7.SP.B.3Common CoreMathStatistics and ProbabilityGrade 7
7.SP.B.3: Comparing Two Distributions by Overlap and MAD
In plain English: 7.SP.B.3 is the Common Core grade 7 math standard that asks students to compare two numerical data sets with about the same spread. Students judge by eye how much the two dot plots overlap, then divide the difference between the means by the mean absolute deviation (MAD) to say how many MADs apart the centers are. The more MADs apart, the less the groups overlap.
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.
Common Core State Standards for Mathematics · Domain: Statistics and Probability (SP) · Cluster: Draw informal comparative inferences about two populations. Also written as 7.SP.3 · Official standard
Students compare two groups of numbers, such as the heights of two sports teams, and decide how different the groups really are. A distribution is the way the values of a data set are spread along the number line, as a dot plot shows. When the dot plots of two groups are drawn on the same scale, they may overlap (share much of the same stretch of the number line) or be separated (sit mostly apart). Judging this by eye is the first half of the standard.
The second half puts a number on it. Students find each group's mean (the sum of the values divided by how many there are) and its mean absolute deviation, or MAD (the average distance of the values from their mean), both from grade 6. They then divide the difference between the two means by the MAD. The answer is a multiple of the MAD: it says how many MADs apart the two centers are. The standard asks for this only when the two groups have similar variability (MADs that are about the same), because then one MAD describes the spread of both. As a rough guide (a rule of thumb, not an official rule, and the same one used in 7.SP.B.4): centers less than 1 MAD apart mean a lot of overlap; between 1 and 2 MADs, one group sits somewhat to one side but the plots still overlap a good deal; 2 or more MADs apart mean a clear, noticeable separation with only some overlap, as in the official example; and at 3 or more MADs there is little or no overlap. 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:
Describe how much two dot plots on the same scale overlap, using words such as "a lot", "some" and "none"
Check that two data sets have similar variability by comparing their mean absolute deviations
Find the difference between two means and express it as a multiple of the MAD
Connect the size of that multiple to the amount of visual overlap, and explain what it means in context
Prior Knowledge Required
Students should already be comfortable with:
Making dot plots and box plots on a number line 6.SP.B.4
Finding the mean, the median, the interquartile range and the mean absolute deviation of a data set 6.SP.B.5
Knowing that a measure of center describes a typical value and a measure of variability describes spread 6.SP.A.3
Sketch two quick dot plots on the board: the ages, in years, of 6 people in two families at a reunion (invented data). Family 1: 8, 10, 11, 12, 13, 14. Family 2: 9, 10, 12, 12, 13, 16. Family 1 has a mean age of about 11.3 years and Family 2 a mean age of 12 years, and each family spreads over 6 or 7 years.
Warm-Up Prompt
"A friend says, 'The Family 2 kids are older.' Their mean age is less than 1 year higher. Would you agree with your friend? What would the dot plots have to look like for you to agree?"
Give pairs two minutes. Students usually notice that the two plots cover the same stretch of the number line, so a 1-year difference in the means is small compared with how spread out the ages are. Collect ideas about "what would convince you": a bigger difference, or plots that sit apart. Tell the class that today they learn to measure a difference in means against the spread, using the MAD.
Direct Instruction20-25 minutes
Part 1: Look first. Show Diagram 1 with the numbers covered. Ask: "Which team is taller in general? Is every basketball player taller than every soccer player?" Students should see that the basketball dots sit to the right, but some soccer players are as tall as some basketball players. That shared stretch is the overlap. Always draw the two dot plots on the same scale, one above the other, or the comparison is not fair.
Part 2: Check the spreads. Recall how to find a MAD: find the mean, find each value's distance from it, add the distances and divide by the number of values. Find both MADs for Example 1. They are both 5 cm, so the two groups have similar variability and the method fits. If one MAD were 3 and the other 11, one number could not describe both spreads, and we would not use this method.
