7.SP.B.4Common CoreMathStatistics and ProbabilityGrade 7
7.SP.B.4: Comparing Two Populations with Random Samples
In plain English: 7.SP.B.4 is the Common Core grade 7 math standard that asks students to compare two populations using random samples. Students find a measure of center (mean or median) and a measure of variability (MAD or IQR) for each sample, compare the difference in centers with the spread, and write a careful conclusion such as "the words in the grade 7 book tend to be longer."
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.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Draw informal comparative inferences about two populations. Also written as 7.SP.4 · Official standard
Students use random samples to compare two groups they cannot measure completely. A population is the whole group you want to know about, such as every word in a book chapter. A sample is the part you actually measure, and a random sample is one where every member of the population has the same chance of being picked. For each sample, students find a measure of center (one number for a typical value: the mean, the sum divided by the count, or the median, the middle value) and a measure of variability (one number for how spread out the values are: the mean absolute deviation, or MAD, the average distance from the mean, or the interquartile range, or IQR, the spread of the middle half).
Then students make an informal comparative inference: a careful conclusion about how the two populations compare, based on the samples and without formal tests. The key question is whether the difference between the two centers is large compared with the spread of the data, and whether it shows up again in new random samples. Conclusions use words like "tend to" and "generally," because samples vary. Every data set on this page is invented for teaching, with realistic values for the objects and people named.
Learning Objectives
By the end of this lesson, students will be able to:
Name the two populations and explain why the samples must be random before comparing them
Find the mean and MAD, or the median and IQR, of two samples and choose the pair that fits the shape of the data
Compare the difference between two centers with the variability, for example "the means differ by about 3 MADs"
Use several random samples to see how sample results vary and whether one population is larger again and again
Write an informal conclusion about two populations that uses "tend to" or "generally" and says how sure it is
Prior Knowledge Required
Students should already be comfortable with:
Finding the mean, median, IQR and MAD of a data set and describing its shape 6.SP.B.5
Reading and drawing dot plots and box plots 6.SP.B.4
Why random samples tend to represent a population, and why small samples vary 7.SP.A.1
Read this claim aloud and give students two minutes to write a response:
Warm-Up Prompt
"My little cousin in grade 4 can type faster than I can. So fourth graders type faster than seventh graders." Is this a good conclusion? What information would you want before you decide?
Collect ideas on the board. Students usually say that one person is not enough, that the people should be chosen fairly, and that you need to see how much typing speeds vary. Name the ideas: the two populations are all fourth graders and all seventh graders; we can only measure a sample of each; the sample should be random; and we need both a typical value and the spread. Tell students that today they learn how to make this kind of comparison carefully.
Direct Instruction20 minutes
Show the four steps students use for every comparison:
Check the samples: name the two populations and make sure each sample was chosen at random and is not tiny.
Summarize each sample: find a measure of center and a measure of variability. Use the mean and MAD when the data are roughly symmetric (about the same on both sides of the middle). Use the median and IQR when the data are skewed (stretched out to one side) or have a value far from the rest.
Compare the difference with the spread: divide the difference between the centers by the larger MAD (or IQR).
Write the inference: a sentence about the populations, with "tend to" or "generally," that says how sure you are.
For step 3, give students this classroom guide (it is a rule of thumb, not an official rule). If the difference between the centers is 2 or more times the larger measure of variability, the two data sets are clearly separated, with only some overlap, and the populations very likely differ. If it is between 1 and 2 times, a difference is likely, but check it with more or larger samples. If it is less than 1 time, the data overlap a lot, and one pair of samples cannot tell you which population is larger.
The official example: word lengths, median and IQR, then repeated samples
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. A student picks 20 words at random from each chapter and counts the letters (Diagram 1). Grade 4: 1, 2, 3, 3, 3, 3, 4, 4, 4, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 8. Grade 7: 1, 2, 3, 3, 4, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 8, 9, 10, 12.
