Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
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
Are the words in a seventh-grade science book generally longer than those in a fourth-grade book? Do students at one school sleep more than students at another? Questions like these compare two whole populations, and seventh graders answer them by taking a random sample from each population and comparing measures of center (mean or median) and measures of variability (MAD or interquartile range).
The comparison is informal but careful. Students compute the statistics for both samples, notice how far apart the centers are, and weigh that against the spreads, as they did in 7.SP.B.3. They choose the median and IQR when data are skewed or contain outliers, such as one very long commute, and the mean and MAD when the data are roughly symmetric. Conclusions are phrased with appropriate caution: 'words in the seventh-grade book tend to be longer' rather than 'every word is longer'. Because the data come from random samples, students also keep in mind that another pair of samples might give slightly different numbers.
Two medians can differ while the data overlap almost completely. Looking at the IQR or MAD alongside the center shows whether the difference stands out.
One 48-minute commute pulls a mean far above most values. The median describes the typical commute better in skewed data.
A sample showing longer words in one book does not prove every word is longer. Phrases like 'tend to' or 'on average' match what the data support.
Ten randomly chosen words from a seventh-grade science book have lengths 3, 4, 5, 6, 6, 7, 8, 9, 10, 12. Ten random words from a fourth-grade book have lengths 2, 3, 3, 4, 4, 4, 5, 5, 6, 7. Compare the medians.
Answer: The seventh-grade sample's median (6.5 letters) is 2.5 letters higher than the fourth-grade median (4 letters), so its words tend to be longer.
Let students collect their own samples from two real sources, such as word lengths from two books or reaction times from two hands, and present a comparison with box plots or dot plots. Real data with a surprising outlier creates a natural discussion about choosing the median.
Test items often give two box plots or two sets of summary statistics and ask which conclusion is supported. Teach students to check three things: the difference in centers, the size of the spreads and whether the samples were random.
Original questions written for this standard. Choose an option or type your answer, then press Check. Every question has a worked explanation.
Answer: 6.6
Add the values: 3 + 8 + 5 + 10 + 7 = 33. Divide by 5: 33 ÷ 5 = 6.6.
Answer: B) Population B tends to have larger values, since the medians differ by more than one IQR
The medians differ by 8, which is more than either IQR, so B's values tend to be higher. The conclusion is informal but supported.
Answer: 7
IQR = Q3 - Q1 = 19 - 12 = 7.
Answer: D) The median
The outlier pulls the mean upward, but the median depends only on the middle values, so it describes the typical commute better.
Answer: 16.5
With 8 values, the median is the mean of the 4th and 5th values: (15 + 18) ÷ 2 = 16.5 minutes.
Answer: B) So each sample is likely to represent its population fairly
Random sampling avoids systematic bias, so differences between the samples are more likely to reflect real differences between the populations.
A full lesson with slides, activities and an exit ticket on comparing two populations with samples, pitched to grade 7 and editable in PowerPoint or Google Slides.
Make a lesson →A printable, differentiated worksheet on 7.SP.B.4 with an answer key, ready in about a minute.
Make a worksheet →Turn comparing two populations with samples into a quiz students answer online that marks itself, with a class summary for you.
Build a test →Use the median and IQR when data are skewed or have outliers, and the mean and MAD when the data are fairly symmetric. Either pair works if it is used consistently for both samples.
7.SP.B.3 focuses on judging overlap by expressing the difference in means as a multiple of the MAD. 7.SP.B.4 broadens this to comparing two populations using any suitable measures from random samples.