Statistics. David W. Scott

Statistics

Lord Rayleigh's Data

In Exploratory Data Analysis, Tukey (1977) demonstrates the box‐and‐whiskers plot using the Lord Rayleigh data, which measure the weight of nitrogen gas obtained by various means; see Table 1.1. Discrepancies in the results led to his discovery of the element argon. Rayleigh made

measurements from 1892 to 1894, with a mean of 2.30584 and a standard deviation of 0.00537. It is common to assume such measurements of a fundamental quantity are normally distributed. Multiple experiments are run and the results averaged in the presumption that a more accurate estimate will result.

Table 1.1 Lord Rayleigh's 24 measurements (sorted) of the weight of a sample of nitrogen. The first 10 came from chemical samples, while the last 14 came from pure air.

2.29816	2.29849	2.29869	2.29889	2.29890
2.29940	2.30054	2.30074	2.30143	2.30182
2.30956	2.30986	2.31001	2.31010	2.31010
2.31012	2.31017	2.31024	2.31024	2.31026
2.31027	2.31028	2.31035	2.31163

Figure 1.3 Displays of Lord Rayleigh's 24 measurements of the atomic weight of nitrogen gas. (Left) Histogram with four bins; (middle) a second histogram; (right) stem‐and‐leaf display using the

command

In the left frame of Figure 1.3, we display a histogram with four (carefully selected) bins. The histogram is shown on a density scale, rather than a frequency scale, so that the area of the shaded region is 1. We shall see in Problem 1 that this is accomplished by dividing the bin counts by

The first histogram in Figure 1.3 hides the interesting structure contained in the small dataset. The second histogram and stem‐and‐leaf plot show the two clusters quite clearly. Charting of data before the 1900s was not common, and looking at a table of the data would typically not reveal this feature. It turned out that Lord Rayleigh had combined various sources of the gas with several purifying agents and extraction methods. The samples originating from “pure air” were “contaminated” with argon. For the discovery of argon, Lord Rayleigh was awarded the Nobel Prize in Physics in 1904.

1.1.3 Discussion

Finding structure in data is a primary goal of data science. Graphical methods are powerful approaches to discovering unexpected or hidden structure. Some of these methods are better suited to small datasets. In a multivariate statistics course, we will learn how to analyze data with more than one variable. Modern genetic datasets often result in more than

variables!

1.2 Exploring Prediction Using Data

The second fundamental task of statistics is prediction. Data for this task are typically ordered pairs,

. The goal is to predict the value of the

variable using the corresponding value of the

variable. For example, we might try to predict a son's height (

) knowing the father's height (

). Or a bank contemplating a mortgage loan may use a person's credit score to predict the probability the person will default on the loan.

The initial step is to plot a scatter diagram of the

data points in order to determine if there is a strong relationship between

and

. The relationship, if it exists, is linear or nonlinear. If knowledge of

does not convey any information about the value of

, then the scatter diagram will have no slope or trends, with

values just scattered around their average.

1.2.1 Body and Brain Weights of Land Mammals

In the left frame of Figure 1.4, we plot the brain and body weights of 62 land mammals from the

MASS library. The relationship, if any, is hard to discern since 59 of the measurements are overplotted near the origin. We might choose to exclude the two elephant measurements and even the human data point as outliers, and then replot.

However, Tukey introduced a power transformation ladder to re‐express a variable

see Problem 3 for an explanation of why

is used in place of

when

In the right frame of Figure 1.4, we use the log function to dramatic effect. There clearly is a strong relationship that allows highly accurate prediction of the log(brain weight) of a land mammal knowing its log(body weight). (The body weight

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