Quartiles, box plots and outliers Problem set A Worked solutions

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

• Sum = 218
• Mean = 218 ÷ 7 = 31.14

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

• In order: 16, 18, 21, 23, 25, 27, 29, 31, 35
• There are 9 values, so the median is the 5th: 25

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

1. (a) write down the five-number summary;

   • In order: 8, 9, 11, 13, 15, 17, 19, 21, 22, 24, 28
   • From the calculator: min 8, Q1 11, median 17, Q3 22, max 28
   • n = 11 is odd: the middle value is left out of both halves

2. (b) find the interquartile range.

   • IQR = Q3 − Q1 = 22 − 11 = 11

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.

   • In order: 5, 9, 12, 14, 18, 21, 23, 27, 30, 36
   • From the calculator: min 5, Q1 12, median 19.5, Q3 27, max 36
   • n = 10 is even: each half has 5 values

2. (b) State the IQR of the second data set.

   • IQR = Q3 − Q1 = 33 − 19 = 14

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

• Mean = 40 ÷ 5 = 8
• Deviations: −5, −3, 0, 2, 6; squared: 25, 9, 0, 4, 36
• Variance = 74 ÷ 5 = 14.8
• σ = √14.8 = 3.85

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.

• Enter the 10 values in cells A1 to A10
• =AVERAGE(A1:A10) gives 29.00
• =STDEV.P(A1:A10) gives 2.89

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

1. (a) Find the lower and upper fences for outliers.

   • IQR = 70 − 52 = 18
   • 1.5 × 18 = 27
   • Lower: 52 − 27 = 25. Upper: 70 + 27 = 97

2. (b) Which of 20, 94 and 99 are outliers?

   • Outliers are below 25 or above 97
   • 20: an outlier
   • 94: not an outlier
   • 99: an outlier

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

• From the calculator: Q1 = 25, Q3 = 30
• IQR = 5; fences 17.5 and 37.5
• Outliers are 9 and 52: each is beyond a fence

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.

   • Mean = 2110 ÷ 7 = 301.43
   • In order: 198, 210, 215, 225, 232, 240, 790; median = 225

2. (b) Which is the better measure of the centre? Give a reason.

   • The value 790 is far above the others and pulls the mean above all but one value
   • The median 225 is close to the typical value, so it is the better measure

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).

1. (a) Find the IQR of each data set.

   • Phone X: 15 − 10 = 5
   • Phone Y: 13 − 11 = 2

2. (b) Compare the two data sets, using the medians and the IQRs.

   • Both medians are 12, so the typical value is the same
   • The IQR of Phone X is larger, so its middle half is more spread out
   • Phone Y is more consistent

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.

   • From the calculator: Q1 = 12.2, Q3 = 12.6
   • IQR = 0.4; fences 11.6 and 13.2
   • 15.9 > 13.2, so it is an outlier

2. (b) Suggest a possible reason for the unusual value.

   • The value 15.9 s is far from the rest. A possible reason: the athlete may have been injured or slipped at the start, or the time may have been recorded wrongly

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.

• From the calculator: min 14, Q1 19.5, median 24.5, Q3 30, max 64
• IQR = 10.5; fences 3.75 and 45.75; 64 is above 45.75: an outlier
• Upper whisker ends at 33, the largest value inside the fence
• Draw the box from 19.5 to 30, the median line at 24.5, whiskers to 14 and 33, a cross at 64

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

1. (a) Find the median.

   • \(n=40\), so the median is between the 20th and 21st values.
   • Running totals: 12, 21, 33, 33, 37, 37, 40, 40
   • Median = 1

2. (b) Find the lower quartile, the upper quartile and the interquartile range.

   • Q1 is between the 10th and 11th values and Q3 between the 30th and 31st values (check with the cumulative frequency).
   • From the calculator: Q1 = 0, Q3 = 2
   • IQR = 2 − 0 = 2

3. (c) Show that a car with 6 faults is an outlier.

   • \(Q_3+1.5\times IQR=2+1.5\times 2=5\)
   • \(6>5\), so it is an outlier.

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

1. (a) Write down the median and the IQR of Desk 1.

   • Median = 10
   • IQR = 15 − 8 = 7

2. (b) Show that the cross on Desk 1 is an outlier.

   • 1.5 × 7 = 10.5
   • Upper fence = 15 + 10.5 = 25.5
   • 38 > 25.5, so it is an outlier

3. (c) Compare the waiting times of the two, using the median and the IQR.

   • Medians: Desk 2 12, Desk 1 10
   • IQRs: Desk 2 5, Desk 1 7
   • Desk 2 has the higher typical value and a smaller spread

4. (d) Describe the skew of Desk 1 and say how its mean and median compare.

   • The median is nearer Q1 and the upper whisker and the outlier are far longer than the lower whisker
   • The data are skewed to the right (positive skew)
   • The outlier pulls the mean above the median

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.

• With x added, n = 10 and x is the largest value, so Q1 = 27 and Q3 = 35 do not depend on x
• IQR = 8; upper fence = 35 + 12 = 47
• x must be greater than 47: the smallest whole number is 48