GCSE · Maths · AQA · Spec 8300 · Higher

Box plots, quartiles, inter-quartile range (Higher)

Two pizza shops both deliver in about half an hour, but only one is reliable. One picture tells you which: a box plot.

Two box plots, one scale

Which pizza shop can you rely on?

The five-number summaries
	Min	LQ	Median	UQ	Max	IQR
Pizza Palace	18	24	30	34	58	10
Slice Shack	15	20	27	38	45	18

Look at the line inside each box. Which shop has the lower median delivery time?

Pizza PalaceSlice Shack

Now look at how wide each box is. Which shop's middle half of delivery times is more consistent?

Pizza PalaceSlice Shack

Which sentence is a proper comparison of the two shops?

Slice Shack's median is lower than Pizza Palace's.Pizza Palace is less reliable, because its range is bigger.Pizza Palace is quicker, because its median is further right.Slice Shack is usually quicker; Palace's typical times vary less.

Read two things from each box plot. The line inside the box is the median: the typical value. The box runs from the lower quartile to the upper quartile (LQ and UQ in the table, also written Q1 and Q3), so its width is the inter-quartile range (IQR): the spread of the middle half of the data. A narrower box means the typical values are more consistent. Then say what each one means for the data, not just which number is bigger.

Where the box comes from: a list

Find the quartiles yourself

Nine students did as many press-ups as they could in one minute: 23, 31, 18, 27, 40, 22, 35, 29, 26. Find the lower quartile, the upper quartile and the inter-quartile range.

There are 9 values: 23, 31, 18, 27, 40, 22, 35, 29, 26.Q1 will be the value with a quarter of the data below it, and Q3 the value with three quarters below it. "Below" only means something once the values are in order.
missing step

Which line is step 2?

The 5th value of the list as written is 40, so the median is 40.Write them in order: 18, 22, 23, 26, 27, 29, 31, 35, 40.Add them up: the total is 251, so the average is about 27.9.

Where the box comes from: a cumulative frequency graph

Time spent on homework (minutes): 50. Cumulative frequency: 18

40 students. Slide the point up the curve until the cumulative frequency is 10, then 20, then 30. Each time, read the time below the point.

Watch out: A quarter means a quarter of the total frequency: 40 ÷ 4 = 10, found on the cumulative frequency axis. Don't go a quarter of the way along the time axis.

Build the box plot

Five values, five lines

From the graph: Q1 = 42, the median = 52 and Q3 = 65 minutes. The quickest student took 6 minutes and the slowest took 97. Drag each of the five values to its place on the scale.

Low — Lowest valueQ1 — Lower quartilePlace MedianQ3 — Upper quartileHigh — Highest value

Predict, then check

Watch what one extreme value does to each measure of spread.

A school bus takes 12, 14, 15, 17, 18, 20 and 21 minutes on seven days. Then one day it breaks down, and the 21-minute trip becomes 60 minutes. The other six times stay the same. What happens to the range and the IQR?

Now write it yourself

Compare two brands of phone

Box plots were drawn for the battery life, in hours, of phones from two brands. Their five values are: Brand P: lowest 9, Q1 12, median 14, Q3 15, highest 19. Brand Q: lowest 7, Q1 11, median 15, Q3 19, highest 22.

Compare the battery life of the phones from the two brands. [2 marks]

Your answer

0 words · your answer stays on this page and is not sent anywhere.

The mark scheme

Compares the medians AND says what that means: Brand Q's median (15 hours) is higher than Brand P's (14 hours), so Brand Q's phones usually last longer. (1 mark)
Compares the IQRs AND says what that means: Brand P's IQR is 15 − 12 = 3 hours and Brand Q's is 19 − 11 = 8 hours, so the middle half of Brand P's battery lives is more consistent. (1 mark)

Model answer

On average, Brand Q's phones last longer: its median battery life is 15 hours, compared with 14 hours for Brand P. But Brand P is more consistent: its IQR is 15 − 12 = 3 hours, compared with 19 − 11 = 8 hours for Brand Q, so the middle half of Brand P's phones have battery lives much closer together.

One sentence about the medians, one about the IQRs, and each says what it means for the phones. 'Brand Q's median is bigger' on its own isn't a comparison yet: it doesn't say what the data was about.

WHAT YOU'VE LEARNED

A quick recap of today's lesson.