Part 3: Measure the difference in MADs. Subtract the means, then divide by the MAD: 10 ÷ 5 = 2, so the centers are 2 MADs apart. Say it in context: "The typical basketball player is 2 MADs taller than the typical soccer player." Work Examples 2 and 3, then show Diagram 2 so students connect the multiple to the picture: 0.5 MAD looks like one group, and 5 MADs looks like two separate groups. Example 4 shows the same idea with the median (the middle value) and the IQR (interquartile range, the width of the middle half of the data), which students may meet on box plots.
Heights of two teams (official example)
The heights of the 10 players on a middle school's grade 8 basketball team and grade 8 soccer team, in centimeters (invented data). Basketball: 162, 162, 164, 167, 170, 171, 172, 175, 178, 179. Soccer: 154, 154, 155, 156, 156, 162, 162, 162, 168, 171.
Equation: Basketball mean = 1,700 ÷ 10 = 170 cm; soccer mean = 1,600 ÷ 10 = 160 cm. Both MADs are 50 ÷ 10 = 5 cm, so the variabilities are the same. Difference of means = 10 cm = 10 ÷ 5 = 2 MADs. On the dot plot (Diagram 1), the separation is noticeable, but the teams still overlap from 162 to 171 cm
Centers less than 1 MAD apart: a lot of overlap
Two gym classes each ran 60 meters, and the teacher timed 10 students from each class to the nearest tenth of a second (invented data). Class A: 8.9, 9.5, 9.7, 9.9, 10.1, 10.1, 10.3, 11.0, 11.1, 11.4. Class B: 9.1, 9.1, 9.3, 9.4, 9.6, 10.0, 10.4, 10.4, 10.7, 11.0.
Equation: Class A mean = 102 ÷ 10 = 10.2 s; Class B mean = 99 ÷ 10 = 9.9 s. Both MADs are 0.6 s. Difference = 0.3 s = 0.3 ÷ 0.6 = 0.5 MAD. The dot plots overlap almost completely (Diagram 2), so the classes run about equally fast
Centers 5 MADs apart: no overlap
A farm weighed 10 chicken eggs and 10 duck eggs on a kitchen scale, in grams (invented data). Chicken: 52, 56, 56, 56, 57, 57, 58, 61, 61, 66. Duck: 67, 67, 70, 71, 71, 73, 73, 74, 76, 78.
Equation: Chicken mean = 580 ÷ 10 = 58 g; duck mean = 720 ÷ 10 = 72 g. Both MADs are 28 ÷ 10 = 2.8 g. Difference = 14 g = 14 ÷ 2.8 = 5 MADs. The heaviest chicken egg (66 g) is lighter than the lightest duck egg (67 g), so the dot plots do not overlap
Using the median and the IQR instead
Box plots of the number of jumping jacks students did in 1 minute (invented data). Grade 6: first quartile (the median of the lower half) 36, median 42, third quartile (the median of the upper half) 48. Grade 8: first quartile 48, median 54, third quartile 60.
Equation: Each interquartile range (IQR, the width of the middle half of the data) is 48 - 36 = 60 - 48 = 12. Difference of medians = 54 - 42 = 12 jumping jacks = 12 ÷ 12 = 1 IQR. The two boxes touch: the middle half of grade 6 ends where the middle half of grade 8 begins, so the groups overlap a fair amount
Guided Practice15 minutes
A school garden club grew carrots in two beds, one with compost mixed into the soil and one without. They weighed 8 carrots from each bed on a kitchen scale (invented data). Work through the steps with the class.
Masses of 8 carrots from each garden bed, in grams (invented data)
Bed
1
2
3
4
5
6
7
8
With compost
81
87
88
98
100
102
103
109
Without compost
68
71
74
76
79
87
92
93
Steps and model answers: (1) Make both dot plots on one scale from 60 to 115 grams. Describe the overlap. (The plots share the stretch from 81 to 93 grams, but most compost carrots are heavier.) (2) Find the means. (768 ÷ 8 = 96 g with compost; 640 ÷ 8 = 80 g without.) (3) Find the MADs. (The distances from the mean add to 64 in both beds, so both MADs are 64 ÷ 8 = 8 g: similar variability.) (4) Express the difference in MADs. (96 - 80 = 16 g = 16 ÷ 8 = 2 MADs.) (5) Write a sentence in context. (The typical compost carrot is 2 MADs heavier, a noticeable difference, although some carrots from the two beds weigh about the same.) Watch for students who divide by the sum of the two MADs, or who forget to subtract the means first.
