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7th Grade · 7.SP.B.3 · 7.SP.B.4

Comparing populations in 7th grade statistics

Comparing populations in 7th grade statistics means deciding whether one group tends to have bigger values than another, using samples from each. One example is whether sunflowers grown in the sun tend to be taller than sunflowers grown in the shade. Your child learns to compare two groups by their centers and their spread, and to judge how much the groups overlap.

What the standards say

The official Common Core wording, quoted exactly. The plain version follows.

7.SP.B.3 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.

7.SP.B.4 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.

What it means, in plain words

How far apart are the centers? 7.SP.B.3

A data set has a center and a spread. The center is a typical value. It can be the mean (add the values and divide by how many there are) or the median (the middle value when the values are in order). The spread, also called variability, shows how far the values sit from the center. Two measures of spread come up in 7th grade. The MAD, or mean absolute deviation, is the average distance of the values from the mean. The IQR, or interquartile range, is the width of the middle half of the data.

Comparing data sets by their centers alone can mislead, because two groups with different centers can still overlap a lot. To judge the overlap, your child divides the gap between the centers by a measure of spread. A gap of less than one MAD means the two groups mostly overlap. A gap of two or three MADs means they barely overlap, and the difference is easy to see on a dot plot.

Your child should be able to:

  • Find the difference between the means or the medians of two data sets
  • Divide that difference by the MAD or the IQR
  • Describe how much two dot plots or box plots overlap

Example

Sunflowers grown in a sunny spot have a mean height of 60 inches. Sunflowers grown in a shady spot have a mean height of 48 inches. Both groups have a MAD of 4 inches. How many MADs apart are the means?

  1. The gap between the means is 60−48=1260 - 48 = 12 inches.
  2. Divide by the MAD: 12÷4=312 \div 4 = 3.
  3. The means are 3 MADs apart, so the two groups barely overlap. The sunflowers in the sun are clearly taller as a group.

Drawing conclusions about two populations 7.SP.B.4

Comparing populations usually means asking what is typical in each one. For example, do 8th graders tend to spend more time on homework than 6th graders? Nobody can ask every student, so your child works from a random sample of each group. Measures of center and variability from the two samples are the evidence.

Your child compares the centers first, then checks them against the spread. When the centers are far apart compared with the spread, it is fair to say one population tends to have larger values. When the gap is small compared with the spread, the samples do not show a clear difference. Conclusions use words like "tends to" or "typically", because some members of the lower group still have higher values.

Your child should be able to:

  • Compare two populations using the means, medians, MADs or IQRs of random samples
  • Read two box plots side by side and describe what they show
  • Write a conclusion with "tends to" instead of "always"
  • Say when two samples do not show a clear difference

Example

Random samples of 20 sixth graders and 20 eighth graders at one school recorded homework time per week. The sixth graders have a median of 3 hours and the eighth graders a median of 7 hours. Both IQRs are 2 hours. Do eighth graders at the school tend to spend more time on homework?

  1. The gap between the medians is 7−3=47 - 3 = 4 hours.
  2. Compare the gap with the spread: 4÷2=24 \div 2 = 2 IQRs.
  3. A gap of 2 IQRs is large, so the two groups overlap only a little.
  4. Yes, eighth graders at this school tend to spend more time on homework than sixth graders.

Where children usually get stuck

  • Your child reads that the sunny sunflowers are 12 inches taller on average and says every sunny sunflower is taller than every shady one.

    A difference between means describes the two groups. It says nothing certain about each single plant.

    What helps. Look at the two dot plots together and find a tall shady plant and a short sunny plant. The groups differ as a whole, but single plants can still go either way.

  • Your child sees means of 62 and 60 and says the first group is clearly bigger, even though the MAD is 8.

    Any gap between two numbers looks like a real difference. With a spread of 8, a gap of 2 is only a quarter of a MAD, and the groups mostly overlap.

    What helps. Ask your child to divide the gap by the MAD before deciding. A result under 1 means there is no clear difference.

  • For the values 2, 4, 6 and 8, your child gets a MAD of 5, which is just the mean again.

    The MAD is an average of distances from the mean. The child averages the values themselves instead.

    What helps. Have your child write the distance of each value from the mean of 5 first: 3, 1, 1 and 3. Their average, 2, is the MAD.

Try four practice questions

Real 7th grade questions from our question bank, easiest first. Answer them right here, check each one, and see a worked solution for any question.

  1. 1Warm-up
    Each row compares two data sets with similar shapes and spreads. The table shows the distance between their centers in MAD units. Based on these distances, which pair would you expect to have more overlap, or would you expect the same overlap?
    PairCenter distance (MAD units)
    Pair A0.3
    Pair B2.5
    A
    Pair A
    B
    Pair B
    C
    Same overlap.
    Show the answer and how to solve it

    The answer is Pair A because 0.3 is a smaller multiple of MAD than 2.5, so Pair A has more overlap.

