MATH 243, LECTURE 3
1. Standard deviation
We have seen that mean and median both measure the “middle” of a set of data. But the mean is easier
to compute, while the median is reliably in the middle.
The situation is similar when measuring the “spread” of data. Last time we defined the inter-quartile
ratio Q3−Q1, which gives us how much of a spread is needed to account for half of the data. The way to
measure spread in a computable way, akin to the average, is with the standard deviation.
Definition 1. Let
x1, .., xn
be a list of data. Let xbe the mean. The standard deviation is given by
σ=v
u
u
t
1
n−1
n
X
i=1
(xi−x)2
Why is this a reasonable measure of spread? If it is small then one expects the quartiles to be close
together, and if it is large then the quartiles should be spread apart.
Example 2 (Excel example).Excel can compute standard deviation with the STDEV command. We can
see how the deviation changes for data sets with larger and smaller “spread.”
1.1. Which description of data is better: five-number summary or mean and standard de-
viation? The mean and standard deviation are always easier to compute; the five-number summary is
always more accurate. Use xand σwhen you have a symmetric distribution of data. Use the five-number
summary otherwise.
If the distribution is approximately symmetric, the median and the mean will be close, and the quartiles
will be about equally placed around the mean. In that case, the mean, and the standard deviation provide
a similar level of information.
2. First manipulations with normal distributions
Last time we were introduced to the all-important “Bell curve,” otherwise known as a normal distribu-
tion. There are only three numbers needed to describe a normal distribution:
The total number of data points. This is usually dealt with by “setting it to one and multiplying at the
end.”
µ, its mean, which is the center of the distribution, around which it is symmetric
and σ, its standard deviation (hinted at towards the end of the first lecture).
We will learn better what σis, but now let’s see how it can be used.
Theorem 3. In a normal distribution, with mean µand standard deviation σ,
(1) 68% of the observations fall within σof µ(within one standard deviation of the mean).
(2) 95% of the observations fall within 2σof µ(within two standard deviations of the mean).
(3) 99.7% of the observations fall within 3σof µ(within 3 standard deviations of the mean).
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