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Measures of central tendency
:- Mean, Median, Mode
Submitted by,
Samuel Roy
S2 MSW
No:- 1425

Measures of central tendency
The tendency of various items in a data to concentrate about a
particular value is called central tendency. Various method is use to
determine the central value in a given series is called measures of
central tendency. The measures of central tendency are usually called
as the averages.
The important measures of central tendency are
 Arithmetic Mean
 Median
 Mode

Arithmetic Mean (AM)
• The mean is simply the sum of the values divided by the total number
of items in the set.
• Mean locate the center of distribution.
• Most common measure.
Basic formula for the Mean

Mean
Ungrouped Data Grouped Data

Example of Mean deviation for Ungrouped data
Q. Daily wages of 12 workers are given below. Estimate there Mean wage
(average wage).
𝑥 =759
n = 12
𝑥 =
759
12
= 63.25
Mean = 63.25
x
48
60
74
55
60
62
40
55
65
80
90
70 .
759

Example of Mean deviation for Grouped data

Median
• Median is the value of the middle most item in an arranged series.
Therefore the number of item belong median will be same to the
number of items above. In other words, while 50% of the items lie
below median, the other 50% lie above it, as the value of median
determined by its position, it is also called positional average.

Measures of central_tendency._mean,median,mode[1]

Calculation of median from continuous frequency table
Median = 𝑳 +
𝒏
𝟐
− 𝒎 ∗ 𝒄
𝒇
• m = cumulative frequency of the class preceding the median class.
• L = lower limit of class interval where the median occurs.
• n = some of total frequency.
• C = class interval.
• f = frequency of the class where median occurs.

Example of Calculation of median from continuous frequency table
Class f c f
20 - 30 3 3
30 - 40 5 8
40 - 50 20 28
50 - 60 10 38
60 - 70 15 43
------------------- ----------------------- ----------------------
43
𝑛
2
=
43
2
= 21.5
Median = 𝑳 +
𝒏
𝟐
− 𝒎 ∗ 𝒄
𝒇
L = 40 m = 8
n = 43 f = 20

Median = 𝐿 +
𝑛
2
− 𝑚 ∗ 𝑐
𝑓
= 40 +
43
2
− 8 ∗ 10
20
= 40 +
21.5 − 8 ∗ 10
20
= 40 +
13.5 X 10
20
= 46.75
Median = 46.75

Mode
Mode is the most frequently occurring value in a
series. In other words, it is the item which repeats the
largest time in a given data .
From ungrouped data, Mode can be
identified by selecting the particular item which
appears the highest number of items.

Mode for grouped data
Mode = 𝑳 𝟏 +
𝒇 𝟏 − 𝒇 𝒐 𝑪
𝟐 ∗ 𝒇 𝟏 − 𝒇 𝒐 −𝒇 𝟐
Where
• 𝑳 𝟏 = 𝒍𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 𝒐𝒇 𝒕𝒉𝒆 𝒎𝒐𝒅𝒆 𝒄𝒍𝒂𝒔𝒔
• 𝒇 𝒐 = is frequency. Just lower mode class
• 𝒇 𝟏 = mode frequency.
• 𝒇 𝟐 = frequency after mode class.
• C = class interval.

Mode = 𝑳 𝟏 +
𝒇 𝟏 − 𝒇 𝒐 ∗ 𝑪
𝟐 ∗ 𝒇 𝟏 − 𝒇 𝒐 −𝒇 𝟐
= 𝟒𝟎 +
𝟐𝟕−𝟐𝟎 ∗ 𝟏𝟎
𝟐 ∗𝟐𝟕 −𝟐𝟎 −𝟏𝟓
= 𝟒𝟎 +
𝟕𝟎
𝟏𝟗
= 43.68
Mode = 43.68

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Measures of central_tendency._mean,median,mode[1]

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