A measure of central tendency is a summary statistic that represents the center point or typical value of a dataset. These measures indicate where most values in a distribution fall and are also referred to as the central location of a distribution. You can think of it as the tendency of data to cluster around a middle value. In statistics, the three most common measures of central tendency are the mean, median, and mode. Each of these measures calculates the location of the central point using a different method.

1. Measure of Central
Tendency
Statistics
Presented By
Salman Khan
AWKUM (Computer Science)
Salm0khan@yahoo.com

2. Measure of Central Tendency
• Central tendency is a statistical
measure that determines a single
value that accurately describes the
center of the distribution and
represents the entire distribution of
scores.

3. Types of Averages
There are five common type, namely;
Arithmetic Mean (AM)
Median
Mode
Geometric Mean (GM)
Harmonic Mean (HM)

4. Arithmetic Mean
• “The sum of all observations divide by the
total number of observation”.
Mean =
𝑆𝑢𝑚 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛
𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑎𝑙𝑙 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

5. Arithmetic Mean
AM for raw data 𝑥 =
𝑥1+𝑥2+𝑥3+⋯…..+𝑥𝑛
𝑛
=
𝑥
𝑛
Find mean for the data
45, 32, 37, 46, 39, 36, 41, 48, 36
n = 9
𝑥 =
45+32+37+46+39+36+41+48+36
𝑛
∙
𝒙 =
𝟑𝟔𝟎
𝟗
= 40
Sample
data
Population data
n N

6. Arithmetic Mean
AM for Grouped data 𝑋 =
𝑓𝑥
𝑓
Classes Frequency
68-87 10
88-107 13
108-127 15
128-147 9
148-167 4
𝑓 = 51
Mid Points (x) fx
77.5 775
97.5 1267.5
117.5 1762.5
137.5 1237.5
157.5 630
𝑓𝑥 = 5402.5
𝑥 =
𝑓𝑥
𝑓
𝑥 =
5402.5
51
𝒙 = 105.9

7. Median
• “A value which divides a data set that
have been ordered into two equal parts”.
OR
• “A median is a value at or below which
50% of ordered data lie”.

8. Median

9. Median
Median for raw data
For even size 𝑥 = The size of
1
2
𝑛
2
𝑡ℎ
item +
𝑛
2
+ 1
𝑡ℎ
item
Find Median for the data
0, 5, 3, 2, 6, 7
0, 2, 3, 5, 6, 7
n = 6
𝑥 = The size of
1
2
6
2
𝑡ℎ
item +
6
2
+ 1
𝑡ℎ
item
𝑥 = The size of
1
2
3 𝑟𝑑
item + 4 𝑡ℎ
item
𝑥 =
1
2
[3+5]
𝒙 = 4
Sample
data
Population data
n N

10. Median
Median for raw data
For Odd size 𝑥 = The size of
𝑛+1
2
𝑡ℎ
item
Find Median for the data
0, 5, 3, 2, 6
0, 2, 3, 5, 6
n = 5
𝑥 = The size of
5+1
2
𝑡ℎ
item
𝑥 = The size of 3 𝑟𝑑 item
𝒙 = 3
Sample
data
Population data
n N

11. Median
Median for grouped data
The size of
𝑛
2
𝑡ℎ
item lies in the class boundary ?
𝑥 = l+
ℎ
𝑓
𝑛
2
− 𝑐
Classes Frequency
68-87 10
88-107 13
108-127 15
128-147 9
148-167 3
𝒇 = 𝟓𝟎
Class boundaries C. Frequency
67.5-87.5 10
87.5-107.5 23
107.5-127.5 38
127.5-147.5 47
147.5-167.5 50
L = Lower Class Boundary = 107.5
H = high boundary – lower boundary = 20
F = frequency of that class = 15
C = Cumulative Frequency of the
preceding class = 23
The size of 25 th
item lies in the
class boundary 107.5 -127.5
𝑥 = 107.5+
20
15
50
2
− 23
𝑥 = 107.5+2.7
𝑥 = 110.2

12. Mode
• “A value which occurs most frequently in
a set of data”.
• A set of data may have more than one
mode or no mode at all when each
observation occurs the same number of
time.

