DENSITY CURVES

■ Describing density curves

■ Normal distributions

■ The 68–95–99.7 rule

■ The standard Normal distribution

■ Finding Normal proportions

■ Using the standard Normal table

■ Finding a value when given a proportion

69 In this chapter, we add one more step to this strategy:

ImageSource/Photolibrary

EXPLORING A 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 devia-tions such as outliers.

3. Calculate a numerical summary to briefly describe center and spread.

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

Figure 3.1 is a histogram of the scores of all 947 seventh-grade students in Gary, Indiana, on the vocabulary part of the Iowa Test of Basic Skills.¹ Scores of many students on this national test have a quite regular distribu-tion. The histogram is symmetric, and both tails fall off smoothly from a

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Iowa Test vocabulary score

2 4 6 8 10 12

F I G U R E 3 . 1

Histogram of the Iowa Test vocabulary scores of all seventh-grade students in Gary, Indiana.

The smooth curve shows the overall shape of the distribution.

E X A M P L E 3 . 1

Our eyes respond to the areas of the bars in a histogram. The bar areas represent pro-portions of the observations. Figure 3.2(a) is a copy of Figure 3.1 with the leftmost bars shaded. The area of the shaded bars in Figure 3.2(a) represents the students with vocabulary scores of 6.0 or lower. There are 287 such students, who make up the pro-portion 287/947  0.303 of all Gary seventh-graders.

Now look at the curve drawn through the bars. In Figure 3.2(b), the area under the curve to the left of 6.0 is shaded. We can draw histogram bars taller or shorter by adjusting the vertical scale. In moving from histogram bars to a smooth curve, we make a specific choice: we adjust the scale of the graph so that the total area under the curve is exactly 1. The total area represents the proportion 1, that is, all the observations. We can then interpret areas under the curve as proportions of the observations. The curve is now a density curve. The shaded area under the density single center peak. There are no large gaps or obvious outliers. The smooth curve drawn through the tops of the histogram bars in Figure 3.1 is a good description of the overall pattern of the data.

7 0 C H A P T E R 3

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2 4 6 8 10 12 Iowa Test vocabulary score

(a)

F I G U R E 3 . 2 ( a )

The proportion of scores less than or equal to 6.0 in the actual data is 0.303.

2 4 6 8 10 12

Iowa Test vocabulary score (b)

F I G U R E 3 . 2 ( b )

The proportion of scores less than or equal to 6.0 from the den-sity curve is 0.293. The denden-sity curve is a good approximation to the distribution of the data.

DENSITY CURVE

A density curve is a curve that

■ is always on or above the horizontal axis, and

■ has area exactly 1 underneath it.

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.

Density curves, like distributions, come in many shapes. Figure 3.3 shows a strongly skewed distribution, the survival times of guinea pigs from Exercise 2.31 (page 60). The histogram and density curve were both created from the data by software. Both show the overall shape and the “bumps” in the long right tail. The density curve shows a single high peak as a main feature of the distribution. The histogram divides the observations near the peak between two bars, thus reducing curve in Figure 3.2(b) represents the proportion of students with scores of 6.0 or lower. This area is 0.293, only 0.010 away from the actual proportion 0.303. The method for finding this area will be presented shortly. For now, note that the areas under the density curve give quite good approximations to the actual distribution of the 947 test scores. ■

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7 2 C H A P T E R 3

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A P P LY Y O U R K N O W L E D G E

3.1 Sketch density curves. Sketch density curves that describe distributions with the following shapes:

(a) Symmetric, but with two peaks (that is, two strong clusters of observations) (b) Single peak and skewed to the left

3.2 Accidents on a bike path. Examining the location of accidents on a level, 5-mile bike path shows that they occur uniformly along the length of the path. Figure 3.4 displays the density curve that describes the distribution of accidents.

(a) Explain why this curve satisfies the two requirements for a density curve.

(b) The proportion of accidents that occur in the first mile of the path is the area under the density curve between 0 miles and 1 mile. What is this area?

(c) There is a stream alongside the bike path between the 0.8-mile mark and the 1.3-mile mark. What proportion of accidents happen on the bike path along-side the stream?

(d) The bike path is a paved path through the woods, and there is a road at each end. What proportion of accidents happen more than 1 mile from either road?

(Hint: First determine where on the bike path the accident needs to occur to be more than 1 mile from either road, and then find the area.)

0 100 200 300 400 500 600

Survival time (days)

F I G U R E 3 . 3

A right-skewed distribution pictured by both a histogram and a density curve.

the height of the peak. A density curve is often a good description of the overall pattern of a distribution. Outliers, which are deviations from the overall pattern, are not described by the curve. Of course, no set of real data is exactly described by a density curve. The curve is an idealized description that is easy to use and accurate enough for practical use.

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