Wiki Page, Math 13 - Perez Sandbox : Module 4 - Histograms (4 of 4)

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2026-01-17T21:15:49-08:00

Using Histograms to Compare Distributions

We now use histograms to compare the distributions of a quantitative variable for two groups of individuals. Previously, we did a similar comparison using dotplots. As before, our descriptions focus on the overall pattern (shape, center, and spread) as well as deviations from the pattern (outliers). We also use percentages to describe and compare different intervals of variable values, since histograms make it easy to do so.

EXAMPLE Smoking and birth weight

Does smoking during pregnancy have an impact on birth weight? To investigate this question, doctors collected data on 189 new mothers who gave birth at a hospital in Massachusetts during the 1980s.

Here we use histograms to compare the distribution of birth weights for mothers who smoked during pregnancy with mothers who did not smoke. The table shows the numbers of mothers with babies in each interval of birth weights. (Left endpoints are included in the bin, so a 1,000-gram baby is in the interval 1,000–1,500 grams.)

Note: For easy and more accurate visual comparisons, both histograms have the same horizontal scale and bin width. Also, the scale on the vertical axis is the same. So we can directly compare the heights of the bars to compare the number of mothers with babies in each interval of birth weights.

Frequency of birth weights

Weight (grams) | Smoker No | Smoker Yes
500-1000 | 0 | 1
1000-1500 | 3 | 1
1500-2000 | 8 | 6
2000-2500 | 18 | 22
2500-3000 | 22 | 16
3000-3500 | 29 | 17
3500-4000 | 27 | 10
4000-4500 | 6 | 1
4500-5000 | 2 | 0
| total = 115 | total = 74

Observations about the shape, center, and spread:

Nonsmokers:The top histogram shows the distribution of birth weights for the nonsmokers. The distribution appears skewed slightly to the left. Nearly half of the babies have a birth weight between 3,000 and 4,000 grams (29 + 27 = 56, 56/115 = 49%) with the number of babies tapering off in the lower weight ranges. There is substantial amount of variability in birth weights. With birth weights as low as 1,000-1,500 grams and as high as 4,500-5,000 grams, we can estimate that the overall range as something in the ballpark of 4,000 grams.

Smokers: The bottom histogram shows the distribution of birth weights for the smokers. The distribution appears slightly skewed to the right. For smokers, nearly half of the babies have a birth weight between 2,000 and 3,000 grams (16 + 22 = 38, 38 / 74 = 51%) with fewer babies in heavier weight ranges. There is substantial amount of variability in birth weights. With birth weights as low as 500-1,000 grams and as high as 4,000-4,500 grams, we can estimate that the overall range as something in the ballpark of 4,000 grams.

Comment: As we have seen, the choice of bin width can affect the shape of a histogram. We also cannot make precise statements about center and spread; we have to use intervals. Bin width may affect our sense of “typical” measurements.

Using a benchmark:

Another strategy for comparing distributions is to use a benchmark. Here are some examples:

1. Doctors define low birth weight as a birth weight below 2,500 grams. Calculate and compare the percentage of smokers and nonsmokers with low-birth-weight babies by this definition.

Nonsmokers: Of babies born to mothers who did not smoke, 3 + 8 + 18 = 29 weighed less than 2,500 grams, so 25% (29 of 115) of the babies born to nonsmokers fit the definition of low birth weight.

Smokers: Of babies born to mothers who smoked, 1 + 1 + 6 + 22 = 30 weighed less than 2,500 grams, so 41% (30 of 74) of the babies born to smokers fit the definition of low birth weight.

2. A condition called macrosomia (also known as big baby syndrome) is defined as a birth weight of 4,000 grams or more. Calculate and compare the percentage of smokers and nonsmokers with babies that fit the definition of macrosomia.

Nonsmokers: Of babies born to mothers who did not smoke, 6 + 2 = 8 weighed 4,000 grams or more, so 7% (8 of 115) of the babies born to nonsmokers fit the definition of macrosomia.

Smokers: Of babies born to mothers who smoked, only 1 weighed 4,000 grams or more, so 1% (1 of 74) of the babies born to smokers fit the definition of macrosomia.

Writing the paragraphs

In this study, we compared mothers who smoked during pregnancy to mothers who did not smoke during pregnancy. The variable is the birth weights of their babies. Both groups had a lot of variability in birth weights, with similar overall range estimates of 4,000 grams.

There was also a lot of overlap in the distributions. Nonsmokers had babies that weighed between approximately 1,000 and 5,000 grams. Smokers had babies that weighed between approximately 500 and 4500 grams.

However, we also observe some subtle differences in the typical ranges of birth weights for the two groups. For nonsmokers, nearly half of the babies have a birth weight between 3,000 and 4,000 grams (56 out of 115, 49%) but only 36% of babies born to smokers (27 out of 74) are this heavy. For smokers, nearly half of the babies have a birth weight between 2,000 and 3,000 grams (40 of 74, 54%) but only 35% of babies born to nonsmokers (40 out of 115) are this light.

If we use the medical definition of low birth weight (under 2,500 grams), we see that smokers in this study have a much higher incidence of low birth weights: 25% (29 of 115) of the babies born to nonsmokers fit the definition of low birth weight, compared to 41% (30 of 74) of the babies born to smokers. In this study, smoking is associated with lower birth weights, though the variability in the data suggests that other variables also contribute to birth weight.

Tips for comparing two distributions and writing your paragraphs

Develop a thesis statement that compares the two groups.

Make the context of the data clear to the reader (individuals, variable with units.)

Synthesize your observations into paragraphs that compare the groups in ways that support your thesis, as opposed to just listing your observations.

Incorporate concepts we have studied: shape, center, spread, relative frequencies (percentages), benchmarks

Mark this page as done, then click the "Next" or > button to continue.

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