1. The document discusses different types of means or averages, including arithmetic mean, geometric mean, and harmonic mean. 2. It provides definitions and formulas for calculating simple arithmetic mean, combined arithmetic mean, and arithmetic mean of grouped data using both direct and shortcut methods. 3. Examples are given to demonstrate calculating the arithmetic mean from both ungrouped and grouped data using the frequency distribution method and the assumed mean method.

1. K.G. ART’S & SCIENCE COLLEGE RAIGARH (C.G.)
SESSION - 2019-20
DEPARTMENT OF ZOOLOGY & RESEARCH CENTER
SUBJECT :-BIOSTASTICS & TOOLS & TECHNIQUES FOR BIOLOGY.
SEMINAR TOPIC :- MEAN .
Guided by Submitted by
Prof. Anita Pandey mam. Girja Prasad Patel
M.Sc. II Sem. Zoology

2. MEAN
 SYNOPSIS :-
1. INTRODUCTION
2. DEFINITION
3. TYPES OF MEAN :-
I. ARITHMETIC MEAN
II. GEOMETRIC MEAN
III. HARMONIC MEAN
4. CONCLUSION
5. REFERENCE

3. 1. INTRODUCTION :-
Average represented mathematically is termed as
mathematically average or mean . It is calculated by taking into account the
value of all items of the series . Therefore , mathematical average is the
mean. It is the measure of central tendency .
2. DEFINITION :- “The sum of the data values divided by the number of data
items.”
Mean =
X1 + X2 + X3+ ......... + Xn
𝑁
=
𝑋
𝑁

4. 3. TYPES OF MEAN OR MATHEMATICAL AVERAGE :-
There can be three different types of means :-
1. ARITHMETIC MEAN :- there are two types :-
i. Simple Arithmetic Mean
ii. Combined Arithmetic Mean
2. GEOMETRIC MEAN
3. HARMONIC MEAN
1. ARITHMETIC MEAN :- ( ) (AM)
Arithmetic mean of a sample or
population is the common average obtained by dividing the sum of values of
all the items of the series by the total number of items of that series .
X

5. TYPES OF ARITHMETIC MEAN :- there are two types :-
i. Simple Arithmetic Mean
ii. Combined Arithmetic Mean
i. Simple Arithmetic Mean :- It is most commonly used of average . It is the value
which we get by dividing the aggregate of various items of the same series by the total
number of observation.
Different formula are used for calculating arithmetic mean of ungrouped data . Both
direct & shortcut methods are used in each case .
A.) Computation of Arithmetic Mean of Ungrouped Data :- We know that
ungrouped data consists of individual observation . This type of arithmetic mean on an
average is calculated by summing up all the individual observation or measurments of
sample and dividing by the total number of items , observation or measurment.

6.  Direct Method :- As discussed above , the simple Arithmetic Mean can be calculated
by using the following formula :-
Arithmetic Mean = Sum of Observation
No. of Observation
=
X1 + X2 + X3+ ......... + Xn
𝑁
=
𝑋
𝑁
=
𝑖=1
𝑛
𝑋𝑖
𝑁
=
𝑋
𝑁
Where :- Individuals observation are represented by = X1 + X2 + X3+ ......... + Xn .
Sum of all the individuals observation is represented by = 𝑿.
Number of observation by “N” and
Their Arithmetic Mean by (“X” bar).
X
X
X

7. Example :- Find the arithmetic mean of the marks by 10 students of class in mathematics
in a certain examination .
The marks obtained are :-
25, 30, 21, 55, 47, 10, 15, 17, 45, 35
Sotution :-
Sum of all the observation 𝑋 = 25+30+21+55+47+10+15+17+45+35
𝑋 = 300
Number of students (N) = 10
Arithmetic Mean ( ) =
𝑋
𝑁
=
300
10
= 30. Ans.
X
X

8. NOTE :- Direct method of calculation of Arithmetic mean is useful only when number of
items in the series is few and the size of value in small.
 Shortcut Method :- Shortcut method for calculating Arithmetic mean is used when the
number of items in the series is very large.
The formula used is as follows :-
= A +
𝑑
𝑁
Here :- = Arithmetic Mean.
A = Assumed Arithmetic Mean.
d = Deviation of items from assumed mean d = (X – A).
𝑑 = Sum of deviation from the assumed mean.
N = Total Number of Observations.
X
X

