Allen L. Shoemaker
Journal of Statistics Education v.4, n.2 (1996)
Copyright (c) 1996 by Allen L. Shoemaker, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the author and advance notification of the editor.
Key Words: Normal distribution; Two-sample t-test; t-statistic; Regression; Confidence interval.
This article takes data from a paper in the Journal of the American Medical Association that examined whether the true mean body temperature is 98.6 degrees Fahrenheit. Because the dataset suggests that the true mean is approximately 98.2, it helps students to grasp concepts about true means, confidence intervals, and t-statistics. Students can use a t-test to test for sex differences in body temperature and regression to investigate the relationship between temperature and heart rate.
1 One population mean that students all "know" is the mean normal body temperature of 98.6 degrees F. What is surprising is that recent medical research has posited that the mean normal temperature is really 98.2 degrees F! Discussions about this finding catch the students' interest, not only because the research contradicts their previous assumption, but also because it is relevant to their experience. Students already understand the "true mean" of body temperature of 98.6 degrees F, and they also have experience with deciding on a cut-point above which the temperature is considered suspect. This dataset on normal body temperature provides the background for such discussions. This dataset is also ideal to demonstrate two-sample t-tests, regression, confidence intervals, histograms, normal curves, and data anomalies.
2 The data were derived from an article in the Journal of the American Medical Association entitled "A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich" (Mackowiak, Wasserman, and Levine 1992). The authors display a histogram of 148 subjects' normal temperatures taken at several different times during two consecutive days, resulting in 700 total readings. The relative frequency histogram is also broken down by gender (122 males and 26 females). I derived the dataset presented here by working backwards from this histogram. I tried as closely as possible to recreate the original data, but with a reduction in sample size to 130 total readings. Relatively more of the female subjects' readings have been represented, so that the number of male and female readings would be equal. This equality helps to keep introductory statistics students from getting sidetracked by weighting issues. The means for both men and women (98.1 and 98.4) are the same as those reported in the article, and the distributions are as close as possible to the original. I also derived heart rates from the regression statistics given in the article, so that equivalent results can be obtained from this dataset.
3 This dataset is not only one to which students can relate, but it also has the benefit of versatility. It can be used to illustrate several statistical principles by addressing the following questions:
4 One of the interesting things about these data is that the overall mean temperature is NOT 98.6 degrees F as one would expect. Instead, the sample mean is 98.25 degrees F. This is one of the main points of the original article -- the traditional mean of 98.6 is, in essence, 100 years out of date. The authors cite problems with Wunderlich's original methodology, diurnal fluctuations (up to 0.9 degrees F per day), and unreliable thermometers.
5 The fact that this sample mean is 98.25 raises the question of whether this result is so unusual that it is unlikely to be due to simple sampling fluctuation. Students can investigate this question by calculating the confidence interval for this sample of 130 subjects. The 95% confidence limits extend from 98.12 to 98.38. Even the 99.9% confidence interval of 98.03 to 98.47 excludes the value 98.6. Ask students what this implies about the true population mean: Could the accepted mean of 98.6 be wrong, or is this just an extremely unusual sample?
6 Another way to investigate whether this sample mean is significantly different from 98.6 is to calculate a t-statistic using the standard error of the mean. The sample standard deviation is .73, so the standard error of the mean is .064. Thus the calculated t (using the sample mean of 98.25) is -5.45. Again, how likely is it that this is due to chance fluctuation?
7 Upon making a histogram of the data, the first thing that one notices is the normal distribution of the temperatures. This also holds true for the males' temperatures and the females' temperatures. Students can also make normal-quantile plots to verify the normality of this sample.
8 Assuming that these data are representative of the population, students can use these new values to answer the age-old question: "When are you really sick?" Individuals or groups can be compared to this dataset using simple t-statistics. However, there are some questions that students must answer first: Should we use a one- or two-tailed comparison? (Are abnormally low body temperatures indicative of sickness or just hypothermia?) What significance level is appropriate? (Would a 5% level create too many Type I errors, especially among truancy-minded kids?) Which has more severe consequences, Type I errors or Type II errors?
9 Because there is a difference between the mean temperatures of males and females in the sample, it is natural to ask if this difference is statistically significant. A two-sample t-test yields a t statistic of 2.29 (p = .024). Here, the results are a little less extreme and more likely to promote discussion about whether there is a true difference. (Beware of innuendoes about which sex is "hotter.")
10 Before analyzing these data, students may be asked to ponder this question and hypothesize about a possible relationship. When they have a fever, do they recall having higher heart rates? Could a higher heart rate be a mechanism of the body to generate higher body temperatures?
11 Correlating body temperatures with heart rates yields a correlation of .25 (p = .004). Given this positive correlation, one might ask what the increase in heart rate is for each degree rise in temperature. Regression analysis with heart rate as the dependent variable gives the equation
heart rate = 2.44 * temperature - 166.3.
The regression coefficient of 2.44 is the same as that reported in the original article.
12 One can introduce this question by asking students whether there is anything unusual about the distribution of temperatures. If students plot the distribution as a histogram with as many categories as possible (each category representing 0.1 degrees F), they will note that there is a periodic rise and fall within the overall normal curve. This occurs around every other category, or about every 0.2 degrees F. Perhaps this is an artifact of rounding, yet it would make little sense to round off at about every 0.2 degrees F. However, if the temperatures had been converted from a Centigrade scale, we would find the periodic rise at every 0.1 degrees Centigrade (because degrees C = (degrees F - 32) * 5/9). This makes more sense -- there would be a tendency to round to the nearest tenth of a degree. Another indication of this is the fact that in the original article, all temperatures are given first in degrees Centigrade, then converted to Fahrenheit.
13 Students have reacted favorably to this dataset, because they can all relate to the ideas that it illustrates. In addition, many statistical concepts can be covered with a relatively simple dataset.
14 The file normtemp.dat.txt contains the raw data. The file normtemp.txt is a documentation file containing a brief description of the dataset.
Columns 1 - 5 Body temperature (degrees Fahrenheit) 9 Gender (1 = male, 2 = female) 14 - 15 Heart rate (beats per minute)
Mackowiak, P. A., Wasserman, S. S., and Levine, M. M. (1992), "A Critical Appraisal of 98.6 Degrees F, the Upper Limit of the Normal Body Temperature, and Other Legacies of Carl Reinhold August Wunderlich," Journal of the American Medical Association, 268, 1578-1580.
Allen L. Shoemaker
3201 Burton St. S.E.
Grand Rapids, MI 49546