Displaying and describing distributions

About this lesson

Grade 6 › Data sets and distributions

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Displaying and describing distributions

Warm-up

Three quick questions from earlier lessons.

Discover

House prices in a town: most around 300 thousand dollars, a few mansions above 2 million dollars. Which is pulled up by the mansions?

  1. The mean
  2. The median

Key idea

Describe a distribution by its shape, center, spread and unusual values.

Shape: symmetric, skewed right (long tail of high values) or skewed left; number of peaks. Center: mean or median. Spread: SD, IQR or range. Outliers: values far from the rest. In a right-skewed distribution the mean is usually above the median.

Worked example

Commute times: median 22 min, mean 29 min, a few 90-min commutes.

Mean > median
The mean is pulled towards the long tail.
Skewed right
The tail is on the high side.
The median describes a typical commute better
Resistant to the tail.

Your turn, guided

Your turn: test scores have mean 61 and median 70. How far below the median is the mean?

99 points: skewed left
70 − 61: the mean is pulled down by a tail of low scores, so the distribution is skewed left.

Practice ★

  1. A distribution of incomes has mean 54 thousand dollars and median 43 thousand dollars. What is its most likely shape?
    1. Skewed right: a long tail of high values pulls the mean up
    2. Skewed left
    3. Symmetric
  2. A distribution of incomes has mean 59 thousand dollars and median 54 thousand dollars. What is its most likely shape?
    1. Skewed right: a long tail of high values pulls the mean up
    2. Skewed left
    3. Symmetric
  1. 43 students name their favorite sport. Which display suits these data?
    1. A bar chart
    2. A histogram
    3. A boxplot

Watch out!

Practice ★★

  1. Which four features should you describe for the distribution of a quantitative variable (here, 42 test scores)?
    1. Shape, center, spread and unusual values (outliers)
    2. The mean, the median, the mode and the sample size
    3. The type of graph, the title, the axes and the colors

Exit ticket

Three questions to check you have got it.

  1. A distribution of incomes has mean 54 thousand dollars and median 43 thousand dollars. What is its most likely shape?
    1. Skewed right: a long tail of high values pulls the mean up
    2. Skewed left
    3. Symmetric
  2. A distribution of incomes has mean 59 thousand dollars and median 54 thousand dollars. What is its most likely shape?
    1. Skewed right: a long tail of high values pulls the mean up
    2. Skewed left
    3. Symmetric
  3. A distribution of incomes has mean 58 thousand dollars and median 46 thousand dollars. What is its most likely shape?
    1. Skewed right: a long tail of high values pulls the mean up
    2. Skewed left
    3. Symmetric

Summary card

Describing a distribution

Shape, center, spread, outliers, in context.

Watch out: skewed right means mean > median (usually).

End of the lesson

Next lesson: Reading and interpreting circle graphs Back to the chapter

Answers: Displaying and describing distributions

A

  1. Skewed right: a long tail of high values pulls the mean up
  2. Skewed right: a long tail of high values pulls the mean up

B

  1. A bar chart

C

  1. Shape, center, spread and unusual values (outliers)

D

  1. Skewed right: a long tail of high values pulls the mean up
  2. Skewed right: a long tail of high values pulls the mean up
  3. Skewed right: a long tail of high values pulls the mean up

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