AP Statistics · Lesson 2

Summary statistics and boxplots

Know Know it

The mean xˉ\bar{x} is the sum of the values divided by how many there are. The median is the middle value in order. The median is resistant: one extreme value barely moves it. The mean is not, so in a skewed distribution the mean is pulled toward the long tail. For data skewed right, the mean is usually greater than the median.

The standard deviation sxs_x is roughly the typical distance of the values from the mean. The range is the maximum minus the minimum. The quartiles split the ordered data into quarters: Q1 is the median of the lower half and Q3 the median of the upper half (leave the overall median out of both halves when the count is odd). The interquartile range is IQR=Q3−Q1IQR = Q_3 - Q_1. The median and IQR are resistant; the mean, standard deviation and range are not.

The five-number summary is the minimum, Q1, median, Q3 and maximum, and a boxplot draws it. By the 1.5 × IQR rule, a value is an outlier if it is below Q1−1.5⋅IQRQ_1 - 1.5 \cdot IQR or above Q3+1.5⋅IQRQ_3 + 1.5 \cdot IQR. A boxplot marks outliers separately and runs each whisker to the most extreme value that is not an outlier.

A z-score, z=x−xˉsxz = \frac{x - \bar{x}}{s_x}, tells how many standard deviations a value lies from the mean, so it lets you compare values from different distributions. Adding a constant to every value shifts the measures of center but leaves the spread unchanged. Multiplying every value by a positive constant multiplies both center and spread by that constant.

Calculator On the exam

You will want a graphing calculator that does statistics for every part of the exam (the digital exam also has a built-in graphing calculator), and you get a sheet of formulas and tables. Use the one-variable statistics feature for the mean, standard deviation and five-number summary, then state each result in context with its units.

Trap A longer piece of a boxplot does not hold more data

Each whisker and each half of the box holds about a quarter of the values. A long whisker or a wide half-box means those values are spread out, not that there are more of them. A boxplot also hides the sample size.

Worked example Easy

Seven fictional shoppers spent these numbers of minutes in a store: 4, 7, 7, 9, 12, 15, 30. What is the median time?

  1. 7 minutes
  2. 9 minutesAnswer
  3. 12 minutes
  4. 17 minutes

How to solve it

  1. The values are already in order, and there are 7 of them.
  2. The middle position is 7 + 1/2 = 4, and the 4th value is 9.

Why each choice is right or wrong

  • A. Incorrect. 7 minutes is the most common value, the mode, but it is the second and third value in order, not the middle one.
  • B. Correct. With 7 values in order, the middle one is the 4th, which is 9 minutes.
  • C. Incorrect. 12 minutes is the mean, 84 divided by 7, which the 30-minute visit pulls above the median.
  • D. Incorrect. 17 minutes is the midrange, halfway between the minimum 4 and the maximum 30, not the middle ordered value.

Explain Explain it

Task verbs: calculate, explain

To flag outliers, show the arithmetic: find the IQR, multiply by 1.5, then subtract from Q1 and add to Q3. Then compare each extreme value with these fences and say what you found in context.

Worked example Medium

Fictional wait times at a clinic have this five-number summary, in minutes: minimum 12, Q1 19, median 24, Q3 28, maximum 50. Above what wait time would a value count as a high outlier by the 1.5 × IQR rule?

  1. 37 minutes
  2. 37.5 minutes
  3. 41.5 minutesAnswer
  4. 42 minutes

How to solve it

  1. IQR = Q3 - Q1 = 28 - 19 = 9 minutes.
  2. 1.5 × IQR = 13.5 minutes.
  3. Upper fence = Q3 + 13.5 = 41.5 minutes, so the 50-minute wait is a high outlier.

Why each choice is right or wrong

  • A. Incorrect. 37 minutes comes from adding just one IQR, 9, to Q3 instead of 1.5 IQRs.
  • B. Incorrect. 37.5 minutes adds 1.5 IQRs to the median, 24, but the rule starts from Q3.
  • C. Correct. The IQR is 28 - 19 = 9, so the fence is 28 + 1.5(9) = 28 + 13.5 = 41.5 minutes.
  • D. Incorrect. 42 minutes is 1.5 times Q3, which multiplies the wrong quantity; the rule adds 1.5 IQRs to Q3.

Explain Your turn

Nine homes in a fictional town sold for these prices, in thousands of dollars: 240, 255, 270, 290, 310, 330, 345, 360, 1308. The mean price is 412 thousand dollars and the median is 310 thousand dollars. Say which of the two better describes a typical sale price here, and explain why the mean and median differ so much.

Write it, then compare. Nothing you type here is saved or graded.

Compare with a model answer

A full answer does this

  • Chooses the median as the better summary of a typical sale price
  • Explains that the one very high price pulls the mean up but not the median

Model answer

The median, 310 thousand dollars, better describes a typical sale, since eight of the nine homes sold between 240 and 360 thousand dollars. The one home that sold for 1,308 thousand dollars pulls the mean up to 412 thousand dollars, above all of the other eight prices, while the median is resistant and stays at the middle sale.

Apply Apply it

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