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Lesson 2.4 Variance and Standard Deviation
Variance
The variance is the average of the squares of the deviations. A deviation is the difference between a value
and the mean and is written as:
Example: {2, 3, 5, 6} is a set of data. The sample mean is 4. The deviations are:
2 - 4 = -2
3 - 4 = -1
5-4=1
6-4=2
The deviations squared are:
(-2) 2 = 4
(-1) 2 = 1
(1) 2 = 1
(2) 2 = 4
An average of the deviations squared is
rounded to 2 decimal places. This is the sample variance. We divide by 3 instead of 4 because, if we add all
the deviations
their sum is exactly 0. Knowing 3 of the deviations determines the 4th one. Only 3 of the squared deviations
can vary freely (can take on different values). So we average all the deviations squared by dividing by 3. The
number 3 is called the degrees of freedom of the variance. For a population variance, divide by the total
number of values in the population.
The sample variance is represented by s2 and the population variance is represented by the Greek letter 2.
Standard Deviation
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The standard deviation is a special average of the deviations. It measures how the data is spread out from
its mean.
The standard deviation is the square root of the variance and has the same units as the mean. The letter s
represents the sample standard deviation and the Greek letter represents the population standard
deviation.
Example: In the variance example above, the sample variance was s2 = 3.33 (to 2 decimal places). The
sample standard deviation is
s=
rounded to one decimal place.
NOTE: The standard deviation is the measure that we use for spread. We use technology to do this
calculation. In today's world, the standard deviation is almost never calculated by hand because technology is
so easy to use.
Think About It
We can relate a value of the data to its sample mean and its sample standard deviation by the equation:
value = mean + (#ofSTDEVs)(standard deviation)
where #ofSTDEVs is the number of standard deviations the value is from the mean.
For example, if a value of data is 7, its mean is 5, and its standard deviation is 2 then,
7 = 5 + (1)(2)
#ofSTDEVs = 1. The equation reads as "seven equals five plus one times two." What the equation means is
that the value 7 is 1 standard deviation above or to the right of (1 multiplied 2) by the mean 5.
Now, suppose in the same data set, we wanted to know how many standard deviations (#ofSTDEVs) the
value 3 is from its mean. Solve the following equation for #ofSTDEVs:
3 = 5 + (#ofSTDEVs)(2)
The first equation reads as " three equals five plus the number of standard deviations times two." If we solve
for the number of standard deviations (the second equation), we get negative one as the answer.
Because #STDEVs is negative, we say that the value 3 is 1 standard deviation below or to the left of the
mean 5
Example: Using the same mean and standard deviation, calculate how far the value 8.5 is from the mean
8.5 = 5 + (#ofSTDEVs)(2 )
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The first equation reads as "eight point five equals five plus the number of standard deviations times two." If
we solve for the number of standard deviations (the second equation), we get one point seven five as the
answer.
Because #STDEVs is positive, we say that the value 8.5 is 1.75 standard deviations above or to the right of
the mean 5.
Example
We often ask what value is within 1 standard deviation of the mean, within 2 standard deviations of the mean,
or within 3 standard deviations of the mean. To find, say, the value that is within 3 standard deviations of the
mean, we would add to the mean and subtract from the mean 3 multiplied by the standard deviation.
Example: If the mean is 5 and the standard deviation is 2, what values are within 3 standard deviations of the
mean ?
Calculate:
(5 + (3)(2) = 11 and 5 - (3)(2) = -1 )
The values that are within 3 standard deviations of the mean are between -1 and 11. Using the same mean
and standard deviation, what values are within 2.5 standard deviations of the mean ?
Bibliography
(1) Dean, S., Illowsky, B. Elementary Statistics, 2004,
http://sofia.fhda.edu/gallery/statistics/lessons/lesson02 - 1.html, under Creative Commons License
Deed, 2004 Foothill- De Anza Community College District & The William and Flora Hewlett
Foundation, accessed June 25, 2009.
Content Developed by Susan Dean and Barbara Illowsky, Licensed under a Creative Commons License
Published by the Sofia Open Content Initiative
2004 Foothill -De Anza Community College District & The William and Flora Hewlett Foundation
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