9. STEP IN CALCULATION :-
Step 1 :- Assumed Mean (A) of the series is calculated by dividing the total of
maximum and the minimum values of the items of a series with two.
Step 2 :- Deviation (d) of different value from the assumed mean is calculated
by subtracting mean from the actual value ( X1-A , X2-A .......)
Step 3 :- Sum of all these deviation (∑d) is calculated by addition.
Step 4 :- All these are placed in the above formula :-
= A +
𝑑
𝑁
X

10. Example :- The table shows the number of colonies of bacteria grown on ten agar plates. Calculate the
arithmetic mean by using shortcut method.
Solution :- Calculation of Arithmetic Mean using Short-cut Method :
Plate No. 1 2 3 4 5 6 7 8 9 10
No. of colonies 60 70 80 95 100 110 115 130 140 160
Sr. No. of Plates No. of bacterial colonies per
plates (X)
Assumed Mean Deviation from Assumed Mean (d)
1 60 60-110= -50
2 70 70-110 = -40
3 80 80-110 = -30
4 95 60+160/2 =110 95-110= -15
5 100 A = 110 100-110= -10
6 110 110-110= 00
7 115 115-110= 05
8 130 130-110= 20
9 140 140-110= 30
10 160 160 -110= 50
N = 10 ∑X =1060 ∑d = -145 +105 = -40

11. Step 1 :- Assumed Mean A = 60+160/2 = 220/2 = 110
Step 2 :- ∑d = -145 +105 = -40 (calculated as shown in the table)
Step 3 :- N = 10
Step 4 :- Put the value in the formula :
= A +
𝑑
𝑁
= 110 +
−40
10
= 110 – 4
= 106 . Ans.
X
X
X
X

12. B.) Computation of Arithmetic Mean of Grouped Data :-
It can be calculated by the
use of following formula :
=
𝒇 𝟏x1+ 𝒇 𝟐x2 + 𝒇 𝟑x3+ ......... + 𝒇 𝒏xn
𝒇 𝟏+ 𝒇 𝟐+ 𝒇 𝟑+ ......... + 𝒇 𝒏
=
𝒇𝒙
𝒇
f = Frequency 𝒇 = Total frequency
x = Value of each item. 𝒇. 𝒙 = Multiplication of frequency & mid-point value.
X
X

13.  Arithmetic Mean of Grouped Data :-
i. Discrete Series :-
If data is in frequency distribution but not in class interval , then it is called as
discrete series.
In the case of discrete series , arithmetic mean of grouped data is calculated by the use of following
formula :-
=
𝑓𝑥
𝑓
or
1
𝑁
𝑓
Here :- N = 𝑓 It is sum of all frequencies (i.e. Total Frequency )
STEP IN CALCULATION :-
Step 1 :- By multiplying each value (X) with the corresponding frequency of its occurance.
(obtaining fx of different observation f1x1 , f2x2 , ............fnxn)
Step 2 :- Adding all the multiplication products to obtain (∑fx).
Step 3 :- Divide this total value ∑fx by total number of observation or total frequency.
𝒇𝒙
𝒇
=
𝑓𝑥
𝑁
X

14. Example :- Rate of respiration of 43 fishes and their respective frequency was recorded as follows :-
Find the arithmetic mean from data.
Solution :- First step is to make a table of three columns. 1st column for variable (Rate of
Respiration) , 2nd for corresponding frequency and 3rd for variable × frequency
=
𝑓𝑥
𝑓
=
𝟏𝟔𝟕𝟐
𝟒𝟑
= 38.88 Ans.
Rate of Respiration 2 16 20 30 39 40 45 49 50 65 70 79 80
Frequency 3 4 7 7 1 3 5 1 2 2 5 1 2
X
X
X

15. Example :- Find the mean from the following :-
Table :- Marks Obtained by Different Number of students :
Calculation :-
=
𝒇𝒙
𝒇
=
995
50
= 19.9 . Ans.
Marks 5 10 15 20 25 30 35 40
No. of Students (f) 5 7 9 10 8 6 3 2
Marks (X) No. of Students (f) fx
5 5 5 × 5 = 25
10 7 10 × 7 = 70
15 9 15 × 9 = 135
20 10 20 × 10 = 200
25 8 25 × 8 = 200
30 6 30 × 6 = 180
35 3 35 × 3 = 105
40 2 40 × 2 = 80
∑f = 50 ∑fx = 995
X
X
X

