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IB Mathematics: Applications and Interpretation HL · 13.3: Statistics · 13.3.4

Name Date

Quartiles, box plots and outliers Problem set A

Find quartiles from the calculator (median of each half) and show the working. Give means to 2 decimal places and standard deviations to 3 significant figures unless told otherwise.

Difficulty: first skills applying them stretch

Skills

  1. Find the mean of 31, 27, 36, 29, 33, 38, 24. Give your answer to 2 decimal places.

  2. Find the median of 23, 31, 18, 27, 35, 21, 29, 16, 25.

  3. For the data 17, 9, 24, 13, 21, 11, 28, 15, 19, 8, 22:

    1. (a) write down the five-number summary;
    2. (b) find the interquartile range.

  4. Two data sets are given. The first is 18, 9, 27, 14, 36, 5, 23, 30, 12, 21. The second has five-number summary 12, 19, 25, 33, 41.

    1. (a) Find the five-number summary of the first data set.
    2. (b) State the IQR of the second data set.

  5. Find the population standard deviation of 3, 5, 8, 10, 14, by hand. Give your answer to 3 significant figures.

  6. Ten parcels have masses in kilograms: 24.8, 31.6, 27.3, 29.9, 33.4, 26.1, 30.2, 28.5, 32.7, 25.5. Use a spreadsheet to find the mean and the population standard deviation, each to 2 decimal places.

  7. A data set has Q1 = 52 and Q3 = 70.

    1. (a) Find the lower and upper fences for outliers.
    2. (b) Which of 20, 94 and 99 are outliers?

  8. Use the 1.5 × IQR rule to identify any outliers in 31, 25, 9, 30, 29, 52, 28, 24, 27, 26.

In context

  1. The asking prices, in thousands of euros, of seven flats for sale in a village are 215, 240, 198, 225, 232, 210, 790.

    1. (a) Find the mean, to 2 decimal places, and the median.
    2. (b) Which is the better measure of the centre? Give a reason.

  2. Box plots show the battery life, in hours, of two phone models. Phone X: 8, 10, 12, 15, 19. Phone Y: 6, 11, 12, 13, 18 (five-number summaries).

    Phone XPhone Y02468101214161820Battery life (hours)
    1. (a) Find the IQR of each data set.
    2. (b) Compare the two data sets, using the medians and the IQRs.

  3. A coach records the 100 m times, in seconds, of 11 athletes: 12.1, 12.4, 11.9, 12.6, 12.3, 12.5, 12.2, 12.4, 12.7, 12.3, 15.9.

    1. (a) Use the 1.5 × IQR rule to find any outliers.
    2. (b) Suggest a possible reason for the unusual value.

  4. A shop records the number of customers in each of 12 hours: 27, 64, 21, 33, 29, 22, 31, 17, 14, 25, 18, 24. Show that there is an outlier, then draw a box plot.

  5. A garage records the number of faults found on each of 40 cars at an inspection.

    Faults01234567Frequency1291204030
    1. (a) Find the median.
    2. (b) Find the lower quartile, the upper quartile and the interquartile range.
    3. (c) Show that a car with 6 faults is an outlier.

Stretch

  1. Box plots show the waiting times, in minutes, at two help desks. The cross on Desk 1 is at 38.

    ×Desk 1Desk 20510152025303540Waiting time (min)
    1. (a) Write down the median and the IQR of Desk 1.
    2. (b) Show that the cross on Desk 1 is an outlier.
    3. (c) Compare the waiting times of the two, using the median and the IQR.
    4. (d) Describe the skew of Desk 1 and say how its mean and median compare.

  2. The nine values 28, 33, 30, 31, 35, 27, 36, 25, 22 are given. A tenth value x is added, and x is larger than all nine. Find the smallest whole number x for which x is an outlier, by the 1.5 × IQR rule.

jmaths.xyz · AI HL · 13.3.4 Quartiles, box plots and outliers