JSO Special - Statistics
Measures of Central
Tendency - Mean
By: Tej Pratap Singh
Sel. Assistant Audit Officer in CAG
Measures of Central Tendency- Common measures of
central tendency – Mean, Median and Mode; Partition
values- Quartiles, Deciles, Percentiles.
Central Tendency- Central Tendency summarises
the data in a single value in such a way that this single
value can represent the entire data.
The measuring of central tendency is a way of
summarising the data in the form of a typical or
representative value.
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The three most commonly used statistical measures
of central tendency are:
1. Arithmetic Mean
2. Median
3. Mode
Arithmetic Mean: It is defined as the sum of the values of all
observations divided by the number of observations and is usually
denoted by
Arithmetic Mean: Arithmetic Mean for Ungrouped Data
1) Direct Method
2) Assumed Mean Method
Where, A = assumed mean
X = individual observations
N = total numbers of observa- tions
d = deviation of assumed mean from individual
observation,
i.e. d = X – A
3) Step Deviation Method
Here C = Common Factor
Que: Calculate Arithmetic Mean from the data showing
marks of students in a class in an economics test: 400,
500, 550, 780, 580.
Que: Plots in a housing colony come in only three sizes: 100 sq. metre, 200 sq. meters
and 300 sq. metre and the number of plots are respectively 200, 50 and 10.
Compute Arithematic Mean using Direct Method, Assumed Mean Method & Step
Deviation Method.
Size (In No. of f.X d=X-A f.d d’=d/u f.d’
Sq. M) Plots
100 200
200 50
300 10
Continuous Series: Here class intervals are given. Class Intervals may
be exclusive (0-10, 10-20, etc.) or Inclusive (0-9, 10-19, etc.) or of
Unequal size (0-10, 10-50, etc.)
Direct Method in Continuous Series:
Here m is mid value of the class intervals.
Step Deviation Method in Continuous Series:
Obtain d' = (m-A)/c
Take A = any arbitrary figure, c = common factor.
The mean of the first 15 natural
numbers is:
1. 9.5
2. 8
3. 9
4. 7.5
SSC CGL 2023
𝒊
If 𝑿𝒊 = + 𝟐, where i= 1.2……..5. then
𝟓
the mean of x1,x2,…..x5 is:
1. 3.6
2. 1
3. 5
4. 2.6
SSC CGL 2023
Use the following table to find the mean.
1. 8.5 Age( in years ) Number of students
0-5 80
2. 7.5
5-10 120
3. 9.5 10-15 160
4. 𝟏𝟎. 𝟓 15-20 40
Total 400
SSC CGL 2017
Arithmetic mean of marks of the
Marks No. of m d fd d’ fd’ students for the given data is:
Stud.
Marks 0- 10-20 20-30 30-40 40-50 50-60
0-10 10
10-20 No of 12 18 27 20 17 6
students
20-30
30-40 1. 38
40-50 2. 48
50-60
3. 28
4. 𝟏𝟖
SSC CGL 2018
For the series 1,2,3,4,5,6,7,8,9,10, the
minimum possible value of
σ𝒏𝒊=𝟏(𝑿𝒊− 𝑨)𝟐 can be attained at:
1. A=6
2. A=10
3. A=5.5
4. 1
SSC CGL 2022
For the mid values 30, 39, 48, 57 and 66, the
second class interval of the distribution is:
1. 34-44
2. 34.5-43.5
3. 33.5-44.5
4. 34-45
SSC CGL 2017
Combined Mean or Composite Mean
𝑛1 𝑥ഥ +𝑛𝟐 𝑥ҧ
Combined Mean =
𝑛1 +𝑛𝟐
10 is the mean of a set of 7 observations and 5
is the mean of a set of 3 observations. The
mean of the combined set is given by:
1. 8.2
2. 8.25
3. 8.5
4. 7.4
SSC CGL 2014
The mean wage of 1000 workers in a factory running
into two shifts 700 and 300 workers is Rs. 500. The
mean wage of 700 workers working in a day shift is Rs.