Independent Practice10-15 minutes
Each student works alone. Eight students typed the same paragraph once on a keyboard and once on a tablet screen, and the class recorded words per minute (invented data). Keyboard: 33, 33, 35, 35, 40, 41, 42, 45. Tablet: 23, 29, 30, 31, 31, 32, 38, 42. (1) Find both means. (304 ÷ 8 = 38 and 256 ÷ 8 = 32 words per minute.) (2) Find both MADs. (Both distance sums are 32, so both MADs are 4 words per minute.) (3) How many MADs apart are the means? (6 ÷ 4 = 1.5 MADs.) (4) Sketch both dot plots on one scale and describe the overlap in one sentence. (The plots overlap a lot from 33 to 42, but the keyboard dots sit a little to the right.)
Closure5 minutes
Exit ticket: in a jump-rope unit, grade 6 students averaged 96 jumps in 1 minute and grade 7 students averaged 110 jumps, and both groups had a MAD of 7 jumps (invented data). How many MADs apart are the means, and what would the two dot plots look like? (14 ÷ 7 = 2 MADs; the grade 7 plot sits noticeably to the right, but the two plots still share some values.) Then ask: why did we check that both MADs are 7 before dividing? (So that one MAD describes the spread of both groups.)
Differentiation Strategies
For Struggling Students
Give a five-row table for each comparison: mean A, mean B, difference, MAD, and a last row "difference ÷ MAD"
Use data sets with whole-number means and MADs first, such as the carrots, before decimals such as the sprint times
Give pre-drawn number lines at the same scale so students can focus on the overlap instead of the scale
For Advanced Students
Ask students to invent two data sets of 6 values with the same MAD whose means are exactly 3 MADs apart, then test a classmate's sets
Compare the same two groups with medians and IQRs and with means and MADs, and explain why the two multiples can differ
Ask what goes wrong when the MADs are very different (for example 1 and 7), and have students draw two dot plots that show the problem
Assessment Guidance
What to Look For
Check that both dot plots are on one scale before students describe the overlap. In calculations, look for the mean difference divided by one MAD (not by the sum of the MADs, and not the MAD divided by the difference), and for a check that the two MADs are close before dividing. In the written conclusion, students should give the multiple ("2 MADs"), connect it to the picture ("noticeable separation, some overlap") and name the context ("the typical compost carrot is heavier"). Watch for "every" statements: a separation of 2 MADs does not mean every value in one group is larger.
02
Classroom Activities
3 Activities
1
Paper Airplane Test Flights
25 minGroups of 4
Groups fold two airplane designs, fly each one 10 times, and decide how different the two designs really are by comparing overlap and the difference of means in MADs.
Procedure
Each group folds one dart (narrow, pointed wings) and one glider (wide wings) from printer paper
Mark a throwing line with masking tape. The same student throws both planes, alternating, 10 times each
Measure each flight from the line to where the nose first touches the floor, to the nearest 10 cm
Make both dot plots on one scale, find each mean and MAD, and express the difference of means in MADs
Sample Group Data (invented, for teacher planning)
Distances in centimeters. Dart: 420, 500, 580, 580, 620, 650, 650, 670, 730, 800. Glider: 300, 350, 380, 410, 460, 490, 490, 550, 580, 590. For these data the means are 620 cm and 460 cm, both MADs are 80 cm, and the difference is 160 cm, or 2 MADs.
Discussion Questions
In the sample data, which design flew farther in general, and by how many MADs? (The dart, by 2 MADs)
Did any glider flight go farther than any dart flight? (Yes: the longest glider flight, 590 cm, beat the 4 shortest dart flights)
Why should the same student throw both planes?