Equation: Grade 4: median 4, IQR 2, mean 4.3. Grade 7: median 5, IQR 3, mean 5.55. The medians differ by 1 letter, which is 1/3 of the larger IQR, so the samples overlap a lot and one pair is not enough. The class takes 5 more pairs of random samples of 20 words. Grade 4 sample means: 4.0, 4.6, 4.2, 4.9, 4.4. Grade 7 sample means: 5.6, 5.0, 5.9, 5.3, 6.0. The grade 7 mean is larger in every pair, so the words in the grade 7 chapter are generally longer, by about 1 letter.
Mean and MAD, a clear difference
Hours of sleep on a school night for a random sample of 10 seventh graders and a random sample of 10 eleventh graders at one school. Grade 7: 7.5, 8, 8, 8.5, 8.5, 8.5, 9, 9, 9, 9. Grade 11: 6, 6.5, 6.5, 7, 7, 7, 7, 7.5, 7.5, 8.
Equation: Grade 7: mean 8.5 hours, MAD 0.4. Grade 11: mean 7 hours, MAD 0.4. Difference 1.5 hours; 1.5 ÷ 0.4 = 3.75 MADs, more than 2. Seventh graders at this school tend to sleep more on school nights than eleventh graders.
Median and IQR for skewed data
Number of pages in a random sample of 11 fiction books and 11 nonfiction books from a school library (Diagram 2). Fiction: 180, 210, 224, 240, 256, 272, 288, 300, 320, 352, 610. Nonfiction: 48, 72, 96, 112, 120, 144, 150, 176, 208, 232, 400.
Equation: The 610-page and 400-page books pull the means up (about 296 and 160 pages), so use medians. Fiction: median 272, IQR 96. Nonfiction: median 144, IQR 112. Difference 128 pages; 128 ÷ 112 is about 1.1 IQRs. Fiction books in this library are likely longer in general, but check with another sample.
When one pair of samples cannot tell
Masses of random samples of 8 apples from two orchards, in grams. Orchard A: 164, 170, 176, 181, 185, 190, 196, 202. Orchard B: 158, 166, 172, 178, 184, 188, 194, 200.
Equation: Orchard A: mean 183 g, MAD 10.25. Orchard B: mean 180 g, MAD 11.5. Difference 3 g; 3 ÷ 11.5 is about 0.26 MADs, less than 1. The samples overlap almost completely, so we cannot say which orchard's apples are heavier. Larger random samples would help.
Diagram 1 shows the official example as two dot plots (a number line with one dot for each value). Point out how much the dots overlap even though the grade 7 dots sit a little further right. That is why the class took more samples: one sample mean can be off by chance, but when the grade 7 sample mean is larger in every pair, the difference is probably real. Diagram 2 shows the library data as box plots (a box from the first quartile, Q1, to the third quartile, Q3, with a line at the median). Ask: "Why is the median a better typical value than the mean for fiction?" (One 610-page book pulls the mean up.)
Stress the wording of conclusions. "Seventh graders tend to sleep more" is supported. "Every seventh grader sleeps more than every eleventh grader" is not: some values overlap. A conclusion is only as good as the samples, so a sample of friends or volunteers cannot support it at all.
Guided Practice15 minutes
Pairs work through three comparisons. For each one, they name the populations, find the difference in centers as a number of MADs or IQRs, and write a one-sentence inference. Circulate and ask each pair which category of the classroom guide their answer falls in.
Guided practice comparisons with answers (invented data)
Comparison
Answer
Backpack mass for random samples of 20 sixth graders and 20 eighth graders: means 5.1 kg and 5.4 kg, MADs 1.2 kg and 1.3 kg
Difference 0.3 kg, about 0.23 of the larger MAD: less than 1, so these samples cannot tell which grade carries heavier backpacks
Height for random samples of 25 first graders and 25 seventh graders: means 117 cm and 157 cm, MADs 4 cm and 6 cm
Difference 40 cm, about 6.7 MADs: seventh graders are clearly taller in general
Minutes of homework last night for random samples of 8 sixth graders (12, 18, 20, 24, 25, 28, 33, 40) and 8 eighth graders (30, 34, 38, 40, 42, 45, 50, 57)
Means 25 and 42 minutes, both MADs 6.5. Difference 17, about 2.6 MADs: eighth graders at this school tend to spend more time on homework
Listen for three errors: comparing the difference with only one sample's spread when the other is larger, writing "all" or "every" in the conclusion, and forgetting to ask whether the samples were random. For the backpack row, ask: "Does 'cannot tell' mean the grades carry the same mass?" (No. It means these samples do not show a difference.)
