This document discusses various measures of central tendency including the mean, median, and mode. It provides definitions and formulas for calculating each measure. The mean is defined as the sum of all values divided by the total number of observations. It can be calculated using direct or shortcut methods for discrete, continuous, and weighted data. The geometric and harmonic means are also defined. The median is the middle value when data is arranged in order. Formulas are provided for calculating the median from discrete and continuous distributions. Examples are included to demonstrate calculating each measure of central tendency.

1. MEASURES OF
CENTRAL TENDENCY
Biostatistics
-Bhumi

2.  Central tendency
-It is a statistical measure that identifies a single value as
representative of the entire set of distribution.
-Central tendency means central value of a statistical series

3. Important characters of measures
of central tendency
Mean Median Mode
• average of numbers
• Sum of all values
divided by total
number of values
• Middle number in
ordered data set
•Most frequent value

4. •Aim
-To provide an accurate description of the entire
data

5. Statistical average
Mathematical average Positional average
Median Mode
Arithmetic
Mean
(͞x)
Geometric
Mean
(G)
Harmonic
Mean
(H)

6. Important characteristics of measures of
central tendency
It should be based on all the observations of the series.
It should be easy to calculate & simple to understand.
Is should not be affected by extreme values.
It should be rigidly defined.
It should be capable of further mathematical treatment.
It should be least affected by the fluctuation of sampling.

7. •Mean
Types of
Mean
Arithmetic
mean
Geometric
mean
Harmonic
mean
Sum of all observations
divided by the total
number of observations.
nth root of the
product of all
observations.
Reciprocal of the arithmetic
mean of the reciprocals of
the observations

8. Arithmetic mean
Arithmetic mean of a set of observations is defined
as the sum of all observations divided by the total
number of observations.
Denoted by ͞x
Called as “average value”.

9. Methods for calculating
arithmetic mean
Direct method Shortcut method

10. For Individual series or Ungrouped data
For e.g., if x1. x2, x3, x4,……xn be the values of n observations, then
the arithmetic mean is,
By direct method
𝑥 =
𝑥
𝑛
By shortcut method
𝑥 = 𝑎 +
𝑑
𝑛
Where,
𝑥 = 𝑥1 + 𝑥2 + 𝑥3 + 𝑥4 … + 𝑥𝑛 =
1
𝑛 𝑖=1
𝑛
𝑥 𝑖
n= total no. of observations
Where,
a= assumed mean
𝑑 = 𝑥 − 𝑎 (deviation)

11. Example 1: Calculate the arithmetic mean of marks obtained in
pharmaceutics by 10 students.
Students A B C D E F G H I J
Marks(x) 06 10 15 06 18 17 12 14 08 14
Here, total no. of students(n)= 10
𝑥 = 06 + 10 + 15 + 06 + 18 + 17 + 12 + 14 + 08 + 14
= 120
⸫
Arithmetic mean 𝑥 =
𝑥
𝑛
=
120
10
= 12
by direct method,

12.  by Shortcut method,
Here, consider assumed mean a=14
Then calculate the deviations from assumed mean, ⸫
d = x- a
Students A B C D E F G H I J
Marks(x) 06 10 15 06 18 17 12 14 08 14
d = x-a -8 -4 1 -8 4 3 -2 0 -6 0
𝑑 = −8 + −4 + 1 + −8 + 4 + 3 + −2 + 0 + −6 + 0 = −20
𝑥 = 𝑎 +
𝑑
𝑛
= 14 +
−20
10
= 12
⸫
𝒙 = 𝟏𝟐

13. For discrete series or ungrouped frequency
distribution
For e.g., distribution with x1, x2,…xn & frequencies f1, f2,…fn then,
𝑥 =
𝑓1𝑥1 + 𝑓2𝑥2 + ⋯ + 𝑓𝑛𝑥𝑛
𝑓1 + 𝑓2 + ⋯ + 𝑓𝑛
𝑖=1
𝑛
𝑓𝑖𝑥𝑖
=
=
𝑓𝑖𝑥𝑖
𝑓𝑖