16. Arithmetic Mean of Grouped Data
ii. Continuous Series :- In the case of grouped continuous series , the arithmetic mean is
calculated after taking into consideration the mid point of various classes.
When there is a continuous class distribution from
X0-X1 , X1-X2 , X2-X3 , ........ With there frequencies as f1 , f2 , f3 , ........ the arithmetic mean is
calculated by using the following formula :-
= 𝒇.𝒎
𝒇
Here :- m = the mid value of various classes.
f = the frequency of each class.
𝒇. 𝒎 = the sum of mid values multiplied by their frequency.
𝒇
= the total frequency .
Mid point is obtained by following method :-
Mid point (m) = 𝒍𝒐𝒘𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍 + 𝒖𝒑𝒑𝒆𝒓 𝒍𝒊𝒎𝒊𝒕 𝒐𝒇 𝒄𝒍𝒂𝒔𝒔 𝒊𝒏𝒕𝒆𝒓𝒗𝒂𝒍
𝟐
X

17. STEP IN CALCULATION :-
Step 1 :- Firstly , mid value (m) of each class is calculated .
Step 2 :- Multiply each mid value by respective frequency (f) to obtain value of fm.
Step 3 :- Add all the fm value to obtain ∑fm or ∑fx .
Step 4 :- Divide this value of ∑fm or ∑fx by the sum total of all frequency . i.e. Total
number of observation ∑(f).
Example :- Value of fecundity (rate of reproduction) of 50 fishes of a species of fish are in
a frequency table. Calculate the mean value of fecundity from grouped data of continuous
series.

18. Table :- Values of Fecundity of 50 Fishes of a Species :-
=
𝒇.𝒎
𝒇
=
𝟏𝟗𝟐𝟓
𝟓𝟎
= 38.5 Ans.
Class
Interval
Mid Value (m) Frequency (f) Multiplication of Frequency & Mid –
point (fm) or (fx)
1-10 (1 + 10)/2 = 5.5 3 5.5 × 3 = 16.5
11-20 (11 + 20)/2 = 15.5 11 15.5 × 11 = 170.5
21-30 (21 + 30)/2 = 25.5 7 25.5 × 7 = 178.5
31-40 (31 + 40)/2 = 35.5 4 35.5 × 4 = 142.0
41-50 (41 + 50)/2 = 45.5 10 45.5 × 10 = 455.0
51-60 (51 + 60)/2 = 55.5 5 55.5 × 5 = 277.5
61-70 (61 + 70)/2 = 65.5 7 65.5 × 7 = 458.5
71-80 (71 + 80)/2 =75.5 3 75.5 × 3 = 226.5
∑f = 50 ∑m×f = 1925
X
X

19. ii. Combined Arithmetic Mean ( 𝑥) :- When two or more distribution are given with
their number of items and their arithmetic means their combined means is called
arithmetic mean . If we are given the mean of two series and their size , then the combined
mean for the resultant series can be the calculated by the formula :-
( 𝑥) = 𝑵 𝟏 𝒙1 + 𝑵 𝟐 𝒙2 / 𝑵 𝟏+ 𝑵 𝟐
Where :-
𝑥 = Combined arithmetic means. ( It is read as ‘X’ double bar )
𝑥1 = Arithmetic means of 1st distribution.
𝑥2 = Arithmetic means of 2nd distribution.
𝑁1 = No. of items of 1st distribution.
𝑁2 = No. of items of 2nd distribution.

20. Example :- If the mean length of a group of 40 earthworms is 65 cm. and the mean length
of other group of 50 earthworms is 60 cm. find the combined mean length of the two groups.
Solution :-
( 𝒙) = 𝑵 𝟏 𝒙1 + 𝑵 𝟐 𝒙2 / 𝑵 𝟏+ 𝑵 𝟐
𝒙1 = 65 , 𝑵 𝟏 = 40
𝒙2 = 60 , 𝑵 𝟐 = 50
𝒙 =
40 × 65+50×60
40+50
𝑥 =
2600+3000
90
=
5600
90
= 62.2 cm. Ans.

