Representation of data

AS Level Mathematics · Probability & Statistics 1. A short explanation of the idea, the rules to remember, the mistake to avoid, a worked example and practice questions with answers.

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The idea

Data can be shown in several ways, and each has a use. A stem-and-leaf diagram keeps every value. A box-and-whisker plot shows the median, quartiles and extremes, and is good for comparing two sets. A histogram shows grouped continuous data: the area of each bar represents the frequency. A cumulative frequency graph lets you estimate the median, quartiles and percentiles.

A set of data is summarised by a measure of central tendency (mean, median or mode) and a measure of variation (range, interquartile range or standard deviation).

The mean and standard deviation use every value, so extreme values affect them. The median and interquartile range are better for skewed data.

Rules to remember

Common mistake

Reading the height of a histogram bar as the frequency. When class widths differ, the height is the frequency density, so multiply by the class width to get the frequency.

Worked example

The masses, x kg, of 20 parcels are grouped: 0 ≤ x < 10 (frequency 4), 10 ≤ x < 20 (frequency 10), 20 ≤ x < 40 (frequency 6). Estimate the mean mass.

  1. Mid-points of the classes: 5, 15 and 30.
  2. Σfx = 4 × 5 + 10 × 15 + 6 × 30 = 20 + 150 + 180 = 350.
  3. Mean ≈ Σfx ÷ Σf = 350 ÷ 20 = 17.5.

Answer: 17.5 kg

Practice questions

Try each one, then open the answer.

1. For a set of 10 values, Σx = 120 and Σx2 = 1690. Find the standard deviation.

  1. A
    25
  2. B
    5
  3. C
    13
  4. D
    2.24
Show answer

Answer: B. Mean = 12. Variance = Σx2/n − mean2 = 169 − 144 = 25, so the standard deviation is √25 = 5. Do not forget to take the square root.

2. Which representation of a set of data allows every original data value to be read back exactly?

  1. A
    a histogram
  2. B
    a box-and-whisker plot
  3. C
    a stem-and-leaf diagram
  4. D
    a cumulative frequency graph
Show answer

Answer: C. A stem-and-leaf diagram keeps every value. A histogram and a cumulative frequency graph group the data, and a box plot shows only five summary values.

3. The annual incomes of the 40 employees of a small company include one very large value, the income of the owner. Which pair of measures is most appropriate to summarise the typical income and its spread?

  1. A
    mean and standard deviation
  2. B
    mean and range
  3. C
    median and interquartile range
  4. D
    mode and range
Show answer

Answer: C. The median and interquartile range depend only on the middle of the ordered data, so one extreme value hardly affects them. The mean, standard deviation and range are all pulled strongly by the outlier.

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