A comparison names shape, center, spread and outliers for both groups. Use comparative language: "A has a higher median than B", never two parallel descriptions. Name the variable and its units. Let the shape decide whether to quote means or medians.
Step 1: Picture It
Start with something you already know.
Two archers at one target
- Two archers each shoot ten arrows at the same target.
- Kai’s arrows land in a tight little cluster, a hand’s width left of the bullseye.
- Rosa’s arrows land all around the bullseye, but scattered far apart.
- Someone asks who shot better, and one word cannot answer it.
- Kai is more consistent; Rosa is closer to the middle on average.
Two groups of data are the two archers: where each group sits is its center, and how scattered it is, is its spread.
A comparison says both, each with "than": Kai’s arrows are centered a hand’s width further left than Rosa’s, and Kai’s are far less spread out than Rosa’s.
Where the picture falls short: data sits on a number line, not a flat target, and a higher center in data is not "better" unless the context says so.
Step 2: Let's Learn
Read it, or press Listen and follow the words.
Try it first
Two coffee shops timed five orders each, in minutes. Shop A: 2, 3, 3, 4, 13. Shop B: 4, 5, 5, 6, 6. Which sentence about a typical wait is right?
Stuck is fine. Let’s Learn shows the way next. Got it first time? You can jump ahead to Watch an Example.
Use comparative language
"Group A has median 12" and "Group B has median 9" is a list. "Group A has a higher median than Group B, 12 against 9 minutes" is a comparison. Write the word "than" into every sentence.
Always in context
"Higher median commute time, in minutes" beats "higher median". A comparison with no nouns from the problem earns nothing on the exam.
Choose measures consistently
For skewed data compare medians and IQRs. For symmetric data compare means and SDs. Do not mix a mean for one group with a median for the other.
Note the overlap
Two groups can differ in center while overlapping heavily. If A runs from 15 to 40 and B from 8 to 45, most of each group could have come from the other. Saying so is a stronger answer than the gap alone.
Outliers in a comparison
Name any outlier and which group it belongs to. "B has one commute of 95 minutes, far beyond its upper fence of 60." An outlier can be the most interesting point in the comparison.
The checklist
Shape of each. Which has the higher center, and by how much. Which has the greater spread. Any outliers. All in context with units. Four features, one paragraph, marks for each.
Step 3: Try It Yourself
Tap and try it out.
Test scores. Class 1: symmetric, mean 72, SD 8, range 55 to 90. Class 2: symmetric, mean 72, SD 15, range 40 to 98, one score at 20. Compare
- 1both roughly symmetric: means and SDsshape decides
Step 0 of 5
- Minimum4
- Lower quartile6
- Median9
- Upper quartile13
- Maximum18
- Interquartile range7
Each of the four sections holds a quarter of the values, however wide it looks. A narrow box means the middle half of the data is packed close together.
Step 4: In Real Life
Two hospitals’ waits
One hospital has a lower median wait but a longer tail. Comparing shape, center and spread together tells a patient which is likely faster and which can be catastrophically slow.
Step 5: Watch an Example
One step at a time.
Watch Amara Compare Two Pizza Shops
Amara timed 11 deliveries from each of two pizza shops, in minutes. Northside: min 18, Q1 22, median 25, Q3 28, max 33. Southside: min 12, Q1 15, median 20, Q3 31, max 58.
- Step 1
Shape comes first, because it decides the measures. Northside’s quartiles sit 3 minutes either side of its median and its ends are about equally far out, so it is roughly symmetric. Southside’s top end stretches out to 58, far past the rest, so it is right-skewed.
Watch Dev Compare Two Camp Stores
Dev collected how much each camper spent at the camp store in a week, in dollars. Camp Pine: min 2, Q1 8, median 12, Q3 18, max 45, right-skewed. Camp Lake: min 9, Q1 13, median 15, Q3 17, max 22, roughly symmetric.
- Step 1
Pine’s spending is right-skewed, so the comparison uses medians and IQRs for both camps, Lake included, so the two are measured the same way.
Step 6: Your Turn
Practice makes it stick.
The Bike Shop
Problem 1 of 15
A bike shop timed repairs, in days, at two locations. Downtown: Q1 = 2, median 3, Q3 = 5. Uptown: Q1 = 1, median 2, Q3 = 9. Both are right-skewed. By how many days is Uptown’s IQR greater than Downtown’s?
days
The Electric Bills
Problem 2 of 15
Monthly electric bills for 30 homes in each of two towns are roughly symmetric with no outliers. Town R: mean $120, SD $15. Town S: mean $135, SD $15. Which sentence is the best comparison?
