AP Statistics · Free-response skill

Compare two distributions in context

The move. Compare two distributions of a quantitative variable by shape, center, variability and unusual values, using numbers and the context in every statement.

How to do it

  1. Write comparisons, not two separate descriptions: say which group is higher or more spread out, and by how much.
  2. Pick summaries that fit the shape. When one distribution is skewed or has an outlier, compare medians and IQRs, which are resistant.
  3. Put a number and the context in every claim: name the variable, the units and the groups, such as Model N phones lasted a median of 19 hours.
  4. When you check for outliers, show the fence arithmetic: IQR, 1.5 times the IQR, and the boundary you compare with.

Practice prompt Try it

Fictional study. A consumer group fully charged 15 phones of each of two new models, Model N and Model S, and recorded how many hours each battery lasted under the same test. The sorted results are shown in the table.

Battery life of 15 phones of each fictional model, in hours, sorted from shortest to longest

ModelHours each battery lasted
Model N12, 14, 15, 16, 17, 18, 18, 19, 20, 20, 21, 22, 23, 24, 26
Model S8, 9, 10, 10, 11, 12, 12, 13, 14, 15, 17, 19, 22, 28, 55

Use the data to answer each part.

  1. Say what shape each model's distribution of battery life has.
  2. Find the median battery life of each model, and say which model's phones usually lasted longer.
  3. Find the interquartile range of each model, and say which model's battery life varied more.
  4. Use the 1.5 × IQR rule to decide whether any Model S phone lasted an unusually long time. Show the boundary you compare with.

Write each part, then check yourself against the checklist. Nothing you type here is saved or graded.

Check yourself and see a model answer

Checklist

Our own checklist of what a full answer does, one line per point. Tick each one your response earns.

0 of 5 points

Model answer, annotated

(a) Model N's battery lives are roughly symmetric, centered near 19 hours with values from 12 to 26 hours. Model S's battery lives are skewed right: most phones lasted 8 to 22 hours, with a long tail out to 28 and 55 hours.

  • Point 1 Calls Model N roughly symmetric and backs it with the center and range.
  • Point 2 Calls Model S skewed right and points to the long tail toward high values.

(b) The median is the 8th of 15 values. For Model N it is 19 hours and for Model S it is 13 hours, so a typical Model N phone's battery lasted 6 hours longer than a typical Model S phone's.

  • Point 3 Gives both medians with units and compares them in context.

(c) For Model N, Q1 = 16 and Q3 = 22 hours, so the IQR is 6 hours. For Model S, Q1 = 10 and Q3 = 19 hours, so the IQR is 9 hours. The middle half of Model S battery lives spread over 9 hours compared with 6 hours for Model N, so Model S varied more.

  • Point 4 Shows the quartiles behind each IQR and says which model varied more.

(d) For Model S, 1.5 × IQR = 1.5 × 9 = 13.5 hours, so the upper boundary is 19 + 13.5 = 32.5 hours. The phone that lasted 55 hours is above 32.5 hours, so it is a high outlier: it lasted unusually long for a Model S phone. The 28-hour phone is below the boundary, so it is not an outlier.

  • Point 5 Shows the fence arithmetic and compares the 55-hour phone with the 32.5-hour boundary.