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