This document discusses various measures of central tendency and dispersion. It defines the mean, median, mode, harmonic mean and geometric mean as measures of central tendency. It provides formulas and examples for calculating the arithmetic mean using direct and shortcut methods. The document also discusses measures of dispersion such as range, quartile deviation, mean deviation and standard deviation. It provides examples of calculating the mean, median and mode from frequency distribution tables.
1. Unit I
1. Measures of Central Value: Characteristics of an
ideal measure;
2. Measures of Central Tendency - mean, median,
mode, harmonic mean and geometric mean.
3. Merits, Limitations and Suitability of averages.
4. Relationship between averages.
5. Measures of Dispersion: Meaning and Significance.
Absolute and Relative measures of dispersion -
Range, Quartile Deviation, Mean Deviation,
Standard Deviation, Coefficient of Variation,
6. Moments, Skewness, Kurtosis.
2. 1. Measures of Central Value:
An average or a measure of central tendency is an effort to
find out one single representative value to describe whole of
figures in a series.
Characteristics of an Ideal/Satisfactory Measure;
Since an average is a single representative value of the series,
it should have following characteristics-
i. Easy to understand
ii. Easy to compute
iii. Based on all the items
iv. Not affected much by extreme observations
v. Rigidly defined- Explained by an algebraic formula.
vi. Capable of further algebraic treatment- like extra use for
finding combined average
vii. Representative Value
3. 2. Measures of Central Tendency - mean, median,
mode, harmonic mean and geometric mean.
An average or a measure of central tendency is an effort to
find out one single representative value to describe whole of
figures in a series. Averages are also known as measure of
location. There are different types of statistical averages, for
the sake of convenience, we divide these measures as below;
1. Mathematical Averages: (a) Arithmetic Mean (b) Geometric
Mean (c) Harmonic Mean
2. Positional Averages: (a) Median (b) Mode
3. Business Averages: (a) Moving Average (b) Progressive
Average (c) Composite Average
4. (a) Arithmetic Mean is the quantity obtained by dividing the
sum of the items in a series by their numbers
Kinds of Arithmetic
Average or Mean
(I) Simple Arithmetic
Mean
(II) Weighted
Arithmetic Mean
5. Calculation of Arithmetic Mean
Simple Arithmetic Mean Weighted Arithmetic
Mean
(I) Individual
Series
(II) Frequency Distribution
(A) Discrete Series (B) Continuous
Series
(a) Direct method (a) Direct method (a) Direct method Direct Method
(b) Short-cut
Method-
(i) Simple
Deviation or
Assumed
Mean Method
(ii) Step Deviation
Method
(b) Short-cut
Method-
(i) Simple
Deviation or
Assumed
Mean Method
(ii) Step Deviation
Method
(b) Short-cut
Method-
(i) Simple
Deviation or
Assumed
Mean Method
(ii) Step Deviation
Method
In this mean weight is given
to different items according
to their individual or relative
importance. The value of
each item is multiplied by
its weight & sum of such
multiplication is divided by
the total of weights.
6. Illustrations for Different Methods
(1). Calculate mean (by direct & assumed mean method), median &
mode of following marks in business statistics obtained by 10
students in mid-term exam :- (Ans- mean-22, med-22.5 & mode-nil)
Roll No. 1 2 3 4 5 6 7 8 9 10
Marks 20 25 15 26 14 30 28 32 12 18
(2). Find out mean (by direct, assumed mean method & step-
deviation method), median & mode from the following data :- (Ans-
34.80, med-30 & mode 30)
Wages (Rs.) 10 20 30 40 50 60 70 80
No. of Workers 4 5 5 4 3 2 1 1
(3). From the frequency distribution given below calculate arithmetic
mean (by direct method, assumed mean method & step-deviation
method), median & mode:- (Ans- mean-28, med-27.4, mode-25.6)
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of Students 12 18 27 20 17 6
7. Some Specific Problems Relating to Arithmetic Mean-
Q.(1) The deviations of six numbers from assumed mean 10 are as
follows. What are the numbers & what is the value of mean:- {on the
basis of deviations in individual series} (Ans-6 12 711 10 20 & 11)
d -4 +2 -3 +1 0 +10
Q.(2): Find Arithmetic Mean from the following data: {combination
of discrete & continuous series} (Ans- 12.33)
Class/Size f
2 1
3 2
4 2
5-7 3
7-10 5
10-15 10
15-20 8
20-25 4
8. Q.(3) Find out arithmetic mean from the following frequency data:-
{on the basis of unequal classes} (Ans-13.81)
CI 0-2 2-5 5-10 10-20 20-40
f 5 8 15 40 12
Q.(4): Two hundred people were interviewed by a public opinion polling
agency. The frequency distribution gives the ages of people interviewed.
Calculate the value of Mean by direct, assumed mean & step-deviation method
from the following data & Median: {inclusive series} (Ans- mean-35.8 years,
med-36 years & mode-42.57)
Age Group Frequency
80-89 2
70-79 2
60-69 6
50-59 20
40-49 56
30-39 40
20-29 42
10-19 32
9. Q.(5): Calculate mean from the following table; {based on open-ended &
less than cumulative frequency distribution}:- (Ans-76.5)
Salary (Rs.) Frequency
Below 50 30
.….. 70 46
…… 100 65
….. 110 85
……120 95
Above 120 5
Q.(6): Find out missing frequency in the following table, if mean is 28
{missing frequency}:- (Ans- 18)
Marks 0-10 10-20 20-30 30-40 40-50 50-60
No. of Students 12 ? 27 20 17 6
10. Q.(7): If mean is 41, find out the missing size from the following:- (Ans-
40)
Size 20 30 ? 50 60 70
No. of Students 8 12 20 10 6 4
Q.(8): The mean of following table is 25. Find the missing frequencies, if
total frequency is 160:- (Ans- f1- 48 & f2 44)
Marks 0-10 10-20 20-30 30-40 40-50
No. of Students 17 f1 32 f2 19
11. Q. Following is the distribution of marks obtained by 50 students of a class
in Political Science. (MBA I Sem. UTU 2014)
Marks (more than) No. of Students
0 50
10 46
20 40
30 20
40 10
50 3
60 0
Calculate the median marks. If 60 percent of the students pass this test, find
the minimum marks obtained by a pass candidate. (14 marks)
12. Some Specific Problems Relating to Median-
Q.(2) Draw ogive from the following data and find out median,
third quartile, seventh decile & 85th percentile:- (Ans 32.5)
Marks 0-10 10-20 20-30 30-40 40-50 50-60
Frequency 8 12 20 40 15 5
13. Some Specific Problems Relating to Mode-
Q.(2) Find value of mode from the following data, algebrically
& graphically:- (Ans 26.7)
Marks 0-10 10-20 20-30 30-40 40-50
Frequency 4 8 16 12 2
