38

We now have a kit of graphical and numerical tools for describing distributions. What is more, we have a clear strategy for exploring data on a single quantitative variable:

Here is one more step to add to this strategy:

Density curves

A density curve is a mathematical model for the distribution of a quantitative variable. Mathematical models are idealized descriptions. They allow us to easily make many statements in an idealized world. The statements are useful when the idealized world is similar to the real world. The density curves that we study give a compact picture of the overall pattern of data. They ignore minor irregularities as well as outliers. For some situations, we are able to capture all of the essential characteristics of a distribution with a density curve. For other situations, our idealized model misses some important characteristics. As with so many things in statistics, your careful judgment is needed to decide what is important and how close is good enough.

Some of these vehicles in our example have been engineered to give excellent fuel efficiency. A marketing campaign based on this outstanding performance could be very effective for selling vehicles in an economy with high fuel prices. Be careful about how you deal with outliers. They may be data errors or they may be the most important feature of the distribution. Computer software cannot make this judgment. Only you can.

39

Here are some details about density curves. We need these basic ideas to understand the rest of this chapter.

The median and mean of a density curve

Our measures of center and spread apply to density curves as well as to actual sets of observations. The median and quartiles are easy. Areas under a density curve represent proportions of the total number of observations. The median is the point with half the observations on either side. So the median of a density curve is the equal-areas point—the point with half the area under the curve to its left and the remaining half of the area to its right. The quartiles divide the area under the curve into quarters. One-fourth of the area under the curve is to the left of the first quartile, and three-fourths of the area is to the left of the third quartile. You can roughly locate the median and quartiles of any density curve by eye by dividing the area under the curve into four equal parts.

40

The situation is different for skewed density curves. Here is an example.

What about the mean? The mean of a set of observations is its arithmetic average. If we think of the observations as weights strung out along a thin rod, the mean is the point at which the rod would balance. This fact is also true of density curves. The mean is the point at which the curve would balance if made of solid material.

41

We can roughly locate the mean, median, and quartiles of any density curve by eye. This is not true of the standard deviation. When necessary, we can once again call on more advanced mathematics to learn the value of the standard deviation. The study of mathematical methods for doing calculations with density curves is part of theoretical statistics. Though we are concentrating on statistical practice, we often make use of the results of mathematical study.

Because a density curve is an idealized description of the distribution of data, we need to distinguish between the mean and standard deviation of the density curve and the mean and standard deviation computed from the actual observations. The usual notation for the mean of an idealized distribution is (the Greek letter mu). We write the standard deviation of a density curve as (the Greek letter sigma).

Normal distributions

One particularly important class of density curves has already appeared in Figure 1.18(a). These density curves are symmetric, single-peaked, and bell-shaped. They are called Normal curves, and they describe Normal distributions. All Normal distributions have the same overall shape. The exact density curve for a particular Normal distribution is described by giving its mean and its standard deviation . The mean is located at the center of the symmetric curve and is the same as the median. Changing without changing moves the Normal curve along the horizontal axis without changing its spread. The standard deviation controls the spread of a Normal curve. Figure 1.22 shows two Normal curves with different values of . The curve with the larger standard deviation is more spread out.

The standard deviation is the natural measure of spread for Normal distributions. Not only do and completely determine the shape of a Normal curve, but we can locate by eye on the curve. Here’s how. Imagine that you are skiing down a mountain that has the shape of a Normal curve. At first, you descend at an ever-steeper angle as you go out from the peak:

The points at which this change of curvature takes place are located along the horizontal axis at distance on either side of the mean . Remember that and alone do not specify the shape of most distributions and that the shape of density curves in general does not reveal . These are special properties of Normal distributions.

43

Why are the Normal distributions important in statistics? Here are three reasons. First, Normal distributions are good descriptions for some distributions of real data. Distributions that are often close to Normal include scores on tests taken by many people (such as GMAT exams), repeated careful measurements of the same quantity (such as measurements taken from a production process), and characteristics of biological populations (such as yields of corn). Second, Normal distributions are good approximations to the results of many kinds of chance outcomes, such as tossing a coin many times. Third, and most important, many of the statistical inference procedures that we study in later chapters are based on Normal distributions.

