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This page sets out the method for box plots in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Box plots topic page.
When you are given (or have found) the minimum, quartiles and maximum, plot them on a scale and build the box.
When two box plots are on the same scale, write two sentences comparing them.
Lower quartile 9 and upper quartile 20 give an interquartile range of 20 − 9 = 11. It is the length of the box.
Put the values in order first. With 7 values the median is the 4th. The lower quartile is the middle of the values below the median, and the upper quartile is the middle of the values above it. For 2, 5, 7, 9, 12, 14, 20 the median is 9, the lower quartile is 5 and the upper quartile is 14.
Draw one scale. Mark the lowest value, lower quartile, median, upper quartile and highest value. Draw a box from the lower quartile to the upper quartile with a line at the median, then draw a whisker from each end of the box out to the lowest and the highest value.
Lowest value 4 and highest value 28 give a range of 24.
The five-number summary is 4, 9, 13, 20, 28: lowest, lower quartile, median, upper quartile, highest. The median is the middle number, 13, and the interquartile range is 20 − 9 = 11.
A larger interquartile range means more spread in the middle half. When you compare two box plots, compare the medians for the average and the interquartile ranges for the spread, and write each in words.
In a five-number summary the third value is the median.
Range uses the ends. Interquartile range uses the quartiles, not the ends.
Order the story of the box from small to large. The minimum is the left end of the left whisker. The lower quartile is the left edge of the box. The median is the line inside the box. The upper quartile is the right edge of the box. The maximum is the right end of the right whisker. Nothing else on the diagram is a new statistic.
The median is not the middle of the box in centimetres on the page unless the scale says so. You read it on the scale, or you use the number the question gives. The box can be uneven because the data can be uneven. A median close to the lower quartile means the second quarter of the data is packed into a short interval and the third quarter is more spread out.
Method: 1. List the five numbers in order: minimum, lower quartile, median, upper quartile, maximum. 2. Draw the box between the lower quartile and the upper quartile. 3. Mark the median with the line inside the box. 4. Draw whiskers out to the minimum and the maximum.
The range uses the ends. For a minimum of 19 and a maximum of 47, the range is 47 - 19 = 28. It answers "from the smallest to the largest, how wide is the whole set?" One very small or very large value changes it, even if everyone else is bunched in the box.
The interquartile range ignores those ends. It is the width of the middle 50 percent. Upper quartile 31 minus lower quartile 18 is 31 - 18 = 13. People mix these up because both are subtractions. The names tell you which pair to use. Range: maximum and minimum. Interquartile range: the two edges of the box.
Method: 1. Range = maximum − minimum. 2. Interquartile range = upper quartile − lower quartile. 3. Do not use the median in either subtraction. 4. Treat a bigger result as more spread on that measure.
Put the two five-number summaries side by side and compare the statistic the question names. A difference of medians is the higher median minus the lower median. A difference of interquartile ranges is the same idea with the two IQR values, after you have calculated each one. Do the subtractions inside each group before you subtract between the groups.
You can compare more than one feature, but each sentence needs its own pair of numbers. "Group B has the higher median, and group A has the smaller interquartile range" is two claims. The first is about the typical value. The second is about consistency. Neither claim follows from the other. A group can be typically higher and also more spread out. In an exam, write each comparison in the context of the question, for example "the median time for group B is higher, so group B typically takes longer".
Method: 1. Write both five-number summaries. 2. Work out the statistic for each group separately. 3. Then subtract those results if the question asks for a difference. 4. Say which group the larger number belongs to, so the comparison has a direction.
Consistent means the middle half is close together. That is a small interquartile range, a short box. It does not mean a high median, and it does not mean a small range, though a small range often goes with a small box. If the question says "more consistent", compare the interquartile ranges. The group with the smaller one is more consistent.
The range can tell a different story because of the whiskers. A group with a short box and one very long whisker has a consistent middle and one extreme value. Quoting only the range would hide the consistent middle. Quoting only the box would hide the extreme. Use the measure the question asked for, and do not swap them to make the groups look more different.
Method: 1. For consistency, compare the interquartile ranges. 2. Call the group with the smaller interquartile range more consistent. 3. Do not use a higher median as proof of consistency. 4. Mention the range only when the question asks about the whole spread, including the ends.
Each section of a box plot holds one quarter of the data. A quarter of the values lie between the minimum and the lower quartile, a quarter between the lower quartile and the median, a quarter between the median and the upper quartile, and a quarter between the upper quartile and the maximum. The box therefore holds half of the data, whatever its length.
Exam questions use this to ask "How many people took longer than the upper quartile?" or to check a claim. If there are 120 values, then a quarter is 30 values and half is 60 values. A student who says "at least 40 people" above the upper quartile is wrong, because exactly a quarter of the data lies above it.
Method: 1. Write down the number of values in the data set. 2. Divide by 4 to find one quarter, and by 2 to find a half. 3. Match the part of the diagram in the question to one, two or three quarters. 4. Compare your count with the claim and state whether it is correct, giving the number.
Tip: The length of the box never changes how many values it holds. A long box is spread out, not larger in number.
When a question gives the raw data, put the values in order first. With n values, the median is at position (n+1)/2, the lower quartile at (n+1)/4 and the upper quartile at (3(n+1))/4. For n = 11 the positions are 6, 3 and 9. For n = 15 they are 8, 4 and 12. Exam lists are chosen so these positions are whole numbers.
Write the five values clearly: minimum, lower quartile, median, upper quartile, maximum. If the question says "Draw a box plot", mark these on the scale, draw a box from the lower to the upper quartile with a line at the median, and join the whiskers to the minimum and maximum. In this lesson you are asked for the five values and what they tell you.
Method: 1. Put the data in order from smallest to largest. 2. Count the values to get n and work out the three positions. 3. Read off the values at those positions, together with the smallest and largest. 4. Work out the range and the interquartile range from the five numbers.
Comparison questions have a pattern. One mark is for comparing a measure of average, usually the median, and one mark is for comparing a measure of spread, usually the interquartile range. Each comparison must use the numbers and the context: "The median journey time for route 2 is 27 minutes, which is 3 minutes longer than route 1, so route 2 is slower on average".
A lower interquartile range means the middle half of the values are closer together, so that group is more consistent. Say so in context: "Route 2 is more reliable because its interquartile range is 9 minutes, compared with 15 minutes for route 1". Avoid vague words such as "bigger" or "better" without saying what is bigger or better.
Method: 1. Work out the median and the interquartile range for each group. 2. Write one sentence about the averages with both values and the context. 3. Write one sentence about the spread, stating which group is more consistent. 4. Check each sentence names the quantity, such as time or mass, and has numbers.
Tip: The interquartile range is usually preferred to the range because one extreme value cannot change it much.
Higher questions often give some of the five numbers and describe the rest. For example, "the median is 4 more than the median of class A" or "the interquartile range is 1.5 times as large". Find the numbers you know for the first group, then use each clue to build the second group, one value at a time.
The interquartile range links the two quartiles: upper quartile = lower quartile + interquartile range. Likewise maximum = minimum + range. After finding all five, check that they are in order, with each value at least as big as the one before. If not, a step has gone wrong, so go back and check the arithmetic of each clue before you write the final answer.
Method: 1. Work out the interquartile range and median of the group you know fully. 2. Apply each clue to get the matching value for the second group. 3. Add the interquartile range to the lower quartile to get the upper quartile. 4. Check the final five values are in increasing order.
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