14. By shortcut method
By direct method
𝑥 =
𝑓𝑥
𝑓
𝑥 = 𝑎 +
𝑓𝑑
𝑓

15. Example 2: Calculate the arithmetic mean from the following table
that shows marks secured in Pharmaceutical engineering by students.
Marks (x) 40 48 52 58 64 69 74 78
Number of
students(f)
5 2 7 8 5 3 2 1
Marks (x) 40 48 52 58 64 69 74 78
Number of
students(f)
5 2 7 8 5 3 2 1
fx 200 96 364 464 320 207 148 78
d= x-a
(assumed
mean a=58)
-18 -10 -6 0 6 11 16 20
fd -90 -20 -56 0 30 33 32 20
Multiply frequency (f) with (x),

16. 𝑥 =
𝑓𝑥
𝑓
𝑥 =
1877
33
= 56.87
So,
• Ʃ𝑓 ≡ 33,
• 𝑓𝑥 = 1877
• 𝑓𝑑 = −51
• a= 58 (assumed mean)
𝑥 = 𝑎 +
𝑓𝑑
𝛴𝑓
𝑥 = 58 +
−51
33
= 56.46
By direct method By shortcut method

17. For Continuous series or grouped frequency
distribution
In this, we take midpoints of each class. If m1, m2,..mn are the
midpoints of the class with frequencies f1, f2,…fn, then arithmetic
mean is calculated as,
By shortcut method
By direct method
𝑥 =
𝑓1𝑚1 + 𝑓2𝑚2 + ⋯ + 𝑓𝑛𝑚𝑛
𝑓1 + 𝑓2 + ⋯ + 𝑓𝑛
͞x=
𝑓𝑖𝑚𝑖
𝑓𝑖
=
𝑓𝑚
𝑛
𝑥 = 𝑎 +
𝑓𝑑
𝑛
Where, d = m-a
a = assumed mean

18. Example 3: Calculate the arithmetic mean of the following
distribution of patient of hypertension as per their weights.
Weight (X) 30-40 40-50 50-60 60-70 70-80 80-90
Number of patients (f) 5 8 7 12 14 17
Solution:
𝑚(𝑚𝑖𝑑 𝑣𝑎𝑙𝑢𝑒) =
𝑙𝑜𝑤𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑟𝑎𝑛𝑔𝑒 + 𝑢𝑝𝑝𝑒𝑟 𝑙𝑖𝑚𝑖𝑡 𝑜𝑓 𝑟𝑎𝑛𝑔𝑒
2
Weight 30-40 40-50 50-60 60-70 70-80 80-90
Number of patients (f) 5 8 7 12 14 17
Mid-value(m) 35 45 55 65 75 85
fm 175 360 385 780 1050 1445
d = m-a (⸫
a = 65) -30 -20 -10 0 10 20
fd -150 -160 -70 0 140 340

19.  𝑓 = 𝑛 = 63
 𝑓𝑚 = 4195
 𝑓𝑑 = 100
By shortcut method
By direct method
𝑥 = 𝑎 +
𝑓𝑑
𝑛
𝑥 =
𝑓𝑚
𝑛
𝑥 =
4195
63
= 66.59
Mean weight of patients
𝑥 = 65 +
100
63
= 66.59
Mean weight of patients

20. Weighted arithmetic mean (𝒙𝒘)
While calculating a simple arithmetic mean, it was assumed
that all items of the data were of equal importance. When
these are not of equal importance, weights are assigned to
them in proportion to their relative importance.
If w1, w2,…wn denote the weights (importance) given to the
variables x1, x2,…xn respectively, then the weighted arithmetic
mean is given by,
𝒙𝒘 =
𝑤1𝑥1 + 𝑤2𝑥2 + ⋯ + 𝑤𝑛𝑥𝑛
𝑤1 + 𝑤2 + ⋯ + 𝑤𝑛
=
𝑤𝑖𝑥𝑖
𝑤𝑖

21.  How to calculate combined mean?
Example : Calculate the combined arithmetic mean of medical
& paramedical staff working in Govt. hospital of Valsad district
in last 3 years.
Year Pharmacist Ward boys Doctors
2018 215 265 817
2019 217 245 854
2020 208 278 912
Solution: First calculate total staff in each category & average staff
Total n1= 640 n2= 788 n3= 2583
Mean 𝑥1 = 213.3 𝑥2 = 262.67 𝑥3 = 861

