Download The Significance of Standard Deviation

The Significance of
Standard Deviation
Consider the volumes of liquid in different cans of a particular brand of soft
drink. The distribution of volumes is symmetrical and bell-shaped. This is
due to natural variation produced by the machine which has been set to
produce a particular volume. Random or chance factors cause roughly the
same number of cans to be overfilled as underfilled.
The resulting bell-shaped distribution is called the
Normal Distribution
If a large sample from a typical bell-shaped data distribution is taken, what
percentage of the data values would lie between 𝑥 − 𝑠 and 𝑥 + 𝑠?
It can be shown that for any measured variable from any population that is
normally distributed, no matter the values of the mean and standard
deviation:
68%
 Approximately
of the population will measure between
1 standard deviation either side of the mean
 Approximately 95% of the population will measure between
2
standard deviation either side of the mean
 Approximately 99.7%of the population will measure between
3 standard deviation either side of the mean
Example
A sample of 200 cans of peaches was taken from a warehouse and the contents of each
can measured for net weight. The sample mean was 486g with standard deviation 6.2g.
What proportion of the cans might lie within:
a. 1 standard deviation from the mean
Three standard
b. 3 standard deviations from the mean
486 ± 6.2
= 479.8 g and 492.2 g
We are 1 standard
deviation away so
about 68%
deviations would fall
within this range
486 ± 3(6.2)
= 467.4 g and 504.6 g
The Normal Curve
The smooth curve that models normally distributed data is asymptotic to the horizontal axis,
so in theory there are no limits within which all the members of the population will fall.
In practice, however, it is rare to find data outside of 3 standard deviations from the mean,
and exceptionally rare to find data beyond 5 standard deviations from the mean.
Note that the position of 1 standard deviation either side of the mean corresponds to the
point where the normal curve changes from a concave to a convex curve.