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```2.6 - Interpreting the standard deviation
Often it makes sense to describe the distance
from the mean in terms of the standard
deviation in order to interpret it.
And we can talk about the percentage of the
data that falls within one standard of the
mean, two standard deviations of the mean
etc.
For a symmetric bell shaped distribution, we
have the following empirical rule:
1) Approximately 68% of the observations lie
within one standard deviations of the mean.
2) Approximately 95% of the observations lie
within two standard deviations of the mean.
3) Approximately all the observations lie within
three standard deviations of the mean.
For a general distribution, however we have the
following:
Chebyshev's Rule : Given a number k ≥ 1, at
least (1 - 1/k2) of the observations lie within
k standard deviations of the mean.
k
1
2
3
1-1/k2
Exercises: 2.86, 2.87, 2.91, 2.95, 2.97, 2.102
2.7 Measures of Relative Standing
Here we are interested in describing the relative
location of a particular meaasurement
within a data set.
One such measure is percentile ranking.
Suppose you scored 80 on an exam - but the
instructor tells you that you scored the 90th
percentile. What does that mean?
Definition : The pth percentile of a data set is
the number (when the data are ordered)
such that p% of the measurements fall
below it and (100-p)% of the measurements
fall above it. This is something that can be
calculated exactly for a large data set only else it has to be approximated. What
percentile is the median?
Recall the job data:
3.1, 3.1, 3.5, 4.14, 4.32, 4.47, 4.5, 4.68, 4.8, 4.9
Another measure of relative standing is the zscore.
The sample z score for a sample is
The population z score is
xx
s
x

How do we interpret this??
e.g. There are two sections of STA 2122 – In
section I for the first exam , the mean is 62
and s = 5 while in Section II , the mean is 62,
but s = 10. Sue is in Section I and scores 72.
Amy is in section II and scores 72 also. Who
did better relative to their classmates?
Exercises: 2.103, 2.108, 2.110, 2.117
```
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