Measures of Central Tendency

Characteristics of a Measure of Central Tendency

Single number that represents the entire set of data (average)

Three Measures of Central Tendency

The Mean

• The sum of the scores divided by the number of scores

Formula for finding the Mean

• Symbolized by M or “X-bar”

N

XM

Characteristics of the Mean

• The mean may not necessarily be an actual score in a distribution

Example of Finding the Mean

• X: 8, 6, 7, 11, 3

• Sum = 35

• N = 5

• M = 7

The Median

• The point in a distribution that divides it into two equal halves

• Symbolized by Md

Finding the Median

1. Arrange the scores in ascending or descending numerical order

2. Calculate the value of (N+1/2)

3. round the (N+1/2)th item

Example of Finding the Median

• X: 6, 6, 7, 8, 9, 10, 11

• Median = 8

• Y: 1, 3, 5, 6, 8, 12

• Median = 5.5

The Mode

• Score or qualitative category that occurs with the greatest frequency

• Always used with nominal data, we find the most frequently occurring category

Example of Finding the Mode

• X: 8, 6, 7, 9, 10, 6

• Mode = 6

• Y: 1, 8, 12, 3, 8, 5, 6

• Mode = 8

• Can have more than one mode

• 1, 2, 2, 8, 10, 5, 5, 6

• Mode = 2 and 5

GROUPED DATA

Mean

Midpoint x CI f fX

95.5 91-100 5 477.5

85.5 81-90 10 855

75.5 71-80 15 1132.5

65.5 61-70 10 655

55.5 51-60 6 333

45.5 41-50 3 136.5

35.5 31-40 1 35.5

N = 50 fX =3625

N

fXMean

M = 3625/50 = 72.5

Merits of Arithmetic Mean

• (1) Simple to understand

• (2) Easy to compute,

• (3) Capable of further mathematical treatment,

• (4) Calculated on the basis of all the items of the series,

• (5) It gives the value which balances the either side,

• (6) Can be calculated even if some values of the series are missing.

• (7) It is least affected by fluctuations in sampling.

Demerits of Arithmetic Mean

• (1) Extreme items have disproportionate effect.

• (2) When data is vast, the calculations become tedious.

• (3) In case of open end classes, mean can only be calculated by making some assumptions.

• (4) A. M. is not representative if series is asymmetrical.

MEDIAN

Exact limit CI f cf

55.5-60.5 56-60 6 60

50.5-55.5 51-55 9 54

45.5-50.5 40-50 15 45

40.5 (L)-45.5 41-45 13 (f) 30

35.5-40.5 36-40 10 17 (M)

30.5-35.5 31-35 7 7

N = 60

302

60

513

)17260(5.40

5.4115.4013

135.405

13

17305.40

cf

mNL

)2(

LOCATION OF THE MEDIAN CLASS

MEDIAN=

Merits of Median

• (1) Easy to calculate,

• (2) Can be calculated even if the data is incomplete,

• (3) It is unaffected in case of asymmetrical series,

• (4) Useful in case the series of qualitative characteristics is given for example beauty, intelligence etc.

• (5) Median is a reliable measure of central tendency if in a series, frequencies do not tend to be evenly distributed.

• (6) Median can be expressed graphically.

Demerits of Median

• (1) Calculation of median requires arraying of items which may be tedious if the data is large,

• (2) It is not suitable for further arithmetic treatment because its value is only positional and not mathematical,

• (3) Affected by number of items and not values,

• (4) It is very unstable. In case of any addition to the series, the value of median would change,

• (5) Items of extremes are given no importance.

MODE

Mode = (3median – 2 mean)

Merits of mode

• (1) Easy to understand,

• (2) Simple to calculate and locate,

• (3) Quantitative data in ranking is possible, mode is very useful

• (4) It is the actual value that is in the series,

• (5) Mode remains unaffected by dispersion of series,

• (6) Not affected by extreme items,

• (7) Can be calculated even if extreme values are not known.

Demerits of Mode

• (1) Mode cannot be subject to further Mathematical treatment, because is not obtained from any algebraic calculations.

• (2) It is quite likely that there is no mode for a series,

• (3) Cannot be used if relative, importance of items have to be considered,

• (4) Choice of grouping has a considerable influence on the value of the mode.