21. Computation of combined Mean by Assumed Mean Method :-
The Arithmetic Mean is given by for formula :
I.) In case of ungrouped data :-
= M’+
𝒅
𝑵
Where :-
= Mean of Observation.
M’ = Assumed mean .
d = Deviation from arbitrary reference point.
N = 𝒅 = Total number of observations.
X
X

22. Example :- Find the mean by short-cut method using the following data :-
59, 65, 71, 67, 61, 63, 69, 73
Solution :- Let us take 65 as assumed mean and prepare the following table :-
Total no. of Observation = N = 8
Assumed mean or mid value = M’ = 65
Deviation from assumed mean = d = (X – M’)
= M’+
𝒅
𝑵
= 65 +
𝟖
𝟖
= 66 cm. Ans.
Observation = X Deviation = d = (X – 65)
59 59 – 65 = - 6
61 61 – 65 = - 4
63 63 – 65 = - 2
65 = M’ (Assumed Mean) 65 – 65 = 0
71 71 – 65 = +6
67 67 – 65 = +2
69 69 – 65 = +4
73 73 – 65 = +8
N = ∑X = 8 ∑d = 20 – 12 = 8
X
X
X

23. II.) In case of grouped data :-
= M’+
𝒇𝒅
𝑵
Where :- f = Frequency
f.d = Product of the frequency and the corresponding deviation.
• Data with Equal Class-interval :- In case of frequency table with equal class-interval
the Arithmetic mean is given by the formula :-
= M’+
𝒇𝒅
𝑵
i
Where :-
i = width of the class-interval .
X
X

24. Example :- In a hospital OPD following patient of different age groups were examined.
Calculate the mean by Assumed Mean Method.
Solution :- Table for the calculation of Assumed Mean, frequency and Deviation of
frequency from mid value.
AGE GROUPS 0-10 10-20 20-30 30-40 40-50 50-60
NO. OF PATEINT 42 44 58 35 26 15
Marks Mid – Value
(M)
d =
M – 35
𝟏𝟎
No. of pateint
frequency (f)
fd
0 - 10 5 - 3 42 -3 × 42 = - 126
10 - 20 15 - 2 44 -2 × 44 = - 88
20 - 30 25 - 1 58 -1 × 58 = -58
30 - 40 35 = M’
(Assumed Mean)
0 35 0 × 35 = 00
40 - 50 45 +1 26 1 × 26 = +26
50 - 60 55 + 2 15 2 × 15 = +30
Class –
interval i = 10
N = 220 ∑fd = - 272-(56)
= - 216

25. Class – interval (i) = 10
Assumed Mean (M’) = 35
Sum of Observation (N) = 220
Deviation of frequency from mid value (∑fd) = - 216
Put the value in the following formula :-
= M’+
𝑓𝑑
𝑁
i
= 35 +[
−𝟐𝟏𝟔
𝟐𝟐𝟎
] × 10
= 35 – 9.8 = 25.2 Ans.
X
X
X

26.  Merit’s and demerit’s of Arithmetic Mean :-
• Merit’s :-
 Certainty :- Arithmetic mean is rigidly defined so, it’s value is always definite and certain.
Mean can never be biased.
 Simplicity :- Arithmetic mean is easy to calculate and simple to understand.
 Stability :- Arithmetic mean is a relatively stable measure.
 Basic of Comparison :- Arithmetic mean is the best measure for comparing two or more
series.
 Accuracy :- The test of accuracy of Arithmetic Mean can be carried out on the basis of
chalier test.
 Balance :- Arithmetic mean balance the value on either side.
• Demerit’s :-
 Arithmetic mean cannot be used for small number of classes.

27.  Arithmetic Mean cannot be used for qualitative characteristics such as colour of flower,
sweetness of oranges or darkness of the colour.
 It can not be determined by inspection nor can be represented graphically .
2. GEOMETRIC MEAN (GM) :-
“The geometric mean is defined as the Nth root of
the product of “n” observations.”
COMPUTATION OF GEOMETRIC MEAN FOR UNGROUPED DATA :-
(Individual Series) :- If X1 , X2 , X3 ,……… Xn are the number of then their
Geometric Mean (GM) = 𝑛
𝑥1. 𝑥2 . 𝑥3 … . 𝑥 𝑛
Or GM = (𝑥1. 𝑥2 . 𝑥3 ….... 𝑥 𝑛)
𝟏
𝒏
Where :-
𝑥1. 𝑥2 . 𝑥3 ….... 𝑥 𝑛 𝑎𝑟𝑒 𝑡ℎ𝑒 𝑵 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒 "x" .

