Box Plots

The median is shown by the line inside the box.

1. In a box plot the vertical line inside the box marks the median.
2. Read across to the scale: the line is at 18 minutes.
3. The median time is 18 minutes.

The quartiles are the two ends of the box.

1. The box runs from the lower quartile to the upper quartile, so the lower quartile is the left-hand end of the box.
2. The left-hand end of the box is at 10 on the scale.
3. The lower quartile is 10 minutes.

The quartiles are the two ends of the box.

1. The box runs from the lower quartile to the upper quartile, so the upper quartile is the right-hand end of the box.
2. The right-hand end of the box is at 26 on the scale.
3. The upper quartile is 26 minutes.

The ends of the whiskers show the shortest and longest times.

1. Range = greatest value - least value.
2. The whiskers end at 4 minutes (the shortest time) and 40 minutes (the longest time).
3. Range =40-4=36 minutes.

The interquartile range is the length of the box.

1. Interquartile range = upper quartile - lower quartile.
2. The box runs from 10 minutes (lower quartile) to 26 minutes (upper quartile).
3. Interquartile range =26-10=16 minutes.

Check both box edges and the median, as well as the whisker endpoints.

1. In a box plot the whiskers end at the least and greatest values, the box runs from the lower quartile to the upper quartile, and the line inside the box is the median.
2. So the plot needs whiskers from 5 to 40, a box from 12 to 25 and a median line at 18.
3. Plot B has all five. Plot D has the right median but its box runs on to 32, so its upper quartile is wrong. Plots A and C have the median at 25, which is the upper quartile.

Find the overall median and then the median of each five-value half.

1. There are 10 values, so the median is halfway between the 5th and 6th values: \frac{8+10}{2}=9.
2. The lower half is 2,4,5,7,8. Its middle value, 5, is the lower quartile.
3. The upper half is 10,12,14,16,20. Its middle value, 14, is the upper quartile.
4. With least value 2 and greatest value 20, the five-number summary is 2,5,9,14,20.
5. The option 2,5,8,14,20 uses the 5th value as the median, but with an even number of values the median lies between the two middle values.

Approximately one quarter of the observations lie above the upper quartile.

1. The upper quartile has about 75\% of values at or below it, leaving about 25\% above.
2. One quarter of 80 is 20 people.

Read each end of the box carefully: each small square on the scale is 5 grams.

1. Interquartile range = upper quartile - lower quartile, which is the length of the box.
2. Each small square is 5 grams. The left end of the box is at 110 g and the right end is at 145 g.
3. Interquartile range =145-110=35 grams.

Use medians for typical time and interquartile ranges for the spread of the middle half.

1. Compare the averages using the medians: Group A's median is 18 minutes and Group B's is 20 minutes, so Group A has the lower median.
2. Compare the spreads using the interquartile ranges (the lengths of the boxes): Group A has 26-10=16 minutes and Group B has 24-14=10 minutes.
3. So Group A has a lower median but a greater interquartile range.

Compare the medians for the average, and the interquartile ranges for how consistent the times are.

1. To compare distributions, compare an average (the median) and a measure of spread (the interquartile range), and say what each means in context.
2. Both median lines are at 20 minutes, so on average the two groups took the same time.
3. Group A's interquartile range is 25-15=10 minutes and Group B's is 32-8=24 minutes.
4. A smaller interquartile range means the middle half of the times are closer together, so Group A's times were more consistent. A longer box means the times are less consistent, not more.

Find each full whisker-to-whisker range before taking their difference.

1. Range = greatest value - least value, so use the ends of the whiskers.
2. Group A: the whiskers end at 4 and 40, so the range is 40-4=36 minutes.
3. Group B: the whiskers end at 2 and 44, so the range is 44-2=42 minutes.
4. The range for Group B is 42-36=6 minutes greater.

A median describes the centre, rather than comparing every individual observation.

1. A lower median does not mean every value is lower.
2. The displayed ranges overlap. In particular A has a maximum near 40 minutes while B has a minimum near 2 minutes.
3. The claim about every time is false.

The interquartile range is the gap between the quartiles, and the range is the gap between the lightest and heaviest values.

Part (a)

1. Interquartile range = upper quartile - lower quartile, so upper quartile = lower quartile + interquartile range.
2. Upper quartile =2.6+1.8=4.4 kg.

Part (b)

1. Range = heaviest - lightest, so heaviest = lightest + range.
2. Heaviest parcel =1.2+5.5=6.7 kg.

With 15 values in order, the lower quartile is the 4th value and the upper quartile is the 12th value.

1. For n=15 values in order, the median is the \frac{15+1}{2}=8th value.
2. The lower quartile is the \frac{15+1}{4}=4th value, which is 22. (This is also the middle of the 7 values below the median.)
3. The upper quartile is the 3\times4=12th value, which is 38.
4. Interquartile range =38-22=16 years.

Identify which quartile the threshold represents in each group, then apply each group size.

1. About 25\% of the values in a box plot lie above the upper quartile.
2. In both groups the right-hand end of the box, the upper quartile, is at 30 minutes.
3. Group A: about \frac14\times80=20 people took more than 30 minutes. Group B: about \frac14\times120=30 people.
4. Estimated total: 20+30=50 people.

Work out the upper quartile and the longest time first, then think about what fraction of the runners lies in each part of a box plot.

Part (a)

1. Range = longest time - shortest time, so longest time = shortest time + range.
2. Longest time =18+30=48 minutes.

Part (b)

1. Upper quartile = lower quartile + interquartile range =24+9=33 minutes.
2. So 33 minutes is the upper quartile, and about one quarter of the runners took longer than this.
3. Estimate: \frac14\times200=50 runners.

The same threshold may be a different quartile in the two groups.

1. For A, 20 minutes is the median, so about half of 60, or 30, took longer.
2. For B, 20 minutes is the upper quartile, so about one quarter of 100, or 25, took longer.
3. The estimated difference is 30-25=5 people.

With 19 values in order, the median is the 10th value and the quartiles are the 5th and 15th values.

Part (a)

1. There are 19 heights, already in order, so the median is the \frac{19+1}{2}=10th value.
2. Counting along the leaves: 12,15,18 are the first 3, then 20,21,24,26,27,29 take us to the 9th. The 10th value is 31.
3. The median is 31 cm.

Part (b)

1. The lower quartile is the \frac{19+1}{4}=5th value and the upper quartile is the 15th value.
2. The 5th value is 21 cm. The 15th value is 41 cm (the first leaf on stem 4, after 14 values on stems 1 to 3).
3. Interquartile range =41-21=20 cm.

Part (c)

1. Compare the medians for the average: 35 cm in B is more than 31 cm in A, so the plants in B were taller on average.
2. Compare the interquartile ranges for the spread: 12 cm in B is less than 20 cm in A, so the heights in B were more consistent.
3. Medians and interquartile ranges can be compared even when the groups are different sizes.

Locate the threshold between the summary values and consider what the plot leaves unknown.

1. The lower quartile is near 10 and the median near 18 minutes.
2. The threshold 15 lies between them, so the percentage below it is between about 25\% and 50\%.
3. The exact percentage cannot be read because the plot gives no distribution within that part of the box.