2. Introduction
•The averages are representatives of a frequency distribution. But
they fail to give a complete picture of the distribution. They do
not tell anything about the scatterness of observations within
the distribution.
•Suppose that we have the distribution of the yields (kg per plot)
of two rice varieties from 5 plots each. The distribution may be
as follows:
Variety I 45 42 42 41 40
Variety II 54 48 42 33 30
3. •It can be seen that the mean yield for both varieties is 42 kg.
•But, we cannot say that the performances of the two
varieties are same.
•There is greater uniformity of yields in the first variety;
whereas, there is more variability in the yields of the second
variety.
•The first variety may be preferred since it is more consistent
in yield performance.
4. •From the above example, it is obvious that a measure of
central tendency alone is not sufficient to describe a
frequency distribution. In addition to it we should have a
measure of scatterness of observations.
•The scatterness or variation of observations from their
average are called the dispersion.
•There are different measures of dispersion. The most
common measures are the range, the variance and the
standard deviation.
5. Range
•This is the simplest possible measure of dispersion and is
defined as the difference between the largest and smallest
values of the variable.
•Range = largest value — smallest value
•In individual observations and discrete series, largest and
smallest values are easily identified.
•In continuous series,
✔Largest value = upper boundary of the highest class
✔Smallest value = lower boundary of the lowest class
6. Example 1
a) The yields (kg per plot) of a cotton variety from five plots
are 8, 9, 8, 10 and 11. Find the range.
Range = largest – smallest = 11 – 8 = 3
a) Calculate the range from the following distribution.
Range = largest – smallest = 75.5 – 60.5 = 15
Size 60.5 — 63.5 63.5 — 66.5 66.5 — 69.5 69.5 — 72.5 72.5 — 75.5
Number 5 18 42 27 8
7. Variance
•The variance is the average of the squares of the distance
each value is from the mean.
•The symbol for the population variance is σ2
, while the
symbol for the sample variance is s2
.
•The formula for the population variance is
8. The Variance
X X – M (X – M)2
8 - 1.2 1.44
9 - 0.2 0.04
8 - 1.2 1.44
10 0.8 0.64
11 1.8 3.24
M = 9.2 0 ∑(X – M)2
= 6.8
10. Example 2
•Find the variance for the following data set.
10, 60, 50, 30, 40, 20
•Variance = 1750/6 = 291.67
•Standard deviation (average distance each value is from the
mean) = √291.67 = 17
11. Standard Deviation
•The standard deviation is the square root of the variance.
•The symbol for the population standard deviation is σ.
•The formula for the population standard deviation is
13. CLASS A = 50 STUDENTS
•Population = all the 50 students (mean age = 28.6)
•Sample = 30 students (mean age = 28.2)
•Sampling error = difference between population parameter and
sample statistic = 28.6 – 28.2 = 0.4
14. Example 3
•Find the sample variance and standard deviation for the
amount of European auto sales for a sample of 6 years
shown. The data are in millions of dollars.
11.2, 11.9, 12.0, 12.8, 13.4, 14.3
17. Example 4
•The following data represent the number of miles run during one
week for a sample of 20 runners. Find the variance and standard
deviation for the data.
Class Frequency
5.5 – 10.5 1
10.5 – 15.5 2
15.5 – 20.5 3
20.5 – 25.5 5
25.5 – 30.5 4
30.5 – 35.5 3
35.5 – 40.5 2