The 68-95-99.7 rule

Although there are many Normal curves, they all have common properties. In particular, all Normal distributions obey the following rule.

Figure 1.23 illustrates the 68–95–99.7 rule. By remembering these three numbers, you can think about Normal distributions without constantly making detailed calculations.

Because we will mention Normal distributions often, a short notation is helpful. We abbreviate the Normal distribution with mean and standard deviation as . For example, the distribution of weights in the previous example is .

The standard Normal distribution

As the 68-95-99.7 rule suggests, all Normal distributions share many common properties. In fact, all Normal distributions are the same if we measure in units of size about the mean as center. Changing to these units is called standardizing. To standardize a value, subtract the mean of the distribution and then divide by the standard deviation.

A -score tells us how many standard deviations the original observation falls away from the mean, and in which direction. Observations larger than the mean are positive when standardized, and observations smaller than the mean are negative when standardized.

If the variable we standardize has a Normal distribution, standardizing does more than give a common scale. It makes all Normal distributions into a single distribution, and this distribution is still Normal. Standardizing a variable that has any Normal distribution produces a new variable that has the standard Normal distribution.

46

Normal distribution calculations

Areas under a Normal curve represent proportions of observations from that Normal distribution. There is no easy formula for areas under a Normal curve. To find areas of interest, either software that calculates areas or a table of areas can be used. The table and most software calculate one kind of area: cumulative proportions. A cumulative proportion is the proportion of observations in a distribution that lie at or below a given value. When the distribution is given by a density curve, the cumulative proportion is the area under the curve to the left of a given value. Figure 1.25 shows the idea more clearly than words do.

The key to calculating Normal proportions is to match the area you want with areas that represent cumulative proportions. Then get areas for cumulative proportions. The following examples illustrate the methods.

There is no area under a smooth curve and exactly over the point 820. Consequently, the area to the right of 820 (the proportion of scores ) is the same as the area at or to the right of this point (the proportion of scores ). The actual data may contain a student who scored exactly 820 on the SAT. That the proportion of scores exactly equal to 820 is 0 for a Normal distribution is a consequence of the idealized smoothing of Normal distributions for data.

How do we find the numerical values of the areas in Examples 1.34 and 1.35? If you use software, just plug in mean 1026 and standard deviation 209. Then ask for the cumulative proportions for 820 and for 720. (Your software will probably refer to these as “cumulative probabilities.” We will learn in Chapter 4 why the language of probability fits.) If you make a sketch of the area you want, you will rarely go wrong.

You can use the Normal Curve applet on the text website to find Normal proportions. The applet is more flexible than most software—it will find any Normal proportion, not just cumulative proportions. The applet is an excellent way to understand Normal curves. But, because of the limitations of web browsers, the applet is not as accurate as statistical software.

48

If you are not using software, you can find cumulative proportions for Normal curves from a table. That requires an extra step, as we now explain.

Using the standard Normal table

The extra step in finding cumulative proportions from a table is that we must first standardize to express the problem in the standard scale of -scores. This allows us to get by with just one table, a table of standard Normal cumulative proportions. Table A in the back of the book gives cumulative proportions for the standard Normal distribution. Table A also appears on the inside front cover. The pictures at the top of the table remind us that the entries are cumulative proportions, areas under the curve to the left of a value .

Now that you see how Table A works, let’s redo the NCAA Examples 1.34 and 1.35 using the table.

49

The area from the table in Example 1.37 (0.8389) is slightly less accurate than the area from software in Example 1.34 (0.8379) because we must round to two places when we use Table A. The difference is rarely important in practice.

Sometimes we encounter a value of more extreme than those appearing in Table A. For example, the area to the left of is not given directly in the table. The -values in Table A leave only area 0.0002 in each tail unaccounted for. For practical purposes, we can act as if there is zero area outside the range of Table A.

Inverse Normal calculations

Example 1.34 through 1.37 illustrate the use of Normal distributions to find the proportion of observations in a given event, such as “SAT score between 720 and 820.” We may, instead, want to find the observed value corresponding to a given proportion.