Modification for Distance Learning
Students fly the planes at home in a hallway, measure with a tape measure or by counting floor tiles of known size, and post their 20 distances in a shared class spreadsheet.
2
Overlap Line-Up Card Sort
15 minPairs
Pairs compute how many MADs apart the means are on 6 cards, then line the cards up from the most overlap to the least and sketch a pair of dot plots for two of them.
Cards (6 cards, invented data)
Card 1, minutes of piano practice per day for two students: means 33 and 30, MAD 10 for both
Card 2, heights of two kinds of tomato plants: means 90 cm and 102 cm, MAD 6 cm for both
Card 3, masses of adult shorthair house cats and Maine coon cats: means 4.2 kg and 7.0 kg, MAD 0.7 kg for both
Card 4, scores on two levels of a video game: means 1,200 and 1,500 points, MAD 200 points for both
Card 5, minutes to finish a corn maze with a map and without one: means 21 and 24, MAD 3 for both
Card 6, foot lengths of grade 3 and grade 7 students: means 19 cm and 24 cm, MAD 1 cm for both
Procedure
For each card, subtract the means and divide by the MAD. Write the multiple on a sticky note on the card
Line up the cards from the most overlap to the least
Pick one card from each end of the line and sketch its two dot plots on one scale
Discussion Questions
Which card has the most overlap, and which has the least? (Card 1, 0.3 MAD; Card 6, 5 MADs)
Card 4 has a difference of 300 points and Card 6 a difference of only 5 cm. Why does Card 6 show less overlap?
On which cards would you say one group is "generally larger"? Where would you draw the line, and why?
Challenge Variation
Pairs write a seventh card whose multiple falls between Card 2 and Card 3, with a realistic context and realistic numbers.
3
Slide the Dots
15 minPairs
Pairs build a data set with sticky dots on a grid-paper number line, copy it, and slide the copy to the right by 1, 3 and 6 MADs to see how the overlap changes.
Materials and Data
Grid paper with two number lines from 0 to 26, one above the other, 1 square per unit
16 round sticky dots in two colors
Set A: 4, 5, 7, 8, 8, 9, 11, 12 (mean 8, MAD 2)
Procedure
Place Set A on the top line. Check the mean and MAD
Make Set B on the bottom line by adding 2 to every value of Set A. Find its mean and MAD, and describe the overlap
Move Set B so that each value is 6 more than in Set A, then 12 more, describing the overlap each time
Discussion Questions
Why does the MAD of Set B stay 2 every time you slide it?
How many MADs apart are the means after each slide? (1, 3 and 6 MADs)
After which slide do the two sets stop overlapping? (After the slide of 12: Set B then starts at 16, above the largest value of Set A, 12)
03
Diagrams & Visual Aids
2 diagrams
Diagram 1: Two Teams, 2 MADs Apart
The heights from Example 1 on the same scale. The dashed lines mark the means, 160 cm and 170 cm, and each blue bar runs 1 MAD (5 cm) on each side of a mean. The means are 10 cm, or 2 MADs, apart: the separation is noticeable, but players from 162 to 171 cm appear on both teams.
Diagram 2: A Small and a Large Multiple of the MAD
Top: the sprint times from Example 2, whose means are only 0.5 MAD apart, cover nearly the same stretch of the number line. Bottom: the egg masses from Example 3, whose means are 5 MADs apart, form two separate groups. In both pairs the two MADs are equal, so one MAD describes the spread of each group.
04
Homework Assignment
~30 min
7.SP.B.3 Homework: Overlap and Differences in MADs
Directions: Show your work. Draw every pair of dot plots on one scale. Check that the two MADs are close before you divide, and write your conclusion in a sentence about the context. All data are invented.
Part 1: Means, MADs and Overlap (Problems 1-3)
A class grew 10 sunflowers in full sun and 10 in part shade and measured their heights in centimeters. Full sun: 168, 171, 173, 174, 177, 177, 183, 190, 191, 196. Part shade: 149, 149, 155, 163, 167, 167, 167, 169, 169, 185. (a) Find the mean and the MAD of each group. (b) Draw both dot plots on one scale and describe the overlap. (c) How many MADs apart are the means?
