use depends

largely on the nature of data and the intended

purpose of the description. This measure is the

measures of position or central tendency, it is

use to see how a large set of raw materials can

be summarized so that the meaningful essential

can be extracted from it.

The most commonly measures of central

tendency are the mean, median, and mode.

Properties of the Arithmetic Mean

easy to compute

easy to understand

valuable in statistical tool

strongly influence by extreme values,

this is particularly true when the number

of cases is small

cannot be compute when distribution

contains open-ended intervals

Uses of Mean

for interval and ratio measurement

when greatest sampling stability is

desired

when the distribution is symmetrical

about the center

when we want to know the “center of

gravity” of a sample

The Arithmetic Mean (X)

The most popular and useful measure of central tendency is the arithmetic mean, which simply refer to as the mean. Widely referred to in everyday usage as the average.

The mean of a set of measurement is defined as,

sum of measurements

mean = -----------------------------------number of measurements In formula form,

Xi

X = --------N

Where:

N = total number of measurements

X = represent the mean

Xi = represents the individual scores

The symbol is a Greek letter sigma, which means sum of. In plain language, the arithmetic is obtained merely by adding the individual scores and dividing the sum by the number of scores.

Mean (ungrouped data)

Xi

X1 + X2 + X3 + X4 +

X5 + … + Xn

X = ---------------=

-----------------------------------------------------N

N

Raw data: 15

Xi

X = ------------------- =

N

16

19

20

18

15 + 16 + 19 + 20 + 18

--------------------------------------------------------------5

= 17.6

Some values sometimes are given

importance than others. In such instance, the

weighted mean is computed.

The formula in computing the weighted

mean is,

WX

X = ----------n

where: W = weight of each item or

value

X

= represent each of the

items

n

= total number of

weights

Example: Determine the weighted mean if,

500 bags were sold at P250.00 each, 350

bags at P200.00 each , 200 bags at P150.00

each , 150 bags at P100.00 each and 50 bags

at

P80.00 each.

Solution:

[500 x 250] + [350 x 200] + [200 x 150] + [150 x 100] + [50 x 80]

X = --------------------------------------------------------------------------------500 + 350 + 200 + 150 + 50 P 244,000.00

X = -------------------1250

X = P195.20

Methods of Computing Mean of

a Grouped Distribution

There are two methods that can

be used in calculating the mean of

a grouped distribution. One

method is called the long method

and the other is called the short or

coded deviation method

Long Method

f midpoint

X = ------------------N

Where:

f midpoint = algebraic sum of all the

product of frequency and the midpoint

N = number of cases or

observations

Example: Find the mean of the following distribution.

X

50

47

44

41

38

35

32

29

26

23

–

–

–

–

–

–

–

–

–

–

f

52

49

46

43

40

37

34

31

28

25

2

2

1

1

5

8

5

1

4

2

N = 31

midpoint

50 – 52

47 – 49

2

2

44 – 46

41 – 43

1

1

38

35

32

29

– 40

– 37

– 34

– 31

5

8

5

1

26 – 28

4

51

4

8

45

42

39

36

33

30

2

2

74

X

fmidpoint

50

47

44

41

38

35

32

29

26

23

–

–

–

–

–

–

–

–

–

–

52

49

46

43

40

37

34

31

28

25

f

2

2

1

1

5

8

5

1

4

2

N = 31

midpoint

51

48

45

42

39

36

33

30

27

24

102

96

45

42

195

288

165

30

108

48

f midpt = 1119

By long method;

f midpoint

X =...

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