Statistics. David W. Scott

Statistics

but not its brain weight.) Moreover, the relationship appears to be linear. In this re‐expressed scatter diagram, the two or three outliers identified in the first plot are no longer outliers.

1.2.2 Space Shuttle Flight 25

The 25th launch in the Space Shuttle program was scheduled for 22 January 1986, but postponed for various reasons each day until 28 January. The temperature had dropped to 28

overnight, and it was 36

when the launch was attempted at 11:38 a.m. During the first 90 s, several O‐rings on the solid rocket boosters failed, leading to a catastrophic explosion and loss of all seven crew members. Scientists knew previous shuttle flights had occasionally experienced one or two O‐ring failures, but a launch had never been attempted at freezing temperatures. Varying opinions of the safety were provided to the launch director, who eventually decided to proceed. One of the data analyses is reproduced in the first row of Figure 1.5.

Graphs depict the plot of the raw and log-transformed body and brain weights of sixty-two land mammals.

Figure 1.4 Scatter diagrams of the raw and

‐transformed body and brain weights of 62 land mammals.

Graphs depict the analysis of the number of O-ring failures for the first twenty-four Space Shuttle launches.

Figure 1.5 Analysis of the number of O‐ring failures for the first 24 Space Shuttle launches; see text.

In the heading of the scatter diagram in first frame, we see a list of the 7 (of the first 24) shuttle flights that experienced 1 or 2 O‐ring failures. Two failures were observed at the lowest temperature of 53 images , which was well above the temperature range of 28–36 images on the day of the disaster. Strangely, two failures had also been observed at the highest temperature of 75 images .

In the second frame, we have jittered the data by adding a little uniform noise. This reveals that there were two data points superimposed at images ; jittering broke that tie. In the third frame, the data are replotted, but with an expanded images ‐axis to include 28 images . Would you have supported the decision to launch? A least‐squares line (discussed in Chapter 8.5) is superimposed. This line suggests that, if anything, lower temperatures might result in fewer O‐ring failures. Thus the launch was attempted.

However, in a re‐analysis of these data, we have included the shuttle flights that experienced no O‐ring failures. Now the final frame suggests that two or more O‐ring failures are quite likely at 28–36 images . The question of including or excluding data is a difficult problem in practice. In other settings, including non‐event data can bias the analysis in the wrong direction. As we saw in the brain‐body weight data, excluding the two or three outliers was not necessary. However, in Rayleigh's nitrogen data, excluding an entire cluster of outliers as bad data would have postponed the discovery of argon.

1.2.3 Pearson's Father–Son Height Data Revisited

We have explored the two variables in this dataset individually, but there is an obvious question of how accurately a son's height can be predicted knowing his father's height. In the first frame of Figure 1.6, we display a scatter diagram of the images pairs. This diagram clearly shows a positive tilt, consistent with the expectation that the sons of tall fathers are tall, and vice versa; however, the strength of the relationship does not seem as strong as in the brain–body weight dataset.

In the top right frame, we have placed a red dot at the location of the average heights of the fathers and sons. We have also drawn a straight line fit using the intuitive equation images . However, the equation images is an improvement, since we observed earlier that sons were 1 inch taller than their fathers on average. As a reference, we have also included a horizontal line at the average heights of the sons. This line would be appropriate if there were no information about a son's height to be gleaned from his father's height; but a positive relationship (correlation) is clear.

Galton (1886) was one of the first to observe that many scatter diagrams observed in nature have an appearance similar to that in Figure 1.6. He noted that the shape appeared elliptical, so he superimposed elliptical contours over the scatter diagram. The bottom left frame in Figure 1.6 shows three (nested) ellipses for these data. Recall that a general ellipse has five parameters: two for the center of the ellipse; two for the horizontal and vertical scales; and a fifth called the eccentricity. Galton focused on this fifth parameter, and the correlation coefficient was the result. Ironically, this parameter is often referred to today as Pearson's correlation coefficient.

Graphs depict the scatter plot of the father and son height data that are collected by Karl Pearson.

Figure 1.6 Father–son height data collected by Karl Pearson.

In the final frame, we take advantage of the large sample size to try to understand if the prediction (as weak as it may be) might be linear or nonlinear. For integer values of the rounded fathers' heights, we compute a three‐point summary of the corresponding sons' heights. The red dots are the arithmetic average of the sons' heights. The vertical lines display the (conditional) interquartile range. The final two red dots on each end are based on only a few points, so that the IQR can not be computed. These four red dots are shown in a smaller font size to indicate that even the averages are not so reliable.

We see that these summary points clearly suggest a linear rather than a nonlinear fit. We also see that the two blue reference lines from the second frame, namely images and images , both miss badly. A new (dashed) line with slope of 1/2 appears to capture the linear trend quite well. The relationship between this slope and the correlation coefficient, as well as a genetic explanation, will be discussed in Chapter 4.1.5.

1.2.4 Discussion

These rather substantial examples illustrate the search for structure in distribution and prediction problems, as well as practical problems and cures that may be encountered. A

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