IQR = Q3 − Q1

The box is the middle half of the data. How wide it is tells you how consistent that half is.

What you need to know

How to find the median, the lower quartile (Q1) and the upper quartile (Q3) from a list of data.
How to read Q1, the median and Q3 off a cumulative frequency graph.
How to draw a box plot from five values, and why the width of its box is the inter-quartile range.
How to compare two box plots using the median and the IQR, in the context of the data.

The big picture

Quartiles cut ordered data into quarters. Q1 has a quarter of the data below it and Q3 has three quarters below it, so the inter-quartile range, Q3 − Q1, is the spread of the middle half. A box plot draws five values: the lowest, Q1, the median, Q3 and the highest. The box is the middle half, so its width is the IQR. To compare two data sets, compare the medians (which is typically higher) and the IQRs (which is more consistent), and say what each means for the data.

Key points

1Q1 has a quarter of the ordered data below it; Q3 has three quarters below it.

2From a list: order the values, find the median, then Q1 and Q3 are the medians of the left and right halves. Two values in the middle? Go halfway between them.

3On a cumulative frequency graph, read Q1 at a quarter of the total frequency, the median at a half and Q3 at three quarters.

4IQR = Q3 − Q1: the spread of the middle 50% of the data. It is the width of the box on a box plot.

5A box plot uses five values: lowest, Q1, median, Q3 and highest.

6A smaller IQR (a narrower box) means the middle half of the data is more consistent. Extreme values change the range but not the IQR.

7To compare, say which median is higher and which IQR is smaller, and what each means for the data.

.we-_R_18slubsnpfiv5ubrb_ [data-we-step]{display:flex !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-solution]{display:block !important}.we-_R_18slubsnpfiv5ubrb_ [data-we-reveal]{display:none !important}

Worked example

Problem

Ten people counted the texts they sent in one day: 8, 15, 3, 22, 11, 9, 30, 14, 6, 17. Find the median and the inter-quartile range.

⚠ Watch out

Using the range to judge consistency. One extreme value can make the range huge while the middle half of the data hasn't changed at all. Judge consistency by the IQR, the width of the box, not by the length of the whiskers.

🧠

Memory hook

Line in the box: typical. Width of the box: consistent. The whiskers only tell you where the extremes are.

✓

Check yourself

The IQR of a data set is 15 and its upper quartile is 32. What is its lower quartile? (Check: Q1 = Q3 − IQR = 32 − 15 = 17.)

Flashcards

(14)

What is the lower quartile (Q1)?

The value with a quarter of the ordered data below it.

What is the upper quartile (Q3)?

The value with three quarters of the ordered data below it.

What must you do to a list before finding its median or quartiles?

Put the values in order.

Once the median is found, where do Q1 and Q3 come from?

Q1 is the median of the lower half; Q3 is the median of the upper half.

The two central values are 4 and 6. What is the median?

5: go halfway between them, (4 + 6) ÷ 2.

Where do you read Q1 and Q3 on a cumulative frequency graph?

Q1 at a quarter of the total frequency, Q3 at three quarters (the median at a half).

How do you calculate the inter-quartile range?

Take the lower quartile away from the upper quartile: IQR = Q3 − Q1.

What does the IQR measure?

How spread out the middle 50% of the data is.

Which five values do you need to draw a box plot?

Lowest value, lower quartile, median, upper quartile, highest value.

On a box plot, what shows the IQR?

The width of the box, which covers the middle half of the data.

What does a narrower box (smaller IQR) tell you?

The middle half of the data is more consistent.

Why doesn't an extreme value change the IQR?

The IQR only looks at the central half of the data, so unusually high or low values don't affect it.

Should an outlier be left out of the range?

No. At this stage, outliers are included in the range.

What two things should a comparison of two box plots include?

A comparison of the medians and a comparison of the IQRs, each saying what it means in the context of the data.

Tap any card to flip it, or use Study as deck to go through them one at a time. In the full lesson these run as a spaced-repetition deck — you rate each card Hard, Good or Easy and the tricky ones keep coming back until they stick.

Learning with Lightbulb is opening soon

You can use this lesson now. Join the waitlist and we'll let you know when the full Lightbulb experience is ready.

Keep me posted

Which pizza shop can you rely on?

Find the quartiles yourself

Where the box comes from: a cumulative frequency graph

Five values, five lines

Compare two brands of phone

More AQA GCSE Maths topics