Statistical software will do this directly. Without software, use Table A backward, finding the desired proportion in the body of the table and then reading the corresponding from the left column and top row.

50

Without software, first find the standard score with cumulative proportion 0.9, then “unstandardize” to find . Here is the two-step process:

Solving this equation for gives

This equation should make sense: it finds the that lies 1.28 standard deviations above the mean on this particular Normal curve. That is the “unstandardized” meaning of . The general rule for unstandardizing a -score is

51

Assessing the Normality of data

The Normal distributions provide good models for some distributions of real data. Examples include the miles per gallon ratings of vehicles, average payrolls of Major League Baseball teams, and statewide unemployment rates. The distributions of some other common variables are usually skewed and, therefore, distinctly non-Normal. Examples include personal income, gross sales of business firms, and the service lifetime of mechanical or electronic components. While experience can suggest whether or not a Normal model is plausible in a particular case, it is risky to assume that a distribution is Normal without actually inspecting the data.

The decision to describe a distribution by a Normal model may determine the later steps in our analysis of the data. Calculations of proportions, as we have done earlier, and statistical inference based on such calculations follow from the choice of a model. How can we judge whether data are approximately Normal?

A histogram or stemplot can reveal distinctly non-Normal features of a distribution, such as outliers, pronounced skewness, or gaps and clusters. If the stemplot or histogram appears roughly symmetric and single-peaked, however, we need a more sensitive way to judge the adequacy of a Normal model. The most useful tool for assessing Normality is another graph, the Normal quantile plot.²⁷

Here is the idea of a simple version of a Normal quantile plot. It is not feasible to make Normal quantile plots by hand, but software makes them for us, using more sophisticated versions of this basic idea.

Any Normal distribution produces a straight line on the plot because standardizing turns any Normal distribution into a standard Normal distribution. Standardizing is a transformation that can change the slope and intercept of the line in our plot but cannot turn a line into a curved pattern.

Figure 1.28 through Figure 1.31 (pages 52-54) are Normal quantile plots for data we have met earlier. The data are plotted vertically against the corresponding Normal scores plotted horizontally. For small data sets, the z axis extends from -3 to 3 because almost all of a standard Normal curve lies between these values. With larger sample sizes, values in the extremes are more likely and the z axis will extend farther from zero. These figures show how Normal quantile plots behave.

52

Figure 1.28 does, of course, show some deviation from a straight line. Real data almost always show some departure from the theoretical Normal model. It is important to confine your examination of a Normal quantile plot to searching for shapes that show clear departures from Normality. Don—t overreact to minor wiggles in the plot. When we discuss statistical methods that are based on the Normal model, we will pay attention to the sensitivity of each method to departures from Normality. Many common methods work well as long as the data are reasonably symmetric and outliers are not present.

The idea that distributions are approximately Normal within a range of values is an old tradition. The remark “All distributions are approximately Normal in the middle” has been attributed to the statistician Charlie Winsor.²⁸

There are several variations on the way that diagnostic plots are used to assess Normality. We have chosen to plot the data on the y axis and the normal scores on the x axis. Some software packages switch the axes. These plots are sometimes called “Q-Q Plots.“ Other plots transform the data and the normal scores into cumulative probabilities and are called “P-P Plots.” The basic idea behind all these plots is the same. Plots with points that lie close to a straight line indicate that the data are approximately

54

BEYOND THE BASICS: Density Estimation

A density curve gives a compact summary of the overall shape of a distribution. Figure 1.17 (page 39) shows a Normal density curve that summarizes the distribution of miles per gallon ratings for 1067 vehicles.

Many distributions do not have the Normal shape. There are other families of density curves that are used as mathematical models for various distribution shapes.

Modern software offers a more flexible option: density estimation. A density estimator does not start with any specific shape, such as the Normal shape. It looks at the data and draws a density curve that describes the overall shape of the data.

55

Density estimates can capture other unusual features of a distribution. Here is an example.

Example 1.43 reminds us of a continuing theme for data analysis. We looked at a histogram and a density estimate and saw something interesting. This led us to speculate. Additional data on the type and location of the seats may explain more about the prices than we see in Figure 1.33.

56