13. Mode

14. Mode
Mode for raw data 𝑀𝑜𝑑𝑒 = 𝑀𝑜𝑠𝑡 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑎𝑛𝑐𝑒
Mode for QUALITATIVE data
Find Mode for the data (Rooms)
D F D F
C W F E
F D F D
F W F C Mode = F
Rooms Frequency
C 2
D 4
E 1
F 7
W 2
𝑓 = 16

15. Mode
Mode for raw data 𝑀𝑜𝑑𝑒 = 𝑀𝑜𝑠𝑡 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑡 𝑜𝑐𝑐𝑢𝑟𝑎𝑛𝑐𝑒
Mode for QUANTITATIVE data
 2, 1, 3, 1, 2, 5, 3, 4, 5, 2
Mode = 2
 2, 1, 0, 5, 2, 6, 5, 4, 2, 5
Mode = 2, Mode = 5
 2, 1, 3, 4, 5, 6, 9, 8, 7, 0
No Mode

16. Mode
Mode for Grouped data 𝑀𝑜𝑑𝑒 = 𝑙 +
𝑓 𝑚−𝑓1
𝑓 𝑚−𝑓1
+ (𝑓 𝑚−𝑓2
)
× h
Class boundaries Frequency
67.5-87.5 10
87.5-107.5 13
107.5-127.5 15
127.5-147.5 9
147.5-167.5 4
l = Lower Class Boundary of the modal class
𝒇 𝒎 = Highest Frequency
𝒇 𝟏 = Preceding frequency of the modal class
𝒇 𝟐 = Following frequency of the modal class
h = Width of class interval𝒇 𝟏
𝒇 𝒎
𝒇 𝟐
= 107.5
= 15
= 13
= 9
= 20
𝑴𝒐𝒅𝒆 = 107.5 +
15 − 13
15 − 13 + (15 − 9)
× 20
𝑴𝒐𝒅𝒆 = 112.5

17. Geometric Mean
• The geometric mean, G, of a set of n
Positive values 𝑥1, 𝑥2,…., 𝑥 𝑛 is defined as
“the positive nth root of their product”.
𝑮 = 𝒏
𝒙 𝟏, 𝒙 𝟐,…., 𝒙 𝒏 or 𝑮 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
𝟏
𝒏
𝑙𝑜𝑔𝒙𝒊
Where x > 0

18. Geometric Mean
Geometric Mean for raw data 𝑮 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
𝟏
𝒏
𝑙𝑜𝑔𝒙𝒊
Find Geometric Mean of the data
45, 32, 37, 46, 39, 36, 41, 48 and 36
n = 9
𝑮 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
1
9
𝑙𝑜𝑔45 + 𝑙𝑜𝑔32 + 𝑙𝑜𝑔37 + 𝑙𝑜𝑔46 + 𝑙𝑜𝑔39 + 𝑙𝑜𝑔36 + 𝑙𝑜𝑔41 + 𝑙𝑜𝑔48 + 𝑙𝑜𝑔46
log 𝐺 =
1
9
1.65321 + 1.50515 + 1.56820 + 1.66276 + 1.59106 + 1.55630 + 1.61278 + 1.68124 + 1.55630
log 𝐺 =
1
9
14.38700
log 𝐺 = 1.59856
𝐺 = 𝑎𝑛𝑡𝑖 − 𝑙𝑜𝑔 1.59856
𝑮 = 𝟑𝟗. 𝟔𝟖

19. Geometric Mean
Geometric Mean for grouped data 𝑮 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
𝟏
𝒏
𝒇𝒊 𝑙𝑜𝑔𝒙𝒊
Weight 𝒇𝒊
65-84 9
85-104 10
105-124 17
125-144 10
145-164 5
51
𝒙𝒊 𝒍𝒐𝒈 𝒙𝒊 𝒇𝒊 𝒍𝒐𝒈 𝒙𝒊
74.5 1.8722 16.8498
94.5 1.9754 19.7540
114.5 2.0589 35.0013
134.5 2.1287 21.2870
154.5 2.1889 10.9445
103.9345
𝑮 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
1
𝑛
𝑓𝑖 𝑙𝑜𝑔𝑥𝑖
𝒍𝒐𝒈 𝑮 =
1
𝑛
𝑓𝑖 𝑙𝑜𝑔𝑥𝑖
𝑮 =
𝟏𝟐𝟑. 𝟒𝟒𝟓𝟐
𝟓𝟏
𝑮 = 𝟏. 𝟒𝟓𝟓𝟖

20. Harmonic Mean
• The Harmonic mean, H, of a set of n values 𝑥1,
𝑥2,…., 𝑥 𝑛 is defined as
“the reciprocal of the arithmetic mean of
the reciprocal of the values”.
𝐻 =
𝒏
𝟏
𝒙 𝟏
+
𝟏
𝒙 𝟐
+⋯+
𝟏
𝒙 𝒏
or 𝐻 =
𝑛
1
𝑥 𝑖
Where x != 0

