| 8.6 | The Normal Distribution |
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- Key Idea 1: The normal distribution, also called the Gaussian distribution, is referred to colloquially as the bell-shaped curve, and fits many distributions of data...
- Gauss noted in 1798 that repeated astronomical measurements of the same quantity had a bell-shaped distribution.
- The French scientist A. Quetelet (1796 - 1874) first noted that if you measure the heights (and certain other physical dimensions) of a large group of people, their distribution is bell-shaped.
- The British scientist Sir Francis Galton (1822-1911) coined the term normal to describe these distributions.
- Of course, one cannot simply assume that a given set of data is normally distributed. Many data sets are, for example, Figure 8.9 and 8.10, many are not, for example, Figure 8.11.
Discussion
- For any given mean and standard deviation, there is one normal curve.
- Figure - Normal curves with the same standard deviation (5) but different means.
- Figure - Normal curves with the same mean (0) but different standard deviations.
- Normal curves are symmetric about their mean; that is, the left half of the curve is a mirror image of the right half. Thus, the mean of a normal density curve is its geometrical center of gravity, or balance point.
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- Hence, points along the x-axis are read as "number of standard deviations from the mean."
- The area, and hence the probability, between the mean and the mean + 1 standard deviation is about 0.34
- Thus, the probability of being within 0.34 standard deviations of the mean (either side) is 0.68 - approximately 2/3.
- Going out another standard deviation from the center adds an additional .135 of area on each side, so that the probability of being within 2 standard deviations of the mean is (0.135 + 0.34 + 0.34 + 0.135) = 0.95.
- Tables 8.8 and 8.9 look at the examples we have seen so far, to see how well the data fit these probability benchmarks for a normal distribution.
- The data that looked bell-shaped when we first saw it (batting averages, heights and temperatures) seem to fit the 67% rule and the 95% rule well. The dogs and cats data, which did not appear bell-shaped, meet the 95% rule but not the 67% rule.
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- Any normally distributed data can be turned into standard normal data by standardizing the data.
- For each observation:
- subtract the mean of the observations
- divide by the standard deviation of the observations
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Here is the formula.
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Such a standardized score is called a z statistic or z score.
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