Chapter 3. The Normal Distributions

Chapter 3. The Normal Distributions

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Chapter 3. The Normal Distributions

Density Curves

Definition. A (probability) density curve is a curve that ? is always on or above the horizontal axis, and ? has area exactly 1 underneath it (that is, the area between the curve and the x-axis).

A density curve describes the overall pattern of a distribution. The area under the curve and above any range of values is the proportion of all observations that fall in that range.

Example. Exercise 3.1 page 67.

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Describing Density Curves

Definition. The median of a density curve is the equalareas point, the point that divides the area under the curve in half. That is, half of the area under the curve is to the left of the median and the remaining half of the area is to its right. The mean of a density curve is the balance point, at which the curve would balance if made of solid material. The median and mean are the same for a symmetric density curve.

Figures 3.4(a) and 3.4(b) page 68. Figure 3.5 page 68.

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Note. The usual notation for the mean of an idealized distribution is ? (mu). The standard deviation of a density curve is denoted (sigma).

Example. Exercise 3.4 page 69.

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Normal Distributions

Note. A VERY common class of density curves is the normal distributions. These curves are symmetric, single-peaked, and bell-shaped. All normal distributions have similar shapes and are determined solely by their mean ? and standard deviation . The points at which the curves change concavity are located a distance on either side of ?. We will use the area under these curves to represent a percentage of observations. (These areas correspond to integrals, for those of you with some experience with calculus.)

Figure 3.8 page 70.

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Note. The text mentions three reasons we are interested in normal distributions:

1. Normal distributions are good descriptions for some distributions of real data. Examples include test scores and characteristics of biological populations (such as height or weight).

2. Normal distributions are good approximations to the results of many kinds of chance outcomes. An example is the proportion of heads in a repeatedly tossed coin experiment.

3. Many statistical inference procedures based on normal distributions work well for other roughly symmetric distributions.

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