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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
If a large sample from a typical bell-shaped data distribution is taken, what percentage of the
data values would lie between 𝑥̅ − 𝑠 and 𝑥̅ + 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:



Approximately
of the population will measure between
deviation either side of the mean
Approximately
of the population will measure between
deviation either side of the mean
Approximately
of the population will measure between
deviation either side of the mean
standard
standard
standard
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
b. 3 standard deviations from the mean
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.
Chapter 23 – Descriptive Statistics
IB Mathematics