Lecture Number 3
Data Description
October 19, 2016
41 / 99
Measures of Variability  Variance and Standard Deviation
The horizontal distances between each dot (measurement) and the
mean ¯
x
will help you to measure the variability.
If the distances are large, the data are more spread out or variable
than if the distances are small.
If
x
i
is a particular dot (measurement), then the deviation of that
measurement from the mean is (
x
i

¯
x
).
Measurements to the right of the mean produce positive deviations,
and those to the left produce negative deviations.
The values of x and the deviations for our example are listed in the
first and second columns of the table on the next slide.
Lecture Number 3
Data Description
October 19, 2016
42 / 99
Measures of Variability  Variance and Standard Deviation
Because the deviations in the second column of the table contain
information on variability, one way to combine the five deviations into
one numerical measure is to average them.
Unfortunately, the average will not work because some of the
deviations are positive, some are negative, and the sum is always zero
(unless roundoff errors have been introduced into the calculations).
Lecture Number 3
Data Description
October 19, 2016
43 / 99
Measures of Variability  Variance and Standard Deviation
Another possibility might be to disregard the signs of the deviations
and calculate the average of their absolute values.
This procedure has been used as a measure of variability in some
statistical methods.
The standard procedure, which leads to calculation of variance and
standard deviation, involves working with the sum of squared
deviations, i.e. each deviation is squared and then a sum is obtained
for all the deviations as shown in column 3 of the table.
From the sum of squared deviations, a single measure called the
variance
is calculated.
To distinguish between the variance of a population and the variance
of a sample, we use the symbol
σ
2
(
σ
is the Greek lower case sigma)
for a population variance and
s
2
for a sample variance.
The variance will be relatively large for highly variable data and
relatively small for less variable data.
Lecture Number 3
Data Description
October 19, 2016
44 / 99
Measures of Variability  Variance and Standard Deviation
We now define the variance for a population and a sample.
Definition of Population Variance
The variance of a population of
N
measurements is the average of the
squares of the deviations of the measurements about their mean
μ
.
The population variance is denoted by
σ
2
and is given by the formula:
σ
2
=
∑
N
i
=1
(
x
i

μ
)
2
N
(6)
In practice, we don’t normally deal with populations but rather
samples which are subsets of populations.
Definition of Sample Variance
The variance of a sample of
n
measurements is the sum of the squared
deviations of the measurements about their mean ¯
x
divided by (
n

1).
You've reached the end of your free preview.
Want to read all 99 pages?
 Fall '18
 F. TAILOKA
 Standard Deviation, Mean