22. 𝑥𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 =
𝑛1𝑥1 + 𝑛2𝑥2 + 𝑛3𝑥3
𝑛1 + 𝑛2 + 𝑛3
=
640 × 213.3 + 788 × 262.67 + (2583 × 861)
640 + 788 + 2583
=
136512 + 206983.96 + 2223963
4011
=
2567458.96
4011
𝑥𝑐𝑜𝑚𝑏𝑖𝑛𝑒𝑑 = 640.1

23. Merits
 Easy to understand and easy to calculate
 It is based upon all the observations
 Capable of further mathematical treatment
 It is affected by sampling fluctuations. Hence it is more
stable.
 It is correctly or rigidly defined.
 Used for further calculations like standard deviation.

24. Demerits
 Affected by extreme values(either low or high)
 It cannot be obtained even if a single value is missing.
 It cannot be determined by inspection
 It cannot be used if we are dealing with qualitative
characteristics, which cannot be measured quantitatively like
caste, religion, sex, etc.

25. Geometric Mean
Geometric mean of a set of observations is defined as the nth
root of product.
 For e.g., if x1. x2, x3, x4,……xn be the values of n observations,
then the geometric mean is,
𝑮 = 𝒏
𝒙𝟏𝒙𝟐𝒙𝟑 … 𝒙𝒏
 Actual calculations can be made with following formula,
𝐺 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
1
𝑁
𝑖=1
𝑛
log 𝑥𝑖

26.  For e.g., distribution with x1, x2,…xn & frequencies f1, f2,…fn
then,
𝐺 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
1
𝑁
𝑖=1
𝑛
𝑓𝑖 log 𝑥𝑖
𝐺 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
𝑛1 log 𝐺1 + 𝑛2 log 𝐺2
𝑛1 + 𝑛2
Where, N = 𝑓𝑖
 If G1 and G2 are the geometric mean of two groups of sizes n1
and n2 then the geometric mean of combination is

27.  If G1 and G2 are the geometric mean of two groups of sizes n
each, then combination geometric mean is
𝐺 = 𝐺1𝐺2

28. Example: The daily income of helpers (in ₹) in one of ayurvedic
formulation industries are given as follow:
700,900,800,850,750
Calculate the geometric mean.
Solution:
Income(x) 700 900 800 850 750 Total
log x 2.8450 2.9542 2.903 2.9294 2.875 14.5066
𝐺 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
1
𝑁
log 𝑥

29. 𝐺 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔
14.5066
5
𝐺 = 𝑎𝑛𝑡𝑖𝑙𝑜𝑔 2.90132
= 796.74
⸫
Geometric mean of daily income is ₹ 796.74

30. Harmonic mean
It is defined as the reciprocal of the arithmetic mean of the
reciprocals of the given items.
H =
𝑛
1
𝑥1
+
1
𝑥2
+ ⋯
1
𝑥𝑛
or
1
𝐻
=
1
𝑛
1
𝑥𝑖

31. If the distribution is discrete or continuous, then
1
𝐻
=
1
𝑁
𝑓𝑖
𝑥𝑖
Where, fi = N

32. Example: The weights of 5 capsules (in gm) are given below.
1.20,1.18,1.23,1.17,1.19
Calculate Harmonic mean.
Solution:
𝑥 1.20 1.18 1.23 1.17 1.19 Total
1
𝑥
0.8333 0.8474 0.8130 0.8547 0.8403 4.1887
𝐻𝑎𝑟𝑚𝑜𝑛𝑖𝑐 𝑚𝑒𝑎𝑛 =
𝑛
1
𝑥
=
5
4.1887
= 1.19
⸫
Harmonic mean of weight for 5 capsules is 1,19 gm.