Compare the Pieces
Settle the shape, use the same measures for both, and say each difference with "than".
Problem 3 of 15
Sunny bed: mean 34 cm, median 33 cm. Shady bed: mean 26 cm, median 27 cm. Both are roughly symmetric with no outliers. By how many centimeters is the sunny bed’s typical height higher, using the center that fits?
cm
Problem 4 of 15
Waiting times at a clinic are right-skewed on Mondays and roughly symmetric on Fridays. Which measures should the comparison use?
Problem 5 of 15
Homework times for one group, in minutes: Q1 = 14, Q3 = 22, and the smallest time is 1 minute. Where is the lower fence, in minutes?
minutes
Problem 6 of 15
Library visits in a month, both groups right-skewed. Group A: Q1 = 4, Q3 = 10. Group B: Q1 = 6, Q3 = 9. Which group’s visits are more spread out?
Problem 7 of 15
Bus rides to school: Route 1 has a median of 14 minutes and Route 2 a median of 10 minutes. Which sentence is a comparison?
Whose Step?
Each page has one line that slipped. Find it, then put it right.
Problem 8 of 15
Leo compared two groups’ times to finish a puzzle, in minutes. One line is a slip. Which?
Leo’s working: which group has the higher typical time?
- 1A: right-skewed, median 20 min, mean 26 min
- 2B: roughly symmetric, median 22 min, mean 22 min
- 3compare A’s mean, 26, with B’s median, 22
- AnswerA has the higher typical time, by 4 minutes
Problem 9 of 15
Comparing both of Leo’s groups by the measure that fits them, which sentence is right?
Problem 10 of 15
Priya checked Group B’s repair times, in days, for a high outlier; the largest is 30. One line is a slip. Which?
Priya’s working: does Group B have a high outlier?
- 1Q1 = 12, Q3 = 20: IQR = 20 − 12 = 8
- 2upper fence = 20 + 8 = 28
- 3the largest value, 30, is beyond 28
- AnswerB has a high outlier
Problem 11 of 15
Where is Group B’s upper fence, in days?
days
Pick the Move
Which move comes first, or which sentence finishes the job? No full working is needed here.
Problem 12 of 15
Two groups of quiz scores need comparing, and nothing has been worked out yet. Which move comes first?
Problem 13 of 15
Two classes’ test scores both run from 10 to 55 points, with IQRs of about 20 points. A student wrote "Class B has a higher median score than Class A, 32 against 30 points." What most needs adding?
Same Story?
Judge each one without writing out a full comparison.
Problem 14 of 15
Class 1’s median score is 70 points and Class 2’s is 64. Select every sentence that says the same thing about the centers.
Problem 15 of 15
Group A’s practice times, in minutes, have Q1 = 10 and Q3 = 22. Select every group below whose IQR is the same as Group A’s.
Step 7: Quick Check
Show what you know.
Question 1 of 2
Q1 = 15, Q3 = 35. What is the IQR?
Question 2 of 2
What makes a comparison complete on the exam?
Ready for more?Optional. Past the lesson, for anyone who wants it.
Stretch 1 of 2
Group A’s five-number summary, in minutes, is 10, 16, 20, 24, 30. Ravi wants to invent Group B with the same median as A but an IQR twice as large as A’s, and he puts B’s Q1 at 12 minutes. Where must B’s Q3 go, in minutes?
minutes
Stretch 2 of 2
Two groups both have a median of 40 minutes. Group A is strongly right-skewed; Group B is roughly symmetric. Which group’s mean is likely higher?
What You Learned
- Compare shape, center, spread and outliers.
- Use comparative language and name the variable in context.
- Let the shape decide whether to quote means or medians, and use the same measure for both groups.
- Note the overlap: groups can differ in center while overlapping heavily.
For the grown-up
The misconception this lesson targets is "two descriptions side by side make a comparison." A student writes "Group A has median 25; Group B has median 18" and believes the job is done, leaving the reader to compare. Two quieter cousins come with it: mixing a mean for one group with a median for the other, and treating a gap between two centers as if every member of one group beat every member of the other. When a student compares, ask for one sentence with "than", the variable and its units, and then ask how much the two groups overlap.