A student dropped 8 new tennis balls and 8 old tennis balls from a height of 100 cm and measured how high each bounced, in centimeters. New: 49, 52, 52, 56, 57, 57, 58, 59. Old: 47, 47, 51, 51, 52, 55, 56, 57. (a) Find the mean and the MAD of each group. (b) Express the difference of means as a multiple of the MAD. (c) Would you say the new balls bounce higher in general? Explain with the overlap.
A farmer measured 50 corn plants in each of two fields. Field A: mean height 214 cm, MAD 6 cm. Field B: mean height 196 cm, MAD 6 cm. (a) How many MADs apart are the means? (b) Sketch what the two dot plots could look like, and describe the overlap.
Part 2: Interpreting the Multiple (Problems 4-6)
Hand spans (thumb tip to little-finger tip, fingers spread) of 10 grade 4 students and 10 grade 7 students, measured to the nearest millimeter, in centimeters. Grade 4: 12.5, 13.2, 14.1, 14.5, 14.7, 15.4, 16.2, 16.3, 16.4, 16.7. Grade 7: 15.9, 16.8, 16.9, 17.1, 17.3, 18.0, 18.6, 19.6, 19.8, 20.0. Both MADs are 1.2 cm. (a) Find both means. (b) How many MADs apart are the means? (c) How many grade 7 hand spans are shorter than the longest grade 4 hand span? Use this to describe the overlap.
Two basketball players each played 10 games. Points scored by Player A: 8, 9, 11, 11, 11, 14, 18, 18, 20, 20. Player B: 6, 10, 11, 13, 16, 16, 16, 18, 20, 24. (a) Find each mean and MAD. (b) How many MADs apart are the means? (c) A fan says Player B is a better scorer. Use the overlap and your answer to (b) to respond.
Make up two data sets of 6 values each that both have a MAD of 2. (a) Choose them so the means are 1 MAD apart, and draw both dot plots on one scale. (b) Change one set so that the means are 3 MADs apart, and draw the new pair. (c) Describe how the overlap changed.
Rubric
Criterion
Full Credit (2 pts)
Partial Credit (1 pt)
No Credit (0 pts)
Dot Plots
Both plots on one scale, correctly placed
Plots correct but on different scales
Plots missing or wrong
Means and MADs
All means and MADs correct, with units
One calculation error
Several errors or no work
Multiple of the MAD
Difference of means divided by the MAD, with the similar-variability check
Correct multiple but no check of the MADs
Divides the wrong numbers or no multiple
Conclusion
Connects the multiple to the overlap and the context
Describes the overlap without the context
No conclusion
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 in the quiz 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
Batteries of Brand X lasted a mean of 9.6 hours in a toy car, and batteries of Brand Y lasted a mean of 8.2 hours. Both brands have a MAD of 0.4 hours. How many MADs apart are the means?
Answer: B
The difference is 9.6 - 8.2 = 1.4 hours, and 1.4 ÷ 0.4 = 3.5 MADs. Choice A is the difference in hours, not in MADs. Choice C divides the MAD by the difference, 0.4 ÷ 1.4. Choice D divides by the sum of the two MADs, 1.4 ÷ 0.8.
Question 2 of 20 · Multiple Choice
Each pair of groups has similar variability. Which pair would show the most overlap on a dot plot?
Answer: A
Divide each difference by its MAD: A is 6 ÷ 8 = 0.75, B is 4 ÷ 2 = 2, C is 12 ÷ 3 = 4 and D is 9 ÷ 6 = 1.5. The smallest multiple, 0.75 MAD, means the most overlap. Choice B has the smallest difference, but its MAD is small too, so its means are 2 MADs apart. Choice C comes from dividing the MAD by the difference (3 ÷ 12 = 0.25), which flips the comparison: C actually has the least overlap.