33. Median
It is defined as the size of the item which lies at the centre
when all items are arranged in either descending or
ascending order of their magnitude,
Median =
N + 1
2
th
item
Where, N = Total number of items

34.  If the distribution is continuous then median is,
Median = L +
N 2 − c. f.
f
∗ I
Where, L= lower limit of median class
c.f.= cumulative frequency of class
f = frequency of the median class
I = interval of median class

35. Example 1: Calculate the median of following marks obtained
in English paper.
10,11,12,8,9
Solution: Arranging all values in ascending order,
8,9,10,11,12
Median =
N + 1
2
th
value
N = Number of items or values
Median =
5 + 1
2
th
=
6
2
th
= 3𝑡ℎ value

36. Example 1: Calculate the median of following weight of tablets.
140,138,130,150,135,145
Solution: Arranging all values in ascending order,
130,135,138,140,145,150
Median = average of
N
2
th
and
N + 1
2
th
= average of
6
2
th
and
6 + 1
2
th
= average of 3rd and 4th value
=
138 + 140
2
=
278
2
= 139

37. Example 3: The daily expenditure of 60 students of Pharmacy is as
follows. Find out median.
Daily Expenditure (₹) 60-70 70-80 80-90 90-100 100-110
Number of students(f) 8 10 12 16 14
Solution: The series is continuous or grouped data with intervals.
Daily Expenditure (₹) 60-70 70-80 80-90 90-100 100-110
Number of students(f) 8 10 12 16 14
Cumulative frequency(c.f.) 8 18 30 46 60
Median (class interval) = 80-90
So, L = lower limit of class = 80
I = width of lass interval = 10

38. c.f. = cumulative frequency of class interval in which
median lies = 18
, f = N = 60
f = frequency of class interval in which median lies = 12
Median = L +
N 2 − c. f.
f
∗ I
= 80 +
60 2 − 18
12
∗ 10
Median = 𝟗𝟎

39. Merits
 It is rigidly defined.
 It is easy to understand and easy to calculate.
 It is not affected by extreme values.
 It can be calculated for distribution with open-end classes.
 It is the only average to be used while dealing with qualitative
data.
 It can be determined by graphically.

40. Demerits
 In case of even number of observations median cannot be
determined exactly.
 It is not based on all the observations.
 It is not capable of further mathematical treatment.

41. Mode
Mode is that value in a series which occurs with the highest
frequency.
• It represents the value occurring most often in a data.
I. For Individual series,
Example 1: Find the mode of following values of weights of 8
tablets in mg.
30,30,25,28,25,30,26,21
Solution: Arranging the given values in the form of frequency
table.

42. values(x) 21 25 26 28 30
Frequency(f) 1 2 1 1 3
So here, maximum frequency is 3.
Hence, the Mode = 30
II. For Discrete series,
Example 2: Fine paid by students of college given in the following
distribution. Calculate the mode.
Fine(₹) 100 120 136 140
Number of students 15 18 19 14
Solution: The maximum frequency is 19.
Hence, the Mode = 136

43. III.For Continuous series,
 In this, class of maximum frequency is called modal class and
value of mode belongs to this class.
 For this, mode is determined by following formula.
Mode = L1 +
f1 − f0
2f1 − f0 − f2
∗ i
Where, L1 = lower limit of modal class
f1= frequency of modal class
f0= frequency of previous class
f2= frequency of the next class
i = class interval of modal class

44. Example 3: Compute the mode from following data.
Price in ₹ 2-6 6-10 10-14 14-18 18-22 22-26 26-30
Frequency 1 9 21 47 52 36 19
Solution: The highest frequency is 52.
Hence the modal class is 18-22 and modal frequency is 52.
L1 = 18, f1= 52, f0= 47, f2= 36, i = 4
Mode = L1 +
f1 − f0
2f1 − f0 − f2
∗ i
= 18 +
52 − 47
2 × 52 − 47 − 36
∗ 4
𝐌𝐨𝐝𝐞 = 𝟏𝟖. 𝟗𝟓

45. Merits
 It is readily comprehensible and easy to calculate.
 It is not affected by extreme values.
 It can be conveniently located even if the frequency
distribution has class intervals of unequal magnitude.
 Open-end classes also do not pose any problem in the
location of mode.
 It is the average to be used to find the ideal size.

46. Demerits
 It is ill defined.
 It is not based upon all the observations.
 It is not capable of further mathematical treatment.
 As compared with mean, mode is affected to a great extent
by fluctuations of sampling.

47. RELATION BETWEEN MEAN,
MEDIAN AND MODE
Mode = 3Median- 2Mean