Independent Practice15 minutes
Students work alone on this comparison. Masses of random samples of 9 eggs from two farms, in grams (invented data, in the normal range for chicken eggs):
Egg masses in grams for two random samples of 9 eggs (invented data)
Farm
Egg masses (g)
Farm A
52, 55, 56, 58, 58, 59, 61, 62, 66
Farm B
60, 62, 63, 64, 65, 66, 68, 69, 72
Tasks: (1) Find the median and IQR of each sample. (2) Find the difference between the medians as a number of IQRs. (3) Write an inference about the two farms. (4) Name one thing that would make you more sure. Answers: Farm A median 58 g, IQR 6 g; Farm B median 65 g, IQR 6 g. The difference of 7 g is about 1.2 IQRs, so Farm B's eggs are likely heavier in general, but it is worth checking with more random samples or larger ones.
Closure5 minutes
Exit ticket: Random samples of 30 seventh graders on the track team and 30 seventh graders not on the team ran 100 m. Team: mean 14 seconds, MAD 1 second. Not on the team: mean 18 seconds, MAD 2 seconds. (1) How many MADs apart are the means? (2) Write one sentence comparing the two populations. (Answers: (1) 4 ÷ 2 = 2 MADs, using the larger MAD. (2) Seventh graders on the track team tend to run 100 m faster than those not on the team.) (3) In one sentence: why would a sample of your friends not work here?
Differentiation Strategies
For Struggling Students
Give a summary table with blanks (center, spread, difference, number of MADs) so students follow the four steps in order
Start with data sets that do not overlap at all on a dot plot, then move to ones that overlap
Provide sentence frames: "The ___ in population A tend to be ___ than in population B, because ___"
For Advanced Students
Ask students to build two samples of 8 values with the same mean but very different MADs, and explain what an inference about each population would say
Ask how the conclusion for the library books changes if the 610-page book is replaced by a 300-page book
Have students plan a real comparison at school (for example, reaction times of two grades), including how to choose a random sample
Assessment Guidance
What to Look For
Check that students name both populations and question how the samples were chosen before they compute anything. Look for the right pair of measures (mean with MAD, median with IQR) and a reason tied to the shape of the data. Students should express the difference between centers as a number of MADs or IQRs, and their conclusion should be about the populations, use words like "tend to," and match the size of the difference: a clear claim for 2 or more, a cautious one below that.
02
Classroom Activities
3 Activities
1
Word Length Hunt
20 minPairs
Pairs carry out the official example with real books. Each pair takes a random sample of 15 words from a chapter of a grade 4 science book and 15 words from a chapter of a grade 7 science book, then the class pools its results.
Procedure
Roll number cubes (or use a random number generator) to pick a page, a line on that page and a word in that line. Repeat until you have 15 words from each chapter.
Count the letters in each word. Skip numbers; count a hyphenated word as one word.
Draw a dot plot for each sample on the same scale, and find each sample's median and IQR.
Write your pair's two medians on sticky notes and add them to the class chart, one column for each book.
Discussion Questions
Did every pair find a larger median for the grade 7 chapter? What does it tell you if most pairs did?
Look at the class chart: which varies less, single word lengths or pair medians?
Why did we pick words with number cubes instead of choosing words that looked typical?
Modification for Distance Learning
Share two chapter scans. Students use an online random number generator to choose page, line and word numbers, and enter their medians in a shared spreadsheet.
2
Can We Tell? Card Sort
15 minPairs
Each pair gets 8 comparison cards and 8 matching answer cards (16 cards in all). For each comparison card, pairs divide the difference in centers by the larger MAD or IQR and sort the card into one of three piles: "Clear difference" (2 or more), "Likely, check more samples" (1 to 2) or "Cannot tell" (less than 1).