Question 3 of 20 · Multiple Choice
Five kiwis from one crate weigh 70, 76, 78, 80 and 86 grams. What is the MAD of their masses?
Answer: C
The mean is 390 ÷ 5 = 78 g. The distances from 78 are 8, 2, 0, 2 and 8, which add to 20, and 20 ÷ 5 = 4 g. Choice A is the range, 86 - 70. Choice B is the mean. Choice D divides 20 by 4 instead of 5.
Question 4 of 20 · Multiple Choice
Two data sets with similar MADs have means that are only a quarter of a MAD apart. What do their dot plots on the same scale look like?
Answer: D
A quarter of a MAD is tiny compared with the spread of each group, so the two plots overlap almost completely. Choice A describes centers many MADs apart. Choices B and C describe separations of a few MADs, not a quarter of one.
Question 5 of 20 · Multiple Choice
A swim team times its swimmers in the 50 m freestyle. Team A has a mean of 41 seconds and Team B a mean of 35 seconds, and both teams have a MAD of 2 seconds. How many MADs apart are the means?
Answer: A
The difference is 41 - 35 = 6 seconds, and 6 ÷ 2 = 3 MADs. Choice B is the difference in seconds. Choice C divides by the sum of the two MADs, 6 ÷ 4. Choice D divides the MAD by the difference, 2 ÷ 6.
Question 6 of 20 · Multiple Choice
Two groups have similar MADs, and their means are 2 MADs apart. Which statement about their dot plots is true?
Answer: B
At about 2 MADs, one group sits clearly to one side, but some values still share the same stretch, as in the team heights of Diagram 1. Choice A fits a multiple well under 1. Choice C fits a much larger multiple, such as the 5 MADs of the eggs. Choice D is an "every" claim that 2 MADs does not support.
Question 7 of 20 · Multiple Choice
The method of this standard needs two groups with similar variability. Which pair of MADs is the best fit for it?
Answer: C
MADs of 3.0 and 3.2 are nearly equal, so either one describes the spread of both groups. In choices A, B and D, one MAD is three or more times the other (9.0 ÷ 2.4 = 3.75), so dividing by either MAD would describe only one of the groups.
Question 8 of 20 · Multiple Choice
Tomatoes per plant for 6 plants in Garden A: 8, 9, 10, 12, 13, 14. For 6 plants in Garden B: 11, 13, 15, 16, 17, 18. Both MADs are 2 tomatoes. How many MADs apart are the means?
Answer: D
Garden A has mean 66 ÷ 6 = 11 and Garden B has mean 90 ÷ 6 = 15. The difference, 4 tomatoes, is 4 ÷ 2 = 2 MADs. Choice A is the difference in tomatoes, not divided by the MAD. Choice B divides by the sum of the MADs, 4 ÷ 4. Choice C divides the MAD by the difference, 2 ÷ 4.
Question 9 of 20 · Multiple Choice
The mean of Group B is 3 MADs greater than the mean of Group A, and the two MADs are equal. Which conclusion is best supported?
Answer: A
Three MADs is a large separation, so Group B is generally larger and the dot plots overlap only a little. Choice B goes too far: a few values may still overlap. Choice C fits a multiple below 1. Choice D reverses the groups.
Question 10 of 20 · Multiple Choice
The means of two groups are 2.5 MADs apart, and the MAD of each group is 3 centimeters. How far apart are the means, in centimeters?
Answer: B
The difference is 2.5 MADs, and each MAD is 3 cm, so the difference is 2.5 × 3 = 7.5 cm. Choice A divides 3 by 2.5. Choice C adds 2.5 and 3. Choice D gives the number of MADs, not centimeters.
Question 11 of 20 · Multiple Choice
Box plots of minutes spent on a science project show one class with a median of 30 and an IQR of 4, and another class with a median of 36 and an IQR of 4. How many IQRs apart are the medians?