21. Harmonic Mean
Harmonic Mean for raw data
𝐻 =
𝒏
𝟏
𝒙 𝟏
+
𝟏
𝒙 𝟐
+ ⋯ +
𝟏
𝒙 𝒏
or 𝐻 =
𝑛
1
𝑥𝑖
Find Harmonic Mean of the given marks
Math = 92, English = 81, Urdu = 70
𝒙 𝟏 = 𝟗𝟐, 𝒙 𝟐 = 𝟖𝟏, 𝒙 𝟑 = 𝟕𝟎 𝒂𝒏𝒅 𝒏 = 𝟑
𝐻 =
𝟑
𝟏
𝟗𝟐
+
𝟏
𝟖𝟏
+
𝟏
𝟕𝟎
=
𝟑
𝟎.𝟎𝟏𝟎𝟖𝟕+𝟎.𝟎𝟏𝟐𝟑𝟓+𝟎.𝟎𝟏𝟒𝟐𝟗
=
𝟑
𝟎.𝟎𝟑𝟕𝟓𝟏
𝑯 = 𝟕𝟗. 𝟗𝟖

22. Harmonic Mean
Harmonic Mean for grouped data
𝐻 =
𝑛
𝑓1
1
𝑥𝑖
Weight 𝒇𝒊
65-84 9
85-104 10
105-124 17
125-144 10
145-164 5
51
𝐻 =
𝑛
𝑓1
1
𝑥𝑖
𝐻 =
51
0.49308
𝑯 = 𝟏𝟎𝟐. 𝟒𝟐 𝑔𝑟𝑎𝑚𝑠
𝒙𝒊 𝒇𝒊
𝟏
𝒙𝒊
74.5 0.12081
94.5 0.10582
114.5 0.14847
134.5 0.07435
154.5 0.03236
0.53405

23. Relations Among Averages
Relation Among Arithmetic Mean, Median and Mode
Mode = 3 Median – 2 Mean
o Symmetrical distribution
o Asymmetrical distribution
1. Symmetrical distribution
 The observations are equally distributed.
 The values of mean, median and mode are always equal.
i.e. Mean = Median = Mode

24. Relations Among Averages
Positively Skewed Negatively Skewed
2. Asymmetrical distribution
The observations are not equally distributed.
Two possibilities are there:

25. Quantiles
• When the number of observation is quite large, the
principle according to which a distribution or an
ordered data set is divided into two equal parts, may
be extended to any number of divisions.
• These are:
1. Quartiles
2. Deciles
3. Percentiles

26. 1. Quartiles
• “The three values which divide the
distribution into four equal parts”.
• These values are denoted by 𝑸 𝟏, 𝑸 𝟐 𝒂𝒏𝒅 𝑸 𝟑.
• 𝑄1 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑓𝑖𝑟𝑠𝑡 𝑜𝑟 𝑙𝑜𝑤𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒.
• 𝑄2 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑠𝑒𝑐𝑜𝑛𝑑 𝑜𝑟 𝑚𝑖𝑑𝑑𝑙𝑒 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒.
• 𝑄3 𝑖𝑠 𝑐𝑎𝑙𝑙𝑒𝑑 𝑡ℎ𝑖𝑟𝑑 𝑜𝑟 𝑢𝑝𝑝𝑒𝑟 𝑞𝑢𝑎𝑟𝑡𝑖𝑙𝑒.

27. Quartiles
𝑸 𝟏 = size of
𝑛
4
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑸 𝟐 = size of
2𝑛
4
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑸 𝟑 = size of
3𝑛
4
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎

28. 2. Deciles
• “The nine values which divide the
distribution into ten equal parts”.
• These values are denoted by 𝑫 𝟏, 𝑫 𝟐, … , 𝑫 𝟗.
• Each Decile contains 10% of the total number
of observations.

29. Deciles
𝑫 𝟏 = size of
𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟐 = size of
2𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟑 = size of
3𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟒 = size of
4𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟓 = size of
5𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟔 = size of
6𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟕 = size of
7𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟖 = size of
8𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
𝑫 𝟗 = size of
9𝑛
10
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎

30. 3. Percentiles
• “The ninety nine values which divide
the distribution into hundred equal
parts”.
• These values are denoted by 𝑷 𝟏, 𝑷 𝟐, … , 𝑷 𝟗𝟗.
• Each Decile contains 1% of the total number
of observations.

31. PERCENTILES
𝑷𝒋 = size of
𝑗𝑛
100
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎
(j = 1 to j = 99)
PERCENTILES
𝑫 𝟑 = size of
3𝑛
100
+ 1 𝑡ℎ 𝒊𝒕𝒆𝒎