Comparison Cards
C1: Weekly hours of exercise, random samples of 40 adults in Town P and Town Q: means 3.2 and 3.5 hours, MADs 1.4 and 1.5 hours
C2: Mass of random samples of 25 adult house cats and 25 adult beagles: means 4.5 kg and 10.5 kg, MADs 0.8 kg and 1.2 kg
C3: Sandwich prices, random samples of 20 sandwiches at two cafes: medians $7.50 and $8.25, IQRs $2.00 and $2.50
C4: Hand span, random samples of 30 fourth graders and 30 adults: means 14 cm and 20 cm, MADs 1.1 cm and 1.4 cm
C5: Battery life, random samples of 15 phones of Model X and 15 of Model Y: medians 610 and 680 minutes, IQRs 40 and 50 minutes
C6: Words per sentence, random samples of 30 sentences from a science magazine and a sports magazine: means 16 and 15 words, MADs 5 and 6 words
C7: Car speeds, random samples of 50 cars on a 25 mph street and on a 55 mph highway: means 27 mph and 58 mph, MADs 3 mph and 4 mph
C8: Daily reading minutes, random samples of 20 fifth graders and 20 seventh graders: medians 25 and 18 minutes, IQRs 5 and 6 minutes
Answer Cards (answer key)
C1: 0.3 ÷ 1.5 = 0.2: cannot tell
C2: 6 ÷ 1.2 = 5: clear difference
C3: 0.75 ÷ 2.5 = 0.3: cannot tell
C4: 6 ÷ 1.4 is about 4.3: clear difference
C5: 70 ÷ 50 = 1.4: likely, check more samples
C6: 1 ÷ 6 is about 0.17: cannot tell
C7: 31 ÷ 4 = 7.75: clear difference
C8: 7 ÷ 6 is about 1.2: likely, check more samples
Discussion Questions
Three cards land in each of the "Clear difference" and "Cannot tell" piles, and two land in the middle pile. For one "Cannot tell" card, what kind of sample would help?
C1 and C6 both use large samples. Why can we still not tell?
Write a conclusion for C2 that uses "tend to". Why is "every beagle is heavier than every cat" not supported by the card?
Challenge Variation
Pairs change one number on a "Cannot tell" card so that it moves to "Clear difference", then explain which number they changed and why.
3
Many Samples from Two Gardens
15 minGroups of 3-4
Each group gets two paper bags. Each bag holds 24 slips, one for every bean plant in a school garden, with the plant's height in centimeters after 4 weeks (invented data). Groups draw random samples of 6 slips from each bag, find the sample medians, and repeat to see how much samples vary.
Shake the bag and draw 6 slips without looking. Record the heights and put the slips back.
Find the median of each sample of 6 and record both medians as a pair.
Repeat four times, so each group has 4 pairs of sample medians.
Add your medians to a class dot plot with one row for North and one for South.
At the end, the teacher reveals the population medians: North 28 cm, South 23 cm.
Discussion Questions
Did any group get a South sample median larger than its North sample median in the same round? How can that happen?
The population medians differ by 5 cm. Do the class dot plots of sample medians show that difference?
Would samples of 12 slips vary more or less than samples of 6? Try one round to find out.
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Word Lengths in Two Science Chapters
Dot plots of the official example: 20 randomly chosen words from each chapter, one dot per word. The dashed lines mark the medians, 4 letters for grade 4 and 5 letters for grade 7. The two samples overlap a lot, so the class took more random samples before deciding. Drawn to scale, 48 pixels per letter.
Diagram 2: Pages in Fiction and Nonfiction Library Books
Box plots of the page counts in Example 3, drawn to scale (0.9 pixel per page). Each box runs from Q1 to Q3, the thick line is the median and the whiskers reach the shortest and longest book. The long right whiskers show the 610-page and 400-page books, which pull the means up but barely move the medians.
04
Homework Assignment
~30 min
7.SP.B.4 Homework: Comparing Two Populations
Directions: Show your work. For each problem, name the two populations, choose a measure of center and a measure of variability, and give a reason for your choice. Express the difference between the centers as a number of MADs or IQRs (use the larger one), and end with a sentence about the populations. All data are invented.