    1. 1The centers in Pair A are 0.3 times the MAD apart. The centers in Pair B are 2.5 times the MAD apart.
    2. 2The smaller multiple of MAD means the centers are closer together compared with the spread, so there is more overlap.
    3. 3Since 0.3 is smaller than 2.5, Pair A has more overlap.

    Common mistake. A common mistake is choosing the larger ratio because it looks like "more." In overlap comparisons, more MADs between the centers means the groups are farther apart, so the overlap is smaller.

    The idea underneath. For data sets with similar shapes and spreads, a smaller distance between centers relative to spread suggests more overlap.

  2. 2On level
    Two random samples were taken, one from Population A and one from Population B. Use the sample summaries in the table to choose the best inference about which population tends to have larger values.
    PopulationSample meanSample MAD
    Population A357
    Population B388
    A
    There is no clear difference in which population tends to have larger values.
    B
    Population B tends to have larger values.
    C
    Population A tends to have larger values.
    Show the answer and how to solve it

    There is no clear difference in which population tends to have larger values because Population B's mean is only 3 higher, while the MADs are 7 and 8.

    1. 1Look at the centers first. Population A has a sample mean of 35, and Population B has a sample mean of 38.
    2. 2Population B's center is only 3 higher.
    3. 3Now compare the spread. Population A has a MAD of 7 and Population B has a MAD of 8, meaning values in each sample are, on average, about 7 to 8 away from the mean.
    4. 4The center gap of 3 is small compared with both MADs (7 and 8). These summaries do not show a clear difference in which population tends to have larger values.

    Common mistake. A common mistake is choosing Population B just because 38 is greater than 35. You also need to compare spread, and here the 3-point center gap is smaller than both MADs of 7 and 8.

    The idea underneath. A difference in centers supports a population inference only when it is large compared with the variability in the samples.

  3. 3On level
    Look at the summary values and James's work. What mistake did James make when finding the variability-unit difference?
    Summary valueValue
    Mean of Group A8.4
    Mean of Group B5.2
    MAD1.5
    Step 1
    ∣8.4−5.2∣=3.2\displaystyle \left|8.4 - 5.2\right| = 3.2
    Step 2
    3.2÷8.4≈0.38\displaystyle 3.2 \div 8.4 \approx 0.38
    A
    Subtracted the centers incorrectly
    B
    Division is incorrect
    C
    Forgot absolute value
    D
    Divided by the wrong number
    Show the answer and how to solve it

    The correct choice is Divided by the wrong number because James found the difference 3.2 correctly in Step 1, but then divided by the mean 8.4 instead of the MAD 1.5.

    1. 1First, read the summary values. The means are 8.4 and 5.2, and the MAD is 1.5.
    2. 2Check James's Step 1. The absolute difference between 8.4 and 5.2 is 3.2, so that line is correct.
    3. 3Now check James's Step 2. The divisor should be the MAD, 1.5, not the mean 8.4. Dividing 3.2 by 1.5 gives approximately 2.13, not 0.38.
    4. 4James's mistake is using 8.4 as the divisor instead of the MAD 1.5.

    Common mistake. A common mistake is to divide 3.2 by 8.4 because that value appeared in the subtraction step. For a variability-unit difference, the divisor must be the variability measure, which here is the MAD 1.5.

    The idea underneath. A variability-unit difference tells how many MADs or IQRs fit into the distance between the centers, so the denominator must be the variability measure, not one of the centers.

  4. 4Stretch
    The table shows random samples of students from two populations. Which population is likely to have the greater typical number of books read last month?
    PopulationSample sizeSample mean (books read)
    Population A356.4
    Population B126.7
    A
    Population A
    B
    Population B
    C
    Neither; the sample means are equal
    Show the answer and how to solve it

    Population B is the best choice because its sample mean, 6.7 books, is greater than Population A's sample mean, 6.4 books.

    1. 1Look at the Sample mean (books read) column. Those are the center values to compare; the sample sizes tell you how many students were surveyed, not which population reads more.
    2. 2Population A has a sample mean of 6.4 books and Population B has a sample mean of 6.7 books. Even though Population A has the larger sample (35 vs. 12), the sample sizes do not change which center is larger.
    3. 3Since Population B has the larger sample mean, Population B is likely to have the greater typical number of books read last month.

    Common mistake. A common mistake is choosing Population A because its sample size of 35 is larger than Population B's sample size of 12. For this question, you compare the sample means, and 6.4 is still less than 6.7.

    The idea underneath. A random sample helps represent its population. Its mean or median estimates the population's typical value. Compare the same center for both samples. A larger sample center suggests a larger population typical value; equal sample centers favor neither population.