450. Then the mean wage of workers working in the
night shift is:
1. Rs. 567.67
2. Rs. 616.67
3. Rs. 570
4. Rs. 543.67
SSC CGL 2015
ഥ , If
The arithmetic mean of n observation is 𝑿
the sum of n-5 observation is a then the mean
of remaining 5 observation is:
ഥ−𝐚
1. n 𝑿
ഥ+𝐚
2. n 𝑿
n 𝑿ഥ −𝐚
3.
𝟓
n 𝑿ഥ +𝐚
4.
𝟓
SSC CGL 2015
Weighted Average Mean
Geometric Mean:
Geometric Mean of 2 numbers a & b = √a×b
Que: Find Geometric Mean of 45 & 5.
Que: Find Geometric Mean of 6, 9 & 4
Harmonic Mean
Harmonic Mean of 2 numbers H = 2ab/(a+b)
For Ungrouped Data, Harmonic Mean
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JSO Special - Statistics
Measures of Central Tendency –
Median & Mode
By: Tej Pratap Singh
Sel. Assistant Audit Officer in CAG
Median: The median is that value of the given number of observations,
which divides it into exactly two parts.
Median of Ungrouped Data:
Arrange the data in ascending (or descending) order, then calculate the
median of ungrouped data as follows:
(i) When the number of observations (n) is odd, the median is the value
(𝑛+1)
of the th observation
2
(ii) (ii) When the number of observations (n) is even, the median is the
(𝑛) 𝑛
mean of the th & ( +1)th observation.
2 2
Que.) The heights (in cm) of 9 students of a class are as follows:
155, 160, 145, 149, 150, 147, 152, 144, 148. Find the median of this data.
Que.) The points scored by a Kabaddi team in a series of matches are as follows:
17, 2, 7, 27, 15, 5, 14, 8. Find the median of the points scored by the team.
Median of Grouped Data:
Discrete Data
The marks obtained by 100 students are given below. Find the
median marks obtained.
Marks No. of Students
Marks No. of Sudents
20 6
29 28
28 24
33 15
42 2
38 4
43 1
25 20
Median of Grouped Data:
Continuous Data:
where l = lower limit of median class,
n = number of observations,
cf = cumulative frequency of class preceding the median class,
f = frequency of median class,
h = class size (assuming class size to be equal).
Continuous Data: The marks obtained by students are given
below. Find the median marks obtained.
Marks Obtained No. of Students
0-10 5
10-20 3
20-30 4
30-40 3
40-50 3
50-60 4
60-70 7
70-80 9
80-90 7
90-100 8
The median of the following data is 525. Find the values
of x & y, if the total frequency is 100.
Class Interval Frequency
0-100 2
100-200 5
200-300 X
300-400 12
400-500 17
500-600 20
600-700 Y
700-800 9
800-900 7
900-1000 4
Find the Mode of the following data:
Length of No. of leaves
leaves
118-126 3
127-135 5
136-144 9
145-153 12
154-162 5
163-171 4
172-180 2
Mode: The mode is that value of the observation which occurs most
frequently, i.e., an observation with the maximum frequency is called the mode.
Mode of Ungrouped Data:
Find the mode of the following marks (out of 10) obtained by 20 students:
4, 6, 5, 9, 7, 7, 6, 5, 4, 9, 3, 4, 7, 6, 9, 9
Mode of Grouped Data:
where l = lower limit of the modal class,
h = size of the class interval (assuming all class sizes to be equal),
f1 = frequency of the modal class,
f0= frequency of the class preceding the modal class,
f2= frequency of the class succeeding the modal class.
Mode of Grouped Data:
Family Size No. of Families
1-3 7
3-5 8
5-7 2
7-9 2
9-11 1
Mode of Grouped Data:
Class No. of Students
Interval
10-25 2
25-40 3
40-55 7
55-70 6
70-85 6
85-100 6
If the median of the observations
2,3,5,6,x,8,9 is 6 then X CANNOT be equal
to:
a) 3
b) 1
c) 10
d) 7
2022
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