Part 1: Summarize and Compare (Problems 1-3)
A weather student picks 10 July days at random from the last 20 years for City A and for City B and records the high temperature (°F). City A: 78, 80, 81, 83, 84, 85, 86, 88, 90, 95. City B: 88, 90, 92, 93, 94, 95, 96, 97, 99, 106. Find the mean and MAD of each sample, find how many MADs apart the means are, and write an inference about July highs in the two cities.
Prices in dollars of random samples of 9 used video games at two stores. Store A: 8, 10, 12, 12, 15, 18, 20, 22, 45. Store B: 15, 18, 20, 24, 25, 28, 30, 32, 60. Explain why the median and IQR fit these data better than the mean and MAD. Then find the median and IQR of each sample and compare the stores.
Words per sentence in random samples of 12 sentences from a picture book and a middle school novel. Picture book: 3, 4, 4, 5, 5, 6, 6, 7, 8, 9, 11, 14. Novel: 11, 12, 13, 14, 15, 16, 16, 17, 18, 19, 21, 26. Find each median and IQR. Are the novel's sentences generally longer? Use the classroom guide to say how sure you are.
Part 2: Drawing and Judging Inferences (Problems 4-6)
A class counts the unpopped kernels in random samples of 30 bags of two brands of microwave popcorn. Brand A: mean 44 kernels, MAD 5. Brand B: mean 32 kernels, MAD 6. How many MADs apart are the means? Write a conclusion that a shopper could use.
Maya takes three random samples of 10 songs from each of two playlists and finds the mean song length each time. Playlist A: 3.4, 3.7, 3.5 minutes. Playlist B: 3.6, 3.3, 3.8 minutes. Does one playlist have longer songs in general? Explain using all six sample means.
Random samples of 15 students from each of two middle schools report minutes of homework per night. Lincoln: mean 48 minutes, MAD 12. Jefferson: mean 44 minutes, MAD 14. Jamal says, "Every Lincoln student does more homework than every Jefferson student." Explain two things wrong with his claim, and write a better conclusion.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Measures
Center and variability correct for both samples, with a fitting choice and a reason
One computation error, or no reason for the choice
Measures missing or mostly incorrect
Comparison
Difference in centers found and expressed as a number of MADs or IQRs
Difference found but not compared with the variability
No comparison
Inference
Conclusion about the populations, uses "tend to" or "generally", and matches the size of the difference
Conclusion only about the samples, or too strong for the evidence
No conclusion or an unsupported one
Sampling
Explains why random samples, repeated samples or larger samples matter where asked
Mentions sampling without explaining it
Ignores how the data were collected
05
Quiz: 20 Questions
Interactive, with answers
Instructions
Work through the questions in order. Your score updates as you answer, and Reset quiz clears everything so you or your students can try again. All data are invented.
Multiple choice: pick an option to check it. Short answer: write your answer, then reveal the model answer.
0 of 20 answered · 0 correct
Question 1 of 20 · Multiple Choice
A park ranger weighs a random sample of 15 trout from Lake Pine and a random sample of 15 trout from Lake Elm. What are the two populations she wants to compare?
Answer: B
The populations are the whole groups the ranger wants to know about: all trout in each lake. Choice A names the samples, not the populations. Choice C is not what she measured, and choosing the heaviest fish would not be random. Choice D includes other kinds of fish, which she did not sample.
Question 2 of 20 · Multiple Choice
A student wants to compare hours of TV per week for sixth graders and eighth graders at her school. Which plan gives random samples?
Answer: C
Drawing names from complete lists gives every student in each grade the same chance of being chosen. Choice A picks students who arrive early, which is not random. Choice B lets students choose themselves, and volunteers may watch more or less TV than others. Choice D picks friends, who may be alike.
Question 3 of 20 · Multiple Choice
Points scored in one game by a random sample of 6 players from each of two youth basketball leagues. League A: 3, 5, 8, 8, 11, 13. League B: 6, 9, 10, 13, 13, 15. How much greater is the mean of League B's sample than the mean of League A's sample?
Answer: D
League A: 48 ÷ 6 = 8. League B: 66 ÷ 6 = 11. The difference is 3 points. Choice A subtracts the totals (66 - 48) instead of the means. Choice B subtracts the medians (11.5 - 8), not the means. Choice C compares only the largest values (15 - 13).