28. Condition :-
a.) GM for two Observation :-
If n = 2, the number of observation are only two the GM can be computed by taking the
square root of their product. Only geometric mean can be computed by taking the square
root of their product.
Using formula :- GM = 𝑛
(𝑥1. 𝑥2 ) or (𝑥1. 𝑥2 )
𝟏
𝒏
Example :- On 1st January weight of a pig was recorded as 14 kg. On 1st March weight of the
same pig was 20 kg. What was the approximately weight of the pig on 1st February.
Solution :- GM = 2
(𝑥1. 𝑥2 ) Or (𝑥1. 𝑥2 )
𝟏
𝟐 for two values
GM = (𝟏𝟒 × 𝟐𝟎)
𝟏
𝟐 = 𝟐𝟖𝟎 = 16.73 kg. Ans.
The weight of the pig was 16.73 kg. approximately on 1st Frebruary.

29. b.) GM for more than two Observation :- But if ‘n’ is more than 2
then the computation of the nth root is difficult.
In that case the calculations are made by making use of log table or calculator.
GM = Antilog
log𝑥1+ log𝑥2 + log𝑥3……..+ log𝑥 𝑛
𝑁
GM = Antilog
∑ log 𝑥
𝑁
.
Computation of GM for Grouped Data :-
1.) Discrete Series :-
GM = Antilog
𝟏
𝑵
∑ 𝒇. 𝒍𝒐𝒈 𝒙

30. Example :- The number of Basophills (a kind of WBCs.) of 30 patient of T. B. was recorded
in frequencies, calculate the GM.
Solution :- A table of four columns is prepared to find the goal. 1st columns for sources for
frequency, 3rd for logx and 4th for f.logx.
GM = Antilog
∑ 𝑓.log 𝑥
∑ 𝑓
GM = Antilog
36.2480
30
GM = Antilog (1.20826)
GM = Antilog of 1.20826 = 16.15 Ans.
No. of Basophills 11 14 17 19 22
Frequencies 5 6 8 7 4
Variable (X) Frequency (f) logx f.logx
11 5 1.0414 5.2070
14 6 1.1461 6.8766
17 8 1.2304 9.8432
19 7 1.2788 8.9516
22 4 1.3424 5.3696
∑f = 30 ∑f.logx = 36.2480

31. 2.) Continuous Series :-
GM = Antilog
𝟏
𝑵
∑ 𝒇. 𝒍𝒐𝒈 𝒎
Where :- m = mid – point of class-interval.
Example :- Calculate the GM from the data obtained in an experiment related to the number of panicles and
number of grains in wheat.
Solution :- First step is to frame table of 5 columns for those components which are required
for formula .
1st column for no. of grains in class-interval.
2nd column for mid point (m) of each class-interval.
3rd column for number of panicles i.e. Frequency.
4th column for logm or (logx).
5th column for frequency multiplied with m.
No. of grains 51 -100 101 - 150 151 - 200 201 - 250 251 - 300
No. of panicles 7 9 10 8 5

32. No. of grains in
class-interval (x)
Mid-point (m) No. of Panicles
(f)
log (m) f.log m
51 - 100 75 7 1.88 13.16
101 - 150 125 9 2.10 18.90
151 - 200 175 10 2.24 22.40
201 - 250 225 8 2.35 18.8
251 - 300 275 5 2.44 12.2
∑f = 39 ∑f.logm = 85.46
GM = Antilog
∑ 𝒇.𝒍𝒐𝒈 𝒎
∑ 𝐟
GM = Antilog
𝟖𝟓.𝟒𝟔
39
GM = Antilog 2.19 = 155.34 Ans.

33.  Merit’s and demerit’s of Geometric Mean :-
• Merit’s :-
 It is based on all the observations.
 It is rigidly defined.
 It is not much affected by fluctuation of sampling.
 It is particularly useful in dealing with ratio, rates and percentages.
• Demerit’s :-
It cannot be used when any of the quantities are zero , negative.
It is difficult to calculate and interpret.
It may come out to be a value which is not existing in the series.

34. 3. HARMONIC MEAN (HM) OR (H) :-
Harmonic mean is the reciprocal of the Arithmetic
mean of the given observation.
For example:- reciprocal of 5 is 1/5 , reciprocal of 9 is 1/9 and soon.
COMPUTATION OF HARMONIC MEAN FOR UNGROUPED DATA :-
(Individual Series) :- For observation X1 , X2 , X3 ,……… Xn
HM =
𝑵
𝟏
𝒙𝟏
+
𝟏
𝒙𝟐
+
𝟏
𝒙𝟑
+⋯……………………+
𝟏
𝒙𝒏
HM =
𝑁
(1/𝑥)

35. Example :- A migratory bird migrates from feeding place to breeding place at a speed of
70km/hour and returns at the speed of 50 km/hour calculate HM.
Solution :- HM =
𝑁
1/𝑥
Here :- N = 2 , X1 = 70 and X2 = 50
HM =
2
1
70
+
1
50
HM =
2
0.0143+0.02
HM =
2
0.343
HM = 58.33 Km/hour. Ans.

