Chapter 13 STATS.docx

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30 Mar 2012
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Chapter 13
Normal Distribution
1) Always plot your data: make a graph, usually a histogram or a stemplot
2) Look for the overall pattern (shape, center, spread) and for striking deviations such as outliers
3) Choose either the five-number summary or the mean and standard deviation to briefly describe
center and spread in numbers
4) Sometimes the overall pattern of a large number of observations is so regular that we can describe
it by a smooth curve
Density Curves
Most histograms show the counts of observations in each class by the heights of their bars and
therefore by the areas of the bars
We now set up curves to show the proportion of observations in any region by areas under the
curve
To do that , we choose the scale so that the total area under the curve is exactly 1
The density curve is intended to reflect the idealized shape of the population distribution
Density curves are smoothed-out idealized pictures of the overall shapes of distributions, they are
most useful for describing large numbers of observations
The center and spread of a density curve
Areas under a density curve represent proportions of the total number of observations
So the median of a density curve is the equal-areas point, the point with half of the area under the
curve to its left and the remaining half of the area to the right
Density curves are idealized patterns, a symmetric density curve is exactly symmetric
The mean is the point at which the curve would balance if made of solid material
The mean and median of a symmetric density curve are equal
Normal Distributions
Normal curves are symmetric, single-peaked, and bell-shaped
Tails fall off quickly so that we do not expect outliers
Normal distributions are symmetric, the mean and median lie together at the peak in the center of
the curve
Normal curves also have the special property that we can locate the standard deviation of the
distribution by eye on the curve
The points at which this change of curvature takes placec are located one standard deviation on
either side of the mean
Normal curves have the special property that giving the mean and the standard deviation
completely specifies the curve
The mean fixes the center of the curve and the standard deviation determines its shape
Normal distributions are good descriptions for some distributions of real data
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