Question 4 of 20 · Multiple Choice
Minutes that a random sample of 5 patients waited at a clinic: 12, 14, 15, 17, 22. The mean is 16 minutes. What is the MAD?
Answer: A
The distances from the mean 16 are 4, 2, 1, 1 and 6. Their sum is 14, and 14 ÷ 5 = 2.8. Choice B adds the distances but forgets to divide by 5. Choice C is the range, 22 - 12. Choice D divides 14 by 4 instead of by the 5 values.
Question 5 of 20 · Multiple Choice
Random samples of 12 house sale prices in two towns each include one mansion that sold for far more than the other homes. Which measures should you use to compare typical prices in the two towns?
Answer: B
One very high price pulls the mean and the MAD up, while the median and IQR depend on the middle values. Choice A is the choice for roughly symmetric data. Choice C uses the range, which depends only on the two most extreme values. Choice D describes the mansions, not typical homes.
Question 6 of 20 · Multiple Choice
Resting heart rates for random samples of 20 swimmers and 20 people who do not exercise: swimmers mean 60 beats per minute, others mean 72 beats per minute. Both samples have a MAD of 4 beats per minute. How many MADs apart are the means?
Answer: C
The difference is 72 - 60 = 12 beats per minute, and 12 ÷ 4 = 3 MADs. Choice A gives the difference in beats per minute, not in MADs. Choice B gives the MAD itself. Choice D divides the wrong way: 4 ÷ 12.
Question 7 of 20 · Multiple Choice
Minutes on the bus for random samples of 30 seventh graders at School P and School Q: School P median 12 minutes, IQR 6 minutes; School Q median 26 minutes, IQR 8 minutes. Which conclusion is best supported?
Answer: D
The medians differ by 14 minutes, which is 14 ÷ 8 = 1.75 IQRs, so a difference is likely (another pair of random samples would make it surer). Of these choices, only the "tend to" statement fits that. Choice A claims every student, but the IQRs show that the samples vary. Choice B gives a cause the data do not show; the students may simply live farther away. Choice C ignores the difference of 14 minutes.
Question 8 of 20 · Multiple Choice
Test scores for random samples of 25 students in two large math courses: Course A mean 81 points, MAD 9; Course B mean 79 points, MAD 10. What can you conclude?
Answer: B
The difference is 2 points, and 2 ÷ 10 = 0.2 MADs, much less than 1. The scores overlap a lot, so these samples cannot tell. Choice A treats any difference in the means as clear. Choice C reverses the samples' direction. Choice D is not true even for the samples: their spreads overlap.
Question 9 of 20 · Multiple Choice
Ravi takes a random sample of 20 adults from a town and finds a mean of 5.2 hours of screen time per day. Tia takes a different random sample of 20 adults from the same town and finds 5.6 hours. What is the best explanation?
Answer: D
Different random samples contain different people, so their means vary a little even when nothing is wrong. Choice A assumes a mistake, but a difference of 0.4 hours is normal sample-to-sample variation. Choice B assumes a change with no reason. Choice C is false: both samples have 20 people.
Question 10 of 20 · Multiple Choice
Which pair of samples gives the most trustworthy comparison of the reaction times of all seventh graders and all adults in a town?
Answer: B
Samples should be random, and larger random samples vary less from sample to sample. Choice A is random but tiny, so its means could be far off. Choices C and D are large but not random: people who visit a booth or volunteer online may react faster or slower than others.
Question 11 of 20 · Multiple Choice
Box plot summaries (minimum, Q1, median, Q3, maximum) of lengths in centimeters of random samples of 15 leaves from two trees. Tree X: 10, 14, 18, 22, 30. Tree Y: 11, 21, 26, 29, 38. What are the difference between the medians and the IQR of each sample?
Answer: A
Medians: 26 - 18 = 8 cm. IQRs: 22 - 14 = 8 cm and 29 - 21 = 8 cm. So the medians are 1 IQR apart. Choice B subtracts Q3 values (29 - 22) instead of medians. Choice C gives the ranges (30 - 10 and 38 - 11), not the IQRs. Choice D subtracts the median from Q3 (22 - 18 and 29 - 26), which is only a quarter of each data set.