Answer: C
The difference is 36 - 30 = 6 minutes, and 6 ÷ 4 = 1.5 IQRs. Choice A is the difference in minutes. Choice B divides the IQR by the difference, 4 ÷ 6. Choice D divides one median by the IQR, 36 ÷ 4, instead of the difference.
Question 12 of 20 · Multiple Choice
Why does this standard compare the difference of means with the MAD only when the two groups have similar variability?
Answer: D
When the MADs are about the same, one MAD is a fair ruler for both groups, so "2 MADs apart" means the same thing for each. Choice A is false: any data set has a MAD. Choice B is false: any two data sets can be drawn on one scale. Choice C is a common mistake: you divide by one MAD, not by the sum of the two MADs.
Question 13 of 20 · Multiple Choice
Students grew two kinds of bean plants. The mean heights are 24 cm and 30 cm, the MADs are equal, and the means are 3 MADs apart. What is the MAD?
Answer: A
The difference is 30 - 24 = 6 cm, and 6 cm is 3 MADs, so one MAD is 6 ÷ 3 = 2 cm. Choice B multiplies 6 by 3 instead of dividing. Choice C is the difference itself. Choice D is the number of MADs.
Question 14 of 20 · Multiple Choice
Two dot plots on the same scale show a clear gap: no values of the two groups overlap. The MADs are similar. Which multiple most likely describes how far apart the means are?
Answer: D
A clear gap means the centers are far apart compared with the spread, so the multiple is large, like 8 MADs (the eggs in Diagram 2 show no overlap at 5 MADs). Choices A, B and C are small multiples that go with a lot of overlap.
Question 15 of 20 · Short Answer
The lengths of 8 peanut shells from Farm A are 31, 31, 32, 34, 34, 35, 37, 38 mm, and from Farm B are 26, 27, 29, 30, 31, 32, 32, 33 mm. Find each mean and MAD, and express the difference of means as a multiple of the MAD. Describe the overlap.
Farm A: mean 272 ÷ 8 = 34 mm; distances 3, 3, 2, 0, 0, 1, 3, 4 add to 16, so MAD = 2 mm. Farm B: mean 240 ÷ 8 = 30 mm; distances 4, 3, 1, 0, 1, 2, 2, 3 add to 16, so MAD = 2 mm. The difference is 4 mm = 4 ÷ 2 = 2 MADs. The Farm A shells are noticeably longer in general, but the plots overlap from 31 to 33 mm.
Question 16 of 20 · Short Answer
The mean battery life of two tablet brands differs by 0.4 MAD, and the MADs are about the same. A store ad says Brand P "lasts longer". Is that a fair claim? Explain using overlap.
Not really. A difference of 0.4 MAD is small compared with the spread of each brand, so the two dot plots would overlap almost completely. Many Brand P tablets would last less time than many tablets of the other brand. The ad's claim is technically true for the means, but the difference would be hard to notice.
Question 17 of 20 · Short Answer
On a middle school's girls' volleyball team, the mean height is 8 cm greater than on the girls' soccer team, and the MAD is about 4 cm on either team. Express the difference as a multiple of the MAD, and describe what the two dot plots would look like.
The difference is 8 ÷ 4 = 2 MADs. On one scale, the volleyball dots would sit noticeably to the right of the soccer dots, but some players on the two teams would have the same heights, so the plots still overlap a little in the middle.
Question 18 of 20 · Short Answer
Group 1 has a mean of 50 and a MAD of 2. Group 2 has a mean of 56 and a MAD of 10. A student says, "The means are 3 MADs apart, so the groups barely overlap." Explain what is wrong.
The student divided by Group 1's MAD only: 6 ÷ 2 = 3. But the groups do not have similar variability: Group 2 is spread out five times as much, so it probably overlaps Group 1 a lot. Using Group 2's MAD gives 6 ÷ 10 = 0.6. The method of expressing the difference in MADs only works when the two MADs are about the same.
Question 19 of 20 · Short Answer
A class weighed pumpkins from two farms. Farm A pumpkins had a mean of 5.4 kg and Farm B pumpkins a mean of 6.6 kg, and both groups had a MAD of 0.8 kg. How many MADs apart are the means? Describe the overlap you would expect.
