Download Measures of Variability

Measures of Variability
• Range
• Standard Deviation
• Variance
Standard Deviation
• Standard Deviation: A measure of the spread of
the scores around the mean.
• Average distance from the mean.
Example: Can you calculate the average distance of
each score from the mean? (X=4)
7, 6, 3, 3, 1 (distance from the mean: 3,2,-1,-1,-3)
3, 4, 4, 5, 4, (distance from the mean: -1,0,0,1,0)
Formula for Standard Deviation
Sigma: sum of what follows
Each individual score
• s = (Xi-X)2
n-1
Mean of all the scores
Sample size
Standard deviation
of the sample
Why n-1?
s (lower case sigma) is an estimate of the population
standard deviation ( :sigma) .
• In order to calculate an unbiased estimate of the
population standard deviation, subtract one from
the denominator.
• Sample standard deviation tends to be an
underestimation of the population standard
deviation.
Variance
• Variance: Standard deviation squared.
• S = (X-X)2
n-1
Not likely to see the variance mentioned by
itself in a report.
Difficult to interpret.
But it is important since it is used in many
statistical formulas and techniques.
X
(x-x)
(x-x)2
110
117
129
300
357
100
500
630
750
-222.55
-215.55
-203.55
-32.55
24.44
-232.55
167.44
297.44
417.44
49530.98
46464.2
41434.86
1059.864
597.5309
54082.09
28037.64
88473.2
174259.9
∑ 483940.2
SD concept in Finance
• Standard deviation is a standard measure of
investment risk. It is a historical statistic
measuring volatility and the dispersion of a set of
data from the mean.
• Standard deviation is a measure of the volatility,
or how far away from the mean the outcomes
will be.
• A small-cap stock will typically have a high
standard deviation compared to a stable blue
chip dividend stock. The small-cap stock may
have a higher expected rate of return but that is
to compensate the owner for a greater amount of
risk.
There are two routes for a worker to reach his office. Both
the routes involve hold ups due to traffic lights. He records
the time it takes over a series of six journeys for each route.
The results are shown in the table.
Route 1
15
15
11
17
14
12
Route 2
11
14
17
15
16
11
Using your answers to Route 1 and Route 2 suggest which
route you would recommend. State your reason clearly.
Grouped Data
X
f fx (X-X )
(X-X)2
f(X-X)2
Coefficient of Variation
The SD is an absolute measure of dispersion that expresses
variation in the same unit as the original data.
The SD cannot be sole basis for comparing two distributions.
If we have SD of 10 and mean of 5 the value is twice as large as
mean itself.
But if we have SD of 10 and mean of 5000 the variation relative
to mean is insignificant.
The CV is relative measure of dispersion.
IT relates SD and mean by expressing SD as percentage of mean.
The unit of measurement is percentage .
• CV (P)=(SD/µ)*100
Technician A completes on an average 40
analysis with a SD of 5.
Technician B completes on an average 160
analysis with a SD of 15.