Question 12 of 20 · Multiple Choice
Random samples of 50 dogs at two shelters: Shelter 1 dogs have a mean age of 6.8 years, MAD 1.1 years; Shelter 2 dogs have a mean age of 2.9 years, MAD 1.3 years. Which statement is the best informal inference?
Answer: D
The means differ by 3.9 years, which is 3.9 ÷ 1.3 = 3 MADs, so the populations very likely differ, and "generally older" says so without claiming every dog. Choice A claims no overlap at all, which the MADs do not show. Choice B invents a cause. Choice C is backward: a large difference is what makes a comparison useful.
Question 13 of 20 · Multiple Choice
Two machines fill cereal boxes. Random samples of 40 boxes: Machine A mean 510 g, MAD 3 g; Machine B mean 512 g, MAD 11 g. What is the best inference about the variability of the fills?
Answer: C
The MAD measures variability: 11 g is almost 4 times 3 g, so Machine B's fills tend to be farther from their mean, and its boxes are less consistent. Choice A uses the mean, which measures center, not variability. Choice B ignores the large gap between the MADs. Choice D reads the MAD backward: a larger MAD means more variability.
Question 14 of 20 · Multiple Choice
A class takes 5 pairs of random samples of 10 students from two schools and finds the median minutes of daily exercise in each sample. School A medians: 31, 34, 30, 33, 32. School B medians: 38, 36, 39, 37, 40. What is the best conclusion?
Answer: A
The sample medians vary, but every School B median (36 to 40) is higher than every School A median (30 to 34), so the difference keeps showing up. Choice B treats normal sample variation as a reason to give up. Choice C talks about every student, but the samples only show typical values. Choice D gives an exact amount, but samples only estimate it.
Question 15 of 20 · Short Answer
Seconds to solve the same puzzle for a random sample of 5 sixth graders (50, 52, 55, 58, 60) and 5 eighth graders (40, 44, 45, 47, 49). Find the mean and MAD of each sample, how many MADs apart the means are, and write a conclusion.
Sixth graders: mean 55 s, MAD 3.2 s (distances 5, 3, 0, 3, 5). Eighth graders: mean 45 s, MAD 2.4 s (distances 5, 1, 0, 2, 4). The difference is 10 s, and 10 ÷ 3.2 is about 3.1 MADs. Eighth graders tend to solve this puzzle faster. The samples are small, so a larger sample would make the conclusion stronger.
Question 16 of 20 · Short Answer
Minutes that random samples of 9 customers waited for a table at two restaurants. Restaurant 1: 5, 6, 8, 9, 10, 12, 13, 15, 40. Restaurant 2: 12, 14, 15, 18, 20, 21, 23, 25, 27. Which measures should you use, and what do they show?
Restaurant 1 has one wait of 40 minutes far from the rest, so use the median and IQR. Restaurant 1: median 10, Q1 7, Q3 14, IQR 7. Restaurant 2: median 20, Q1 14.5, Q3 24, IQR 9.5. The difference, 10 minutes, is about 1.05 of the larger IQR, so customers at Restaurant 2 likely wait longer in general, but more samples would help.
Question 17 of 20 · Short Answer
To compare the heights of all seventh graders and all eighth graders at a school, Leo measures the players on the seventh-grade and eighth-grade volleyball teams. Explain why his comparison may not be valid and how to fix it.
The volleyball players are not a random sample: volleyball players may be taller than other students, so the samples may not represent the two grades. Leo should draw names at random from a list of all seventh graders and a list of all eighth graders, then compare the centers and spreads.
Question 18 of 20 · Short Answer
Random samples of 40 pumpkins from two farms: Farm 1 mean 6.2 kg, MAD 0.8 kg; Farm 2 mean 4.1 kg, MAD 0.7 kg. How many MADs apart are the means? Write an inference.
The difference is 6.2 - 4.1 = 2.1 kg, and 2.1 ÷ 0.8 is about 2.6 MADs (using the larger MAD). That is more than 2, so pumpkins from Farm 1 tend to be heavier than pumpkins from Farm 2.