The difference is 6.6 - 5.4 = 1.2 kg, and 1.2 ÷ 0.8 = 1.5 MADs. The Farm B dot plot would sit somewhat to the right, but the two plots would overlap a good deal: many Farm A pumpkins weigh as much as Farm B pumpkins.
Question 20 of 20 · Short Answer
Make two data sets of 4 values each, both with a MAD of 2, whose means are 2 MADs apart. Show that your sets work.
Sample answer: Set A is 2, 2, 6, 6 and Set B is 6, 6, 10, 10. Set A has mean 16 ÷ 4 = 4, and every value is 2 from 4, so its MAD is 2. Set B has mean 32 ÷ 4 = 8 and MAD 2 the same way. The difference of means is 8 - 4 = 4 = 2 MADs. Any pair with both MADs equal to 2 and means 4 apart earns full credit.
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06
Frequently Asked Questions
10 Questions
What does 7.SP.B.3 mean?
7.SP.B.3 means students compare two data sets that have about the same spread by looking at how much their dot plots overlap and by saying how many MADs apart their means are. For example, if two teams' mean heights differ by 10 cm and the MAD is 5 cm, the means are 2 MADs apart and the separation is noticeable.
Is 7.SP.B.3 taught before or after 7.SP.B.4?
It is usually taught just before 7.SP.B.4. In 7.SP.B.3 students compare two data sets they are given. In 7.SP.B.4 they compare two populations using random samples, and use the same measures of center and variability to decide whether a difference is real.
How do you express a difference of means as a multiple of the MAD?
Subtract the smaller mean from the larger one, then divide by the MAD. If the means are 58 g and 72 g and the MAD is 2.8 g, the difference is 14 g and 14 ÷ 2.8 = 5, so the means are 5 MADs apart.
What if the two MADs are not exactly equal?
If the MADs are close, such as 4.1 and 4.4, divide by either one or by a value between them; the multiples will be almost the same. If one MAD is much larger than the other, such as 1.5 and 6, the groups do not have similar variability, and the standard's method does not describe them well.
What does "visual overlap" mean?
Visual overlap is how much of the number line the two dot plots share when they are drawn on the same scale, one above the other. A lot of overlap means the groups are hard to tell apart. Little or no overlap means one group is clearly larger.
How many MADs apart is "a big difference"?
There is no official cutoff. As a rough guide (the same rule of thumb used in 7.SP.B.4): less than 1 MAD apart means a lot of overlap; 1 to 2 MADs means one group sits somewhat to one side but the plots still overlap a good deal; 2 or more MADs means a clear, noticeable separation with only some overlap, as in the official example; and 3 or more MADs means little or no overlap. Students should always look at the dot plots as well.
Can students use the median and IQR instead of the mean and MAD?
Yes. The standard says "a measure of variability", so students may divide the difference of medians by the interquartile range, especially when they have box plots. The idea is the same: compare the difference of centers with the spread.
Why not just compare the two means?
A difference of means alone does not say whether it is large. On Card 6 of the card sort, a 5 cm difference in foot length is large because each group's foot lengths vary by only about 1 cm, while on Card 4 a 300-point difference in game scores is only 1.5 MADs because scores vary by 200 points. Dividing by the MAD puts both on the same footing.
What mistakes do students make with 7.SP.B.3?
A frequent mistake is dividing the MAD by the difference instead of the difference by the MAD. Others are dividing by the sum of the two MADs, drawing the two dot plots on different scales, and saying "every" value in one group is larger when the plots still overlap.
How does 7.SP.B.3 connect to high school statistics?
In high school, students compare the center and spread of two or more data sets with the mean and the standard deviation, a measure of spread similar to the MAD (HSS.ID.A.2), and interpret differences in shape, center and spread in context (HSS.ID.A.3). The idea of measuring a difference against the spread stays the same.
07
Related Standards
5 standards
These standards connect to 7.SP.B.3: 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 in context with measures of center and variability
