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Understanding the Basics of Standard
Deviation
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STANDARD DEVIATION
Standard deviation is the square root of variance and is the most commonly used
measure of dispersion. The standard deviation represents the average amount by
which the values in the distribution deviate from the mean. When sufficient large
amount of data is used, the patterns of deviations from the mean will be spread
symmetrically on either side and if the class interval are small enough the resultant
frequency distribution curve may look like a bell shaped curve. A low standard
deviation indicates that the data points tend to be very close to the mean, whereas
high standard deviation indicates that the data are spread out over a large range of
values.
In other words, standard deviation is a statistic that tells you how tightly all the various
observations are spread around the mean in a set of data. When the observations are
pretty tightly bunched together and the bell-shaped curve is steep, the standard
deviation is small. When the observations are spread apart and the bell curve is
relatively flat, that implies a relatively large standard deviation. The identical
distribution is the normal distribution where the bell-shaped curve is formed.
Graphically the normal distribution is presented in figure 1
Figure 1: Normal distribution
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In a normal distribution:
• Approximately 2/3 or 68,26% of the observations will be within 1 standard
deviation either side of the mean
•
Approximately 95,5% of all the observations will be within 2 standard deviations
either side of the mean
•
Approximately 99,75% of all the observations will be within 3 standard
deviations either side of the mean
As a simple example we can examine the average height for adult men in a small town
which is estimated at 178 cm, with a standard deviation of around 8cm. This means
that most men (about 68 percent, assuming a normal distribution have a height within
8cm of the mean (170–185 cm) – one standard deviation, whereas almost all men
(about 95%) have a height within 16 cm of the mean 162–194 cm – 2 standard
deviations. If the standard deviation were zero, then all men would be exactly 178 cm
high. If the standard deviation was higher then there would be higher variance, namely
if the standard deviation were 51 cm, then men would have much more variable
heights, with a typical range of about 127 to 229 cm. If this curve was flatter and more
spread out, the standard deviation would have to be larger in order to account for those
68 percent or so of the people.
The formula for standard deviation (SD) is
SD=
|
̅|
where Σ means "sum of", x is a value in the data set, x is the mean of the data set,
and n is the number of data points (sample).
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