Question 19 of 20 · Short Answer
Describe how you would use random samples to decide whether cars on Main Street generally go faster than cars on Oak Street. Say what you would measure, how you would choose the cars, and what result would convince you.
Sample answer: measure the speeds of cars on each street (for example with a speed sign or a radar gun, with an adult). Choose the cars at random, for example the cars passing at randomly chosen times on different days, and use the same number for each street, such as 40. Find the mean and MAD (or the median and IQR if a few cars are very fast). A difference in centers of 2 or more MADs, or one that shows up again in new samples, would convince me.
Question 20 of 20 · Short Answer
Random samples of 12 daily arrival delays in minutes for two bus routes: Route 5 median 4, IQR 2; Route 9 median 4, IQR 9. Compare the two routes. Which one would you choose if you must not be late?
Both routes have the same typical delay, a median of 4 minutes, so the centers do not differ. The variability does: Route 9's IQR of 9 minutes is much larger than Route 5's IQR of 2 minutes, so Route 9's delays tend to vary more. Choose Route 5, because its delays are more predictable.
0 of 20 answered · 0 correct
06
Frequently Asked Questions
10 Questions
What does 7.SP.B.4 mean?
7.SP.B.4 means students use random samples to compare two populations. For each sample they find a center (mean or median) and a spread (MAD or IQR). Then they decide whether one population tends to have larger values, as in the official example about word lengths in a grade 7 and a grade 4 science book.
What grade is 7.SP.B.4, and what comes after it?
It is a grade 7 standard in the Statistics and Probability domain. It builds on grade 6 work with center and spread (6.SP.B.5). In high school, students compare data sets with the standard deviation (another measure of spread) (HSS.ID.A.2) and use randomized experiments and simulations to judge whether a difference is real (HSS.IC.B.5).
What is an informal comparative inference?
It is a careful conclusion about how two populations compare, made from samples without formal statistical tests. "Informal" means students reason from centers, spreads, plots and repeated samples. A good inference sounds like "The words in the grade 7 chapter are generally longer, by about 1 letter," not "every word is longer."
Why do the samples have to be random?
Random samples tend to represent their populations, so what is true of the samples is likely true of the populations. A sample of friends, volunteers or the first people in line can be very different from the whole group. If the samples are not random, no amount of computing makes the comparison trustworthy (7.SP.A.1).
Should students use the mean and MAD or the median and IQR?
Use the mean and MAD for roughly symmetric data, and the median and IQR for skewed data or data with values far from the rest. For example, one very long book pulls the mean page count up, while the median barely moves. Always use the same pair of measures for both samples.
How big does the difference between two samples have to be?
There is no official cutoff, but a useful classroom guide compares the difference in centers with the larger MAD or IQR. Two or more means a clear difference, 1 to 2 means a likely difference to check with more samples, and less than 1 means the samples cannot tell. Repeated random samples that all point the same way also count as strong evidence.
Why do two random samples from the same population give different results?
Each random sample contains different members of the population, so its mean or median changes a little from sample to sample. This is normal and is called sampling variability. Larger samples vary less, which is why they give more trustworthy comparisons.
Is 7.SP.B.4 the same as 7.SP.B.3?
No, but they work together. 7.SP.B.3 asks students to judge how much two data distributions overlap and to express the difference in centers as a multiple of a measure of variability. 7.SP.B.4 uses those tools on random samples to draw conclusions about the populations the samples came from.
How can parents help with 7.SP.B.4 at home?
Ask comparison questions about everyday things and talk about how to answer them fairly. For example: "Are the songs on your playlist longer than the songs on mine?" Pick 10 songs from each list at random, find the median length of each, and talk about whether the difference is big compared with how much the lengths vary.
What mistakes should teachers watch for?
A common mistake is treating any difference between two sample means as proof that the populations differ, even when the data overlap a lot. Students also write conclusions about every member ("all eighth graders") instead of typical values, mix a mean with an IQR, or forget to ask whether the samples were random.
07
Related Standards
6 standards
These standards connect to 7.SP.B.4: prerequisites to review first, parallel standards at the same level, and next steps that build on it.
Before this lesson
6.SP.B.5Prerequisite
Summarize numerical data sets with center, variability and shape in context
