Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.
Official wording from the Common Core State Standards for Mathematics (© 2010 National Governors Association Center for Best Practices and Council of Chief State School Officers). View on thecorestandards.org
Is a 6-centimeter difference in average plant height a big difference? It depends on how much the heights vary within each group. If the plants in each group differ from their group's mean by only a centimeter or two, a 6-centimeter gap separates the groups clearly. If heights within each group vary by 10 centimeters, the same gap is lost in the noise. Seventh graders learn to judge differences this way.
When two numerical data sets have similar variability, students measure the difference between their centers and express it as a multiple of a measure of variability, usually the mean absolute deviation (MAD). A gap of 2 MADs means noticeable separation on a dot plot; a gap of half a MAD means the two distributions overlap heavily. This is an informal step toward the formal significance testing of high school statistics, and it trains students to look at both the centers and the spreads before declaring that two groups really differ.
Students conclude one group is better because its mean is higher, even by a tiny amount. Expressing the gap in MADs shows whether it stands out from normal variation.
The MAD describes spread, not center. The comparison here is the difference between centers, measured in units of the MAD.
The method works best when both groups have similar spreads. If one MAD is much larger, students should describe the spreads separately rather than divide by one of them.
Plants given fertilizer A have a mean height of 24 cm with a MAD of 3 cm. Plants given fertilizer B have a mean height of 30 cm with a MAD of 3 cm. How much do the distributions overlap?
Answer: The means differ by 2 MADs, so the two groups are noticeably separated with limited overlap.
Show pairs of dot plots on the same scale, some with means 0.5 MAD apart and some 3 MADs apart, and ask students to rank them by overlap before calculating anything. The calculation then confirms what their eyes already noticed, which makes the ratio feel meaningful rather than arbitrary.
Assessment items typically give two dot plots or summary statistics and ask students to compute the difference in means as a multiple of the MAD and interpret it. Insist on a sentence that links the number back to the overlap of the data.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 2
The difference is 20 - 15 = 5. Divide by the MAD: 5 ÷ 2.5 = 2 MADs.
Answer: A) The means differ by 4 MADs
The bigger the gap measured in MADs, the less the two distributions overlap. A 4-MAD gap separates them the most.
Answer: 2
The mean is 20 ÷ 4 = 5. The distances from 5 are 3, 1, 1 and 3, which total 8. MAD = 8 ÷ 4 = 2.
Answer: C) The distributions overlap a lot, so the difference is small compared with the variation
A gap of only half a MAD is smaller than the typical distance of data from its mean, so the dot plots overlap heavily.
Answer: 1.5
The difference is 78 - 72 = 6. Then 6 ÷ 4 = 1.5 MADs.
Answer: A) Because a difference only matters relative to how spread out the data are
A 6-point gap is large when scores barely vary but small when they vary widely. Dividing by the MAD puts the gap in context.
A full lesson with slides, activities and an exit ticket on comparing two distributions with overlap, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.SP.B.3 with an answer key, ready in about a minute.
Make a worksheet →Turn comparing two distributions with overlap into a quiz students answer online that marks itself, with a class summary for you.
Build a test →It is the average distance of the data values from their mean. It describes how spread out the data are, in the same units as the data.
Not in this standard. Students make informal judgments: a gap of 2 or more MADs usually shows clear separation, while a gap well under 1 MAD shows heavy overlap.