Representing and Comparing Data Distributions
Students create aligned dot plots or box plots for two numerical data sets and use centers, variability, and visual overlap to compare the distributions.
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Review Data Displays
A numerical data distribution shows the values in a data set and how often they occur. A dot plot places one dot above a number for every observation. A histogram groups values into intervals, while a box plot summarizes a distribution using the minimum, first quartile, median, third quartile, and maximum. Look for the center, spread, shape, and any unusual values. For example, the plant heights 4, 5, 5, 6, 6, 6, 7, and 8 inches have a median of 6 inches and a range of 4 inches. A dot plot reveals the cluster at 5 and 6 inches. A box plot uses less detail but makes the center and overall spread easy to see.
Choose an Appropriate Graph
Choose a graph based on the data and the comparison you need to make. Dot plots work well for small data sets because they show every value, repeated values, gaps, and possible outliers. Box plots are useful for larger data sets because they quickly compare medians and interquartile ranges. Histograms show overall shape, but grouping values can hide exact observations. Suppose Class A has quiz scores of 6, 7, 7, 8, 8, 8, 9, 9, and 10, while Class B has scores of 8, 9, 9, 10, 10, 10, 11, 11, and 12. Because each class has only nine scores, aligned dot plots are a strong choice. They preserve every score and make the shift between the two distributions visible.
Create Aligned Comparative Graphs
To compare two distributions fairly, draw both graphs with the same scale, intervals, and horizontal orientation. Place one graph directly above the other so equal values line up vertically. For the class quiz scores, label a shared horizontal axis from 6 through 12. For Class A, stack one dot at 6, two at 7, three at 8, two at 9, and one at 10. For Class B, stack one dot at 8, two at 9, three at 10, two at 11, and one at 12. Do not change the spacing or shorten one axis, because that could make one distribution appear more or less spread out. Include a title, class labels, and the score unit so the display can be interpreted correctly.
Identify Center and Variability
Center describes a typical location in a distribution, and variability describes how spread out the values are. For the two classes, the mean score for Class A is 8 and the mean for Class B is 10. The median scores are also 8 and 10. One measure of variability is the mean absolute deviation, or MAD, which is the average distance of all values from the mean. For Class A, the absolute distances from 8 total 8, so the MAD is 8 divided by 9, or about 0.89. Class B has the same pattern shifted two points higher, so its MAD is also about 0.89. The difference between the means is 2 points. Dividing 2 by 0.89 shows that the centers differ by about 2.25 MADs.
Analyze Visual Overlap
Visual overlap is the region where both distributions contain values. Begin by checking that the distributions have similar variability; otherwise, a center difference alone can be misleading. The class distributions have equal MADs and the same overall dot pattern, so comparing their overlap is reasonable. Both classes have scores at 8, 9, and 10, creating an overlap region from 8 through 10. Class A also has lower scores at 6 and 7, while Class B has higher scores at 11 and 12. The overlap is partial rather than complete. Because the means are about 2.25 MADs apart, the shift in center is noticeable compared with the typical distance of scores from each mean. Generally, greater center separation relative to variability produces less visual overlap.
State a Data-Based Comparison
A strong comparison names the context, compares centers, compares variability, and describes overlap using evidence. It should also avoid claiming that the data prove a cause. For example: In these samples, Class B had a higher typical quiz score than Class A. The mean and median were 10 for Class B and 8 for Class A, a difference of 2 points. Both distributions had a MAD of about 0.89 point, so their variability was similar. The centers were separated by about 2.25 MADs. The distributions overlapped at scores 8 through 10, but Class A extended lower and Class B extended higher. This statement is more informative than saying only that Class B did better because it explains the center, spread, and degree of overlap.