36. COMPUTATION OF HARMONIC MEAN FOR GROUPED DATA :-
1.) Discrete Series :-
Following is used to calculate Harmonic mean in discrete series :
suppose frequencies of a score X1 , X2 , X3 ,……… Xn are 𝒇𝟏 + 𝒇𝟐 + 𝒇𝟑 + ⋯ … . . 𝒇𝒏
respectively.
HM =
𝒇𝟏+𝒇𝟐+𝒇𝟑+⋯………………………..𝒇𝒏
𝒇𝟏
𝒙𝟏
+
𝒇𝟐
𝒙𝟐
+
𝒇𝟑
𝒙𝟑
+ …………………….
𝒇𝒏
𝒙𝒏
HM =
𝒇
(𝒇/𝒙)

37. Example :- Hb % of 10 persons of a family was recorded and placed frequency distribution
discrete series.
Solution :- The reciprocal of the various values should be taken first and then the reciprocal
multiplied by the respective frequency and total (𝒇/𝒎)is obtained
A table of 4 columns is prepared . 1st for values of variables , 2nd for frequency , 3rd for
reciprocal of X and 4th for f/x .
HM =
𝒇
(𝒇/𝒙)
HM =
𝟏𝟎
𝟎.𝟕𝟒𝟔
HM = 13.39 Ans.
Hb % (mg/100ml)
Values (X)
Frequency(f) 1/(x) f/x
12 3 0.083 0.25
13 3 0.076 0.23
14 1 0.071 0.07
15 2 0.066 0.13
16 1 0.062 0.06
𝒇 =10 𝒇
𝐱
= 0.746

38. 2.) Continuous Series :- In a continuous series , we take the mid-points of class-intervals.
Here we take reciprocal of the mid-points following formula is used :-
HM =
𝒇
(𝒇/𝒎)
Where :- m = mid-point of class-interval.
Example :- Length of 68 plants was recorded and presented in following frequency
distribution continuous series .
Solution :- A table of 4 columns is prepared . 1st for class-interval ,2nd for mid-points , 3rd for
frequency and 4th for 𝒇/𝒎.
HM =
𝒇
(𝒇/𝒎)
HM =
𝟔𝟖
𝟏.𝟗𝟔
HM = 34.69 Ans.
Classes 10 -20 20 – 30 30 - 40 40 - 50 50 - 60 60 - 70 70 - 80
Frequency 6 12 14 16 8 8 4

39. Class-Interval (x) Mid-Point (m) Frequency (f) (f/m)
10 – 20 15 6 0.40
20 - 30 25 12 0.48
30 - 40 35 14 0.40
40 - 50 45 16 0.36
50 - 60 55 8 0.15
60 - 70 65 8 0.12
70 - 80 75 4 0.05
𝒇 = 68 (𝒇/𝒎) =1.96
Table :-

40.  Merit’s and demerit’s of Harmonic Mean :-
• Merit’s :-
 It is highly rigidly defined.
 It is based on all the observations of a series.
 It gives greater weightage to the smaller items.
 It is useful to study the respiration, rate of pulse , heart beat etc. in unit time.
 It is not much affected by sampling fluctuation.
• Demerit’s :-
 It is not easy to calculate fluctuations.
 It cannot be calculated if one value zero.
 It cannot be calculated if negative and positive value are given in a series.

41. 4. CONCLUSION :- Mean is method of the measure of mathematical average . It is a type of
the mathematical average . It’s carried both merit’s & demerit’s properties. Three type of
mean :- Arithmetic , Geometric & Harmonic mean use measures of central tendency . Mean is
most common measure of central tendency . Mean represent centre of a set of values. Mean
is affected by extreme value . Different formula used in each case.
5. REFERENCE :-
 BIOSTASTISTICS :- Veer Bala Rastogi.
 ELEMENTS OF BIOSTASTISTICS :- Satguru Prasad ( 3rd edition )
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