1. Measures of Central Tendency
Arithmetic Mean
Geometric Mean
Harmonic Mean

2. Meaning of Average
An average is a value which can be taken as a representative of the whole distribution.
“Average is a value which is typical or representative of a set of data” Murray R. Spiegal

3. Measures of An Average
Mathematical Average
Arithmetic Mean
Geometric Mean
Harmonic Mean
Quadratic Mean
Positional Average
Median
Mode
Commercial Average
Moving Average
Progressive Average
Composite Average

4. Requisites of a Good Average
Easy to Understand & Simple to compute
Based on all the items
Not be unduly affected by extreme observations
Rigidly Defined
Capable of further Algebraic Treatment
Sampling Stability

5. Arithmetic Mean
Simple arithmetic mean
Weighted arithmetic mean

6. Mathematical Properties of Arithmetic mean
The sum of deviations of the observations from the AM is always zero
The sum of the squared deviations of the observations from AM is minimum
AM is capable of being treated algebraically

7. Mathematical Properties of Arithmetic mean

8. Mathematical Properties of Arithmetic mean
If a constant B is added (subtracted) from every observation, the mean of these observations also gets added (subtracted) by it.
If every observation is multiplied (divided) by a constant β, the mean of these observations also gets multiplied (divided) by it.

9. Calculation of Arithmetic Mean
Direct Method
Indirect Method
Individual series
Discrete series
Continuous series

10. Convert individual series into discrete and continuous series1 4 9 5 2 12 3 6 7 8

11. Calculate mean (Individual series)
Monthly income of 10 employees in an office is given below. Calculate arithmetic mean of monthly income.
Income (Rs.)
1780
1760
1690
1750
1840
1920
1100
1810
1050
1950

12. Calculate mean (Discrete Series)
In an examination of 146 candidates, the examiner supplied the following information. Calculate mean marks obtained.
Marks 10 15 20 25 30 35 40
Frequency 8 12 36 28 18 9

13. Calculate mean (Continuous Series)
The HR manager at a city hospital began a study of the overtime hours of the registered nurses. 50 nurses were selected at random, and following overtime hours during a month were recorded. Calculate mean of overtime hours during a month.
Overtime Hours
Number of employees
0-4 5
5-9 22
10-14 13
15-19 8
20-14 2

14. Some special types of problems

15. Unequal class interval
Inclusive series
Open ended series
More than or less than cumulative frequency
Correcting incorrect value
Combined mean

16. Frequency distributions with open-ended class
Salary (Rs ’000)
Number of employees
below10 7
10-20 39
20-30 95
30- 40
201
40 and above
381

17. Frequencies are given in cumulative form (either more than type or less than type)
Marks obtained
Number of students
Less than 10 7
Less than 20 39
Less than 30 95
Less than 40
201
Less than 50
381
Less than 60
545
Less than 70
631
Less than 80
675

18. Correcting Incorrect Values
The mean marks of 100 students were found to be 40. Later on it was discovered that a score of 53 was misread as 83. Find the correct mean.

19. Combined Arithmetic Mean
Mean height of 25 male workers in a factory is 61 inches & the mean height of 35 female workers in the same factory is 58 inches. Find Combined mean.

20. Advantages of Arithmetic mean
It is rigidly defined.
It is easy to calculate and simple to follow.
It is based on all the observations.
It is readily put to algebraic treatment.
It is least affected by fluctuations of sampling.

21. Disadvantages of Arithmetic mean
The arithmetic mean is highly affected by extreme values.
It cannot average the ratios and percentages properly.
It cannot be computed accurately if any item is missing.
The mean sometimes does not coincide with any of the observed value.
It cannot be computed accurately for unequal and open-ended class intervals.

22. Weighted Arithmetic Mean
It is used where the relative importance of the different items is not same.
Useful in problems relating to
Construction of index numbers
Standardized birth and death rates

23. Weighted Arithmetic Mean
A train runs 25 miles at a speed of 30 m.p.h., another 50 miles at a speed of 40 m.p.h., then due to repairs of track travels for 6 minutes at a speed of 10m.p.h. and finally covers the remaining distance of 24 miles at a speed of 24 m.p.h. What is the average speed in miles per hour.

24. Weighted Arithmetic Mean
Course
Pass %
No. of students (in hundred)
M. A. 71 3
M. Com. 83 4
MBA 73 5
MSc 74 2
BBA 65
B. Com. 66

25. Geometric mean
Geometric mean is defined as the Nth root of the product of N items.

26. Geometric mean
Used to find average per cent increase in sales, production, population, etc.
Construction of index numbers

27. Geometric mean
The Price of a commodity increased by 5% from 2006 to 2007, 8% from 2007 to2008 and 77% from 2008 to Calculate average increase in Prices.

28. Harmonic Mean
It is defined as the reciprocal of the arithmetic mean of the reciprocal of individual observations.

29. Harmonic Mean
An Aeroplane covers four sides of a square at a speed of 1000,2000,3000 and 4000Km per hour. Calculate average speed.

30. Relationship between A.M., G.M. & H.M