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Histograms and Normal Distributions |
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The histogram of average scores when rolling 5 dice has a shape
like the normal distribution. The approximate normal curve is
shown as an overlay line for comparison. Average die rolls of
a finite number of dice cannot give an actual continuous range
of values.
Results generated in a statistics workshop are shown at the right. The participants completed 420 rolls of 5 dice and found the average of the 5 for each roll.
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A roll of 2, 2, 3, 4, 6 would produce an average of 17 / 5 = 3.4. A roll of 1, 1, 1, 1, 4 would produce an average of 8 / 5 = 1.6 . Notice there are a lot more roll averages around the overall average than at the extremes. There are zero averages of 1 or 6, even in 420 attempts. The bars show actual results. The shape of a bell curve has been superimposed on the data. |
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| The actual results do not exactly trace a bell curve because there
was chance variation-- for example more than the expected number
of averages in the 3.8, 4.0 class. Sampling will always produce
some deviation from theory. Sampling error is due to chance deviation,
and not in the usual sense of making mistakes. Actual data frequencies
can vary from theoretical frequencies, while theoretical frequencies
relfect the long term behavior of the data. |
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One way to see if your data are approximately normally distributed is to make a histogram of the scores. The histogram should be balanced and bell-shaped with most scores clustered about the mean, and tapering off symmetrically on both sides.
Balanced distributions like the one shown are not necessarily
normal distributions-- that distribution must conform to certain
specific proportions, of which the empirical rule is an example.
Other distribution evidence comes from the situation. Lengths
from a cut-off saw between the blade and a stop will not be normally
distributed because the lengths cluster at the maximum length.
A cut-off saw could be expected to produce data following a skewed
distribution (below).
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 A balanced (symmetric) histogram with nearly normal proportions.
This distribution is slightly heavy in the tails, i.e. there are
more individuals at high and low values than in a normal distribution.
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