Maths · 10 Statistics

10.2 Interpreting statistical data

Before you start

Finding the mean, median, mode and range of a short list of numbers (calculated in full in 10.4).
Reading a frequency table and a two-way table, and checking that rows and columns add up (10.1).
Writing one number as a fraction or a percentage of another.
Reading values from bar charts, pie charts and stem-and-leaf diagrams (drawn in full in 10.6).

Syllabus objectives covered (3)

Read, interpret and draw inferences from tables and statistical diagrams.
Compare sets of data using tables, graphs and statistical measures.
Appreciate restrictions on drawing conclusions from given data.

Part 1 of 7

Read the frame before you read the numbers

Every table and chart has a frame: a title, labels, units, a scale and sometimes a key. Read all of it before you read a single value. The frame tells you what each number actually means.
On a bar chart or line graph, work out the scale first. If five small squares cover a step of $10$ on the axis, each small square is worth $2$ , not $1$ .
Check what the vertical axis counts. An axis labelled 'Frequency' gives numbers of people or items. An axis labelled 'Percentage' gives shares of a total. They answer different questions.
A stem-and-leaf diagram is useless without its key. Here the key says $1 | 4$ means $14$ minutes, so the first row holds $14$ , $17$ and $18$ .
$\begin{matrix} 1 & 4 7 8 \\ 2 & 0 3 5 5 9 \\ 3 & 1 6 \end{matrix} Key: 1 | 4 means 14 minutes$
A two-way table sorts each item by two things at once, for example by gender and by favourite activity. Each inside cell is the count for one combination. The row totals, column totals and grand total must all agree.
Use those totals to fill gaps. A missing cell is the row total minus the other cells in that row. Then check it against its column total too, because one check by row and one by column catches an arithmetic slip.
Watch the phrase 'of the …'. 'What fraction of the girls chose music?' makes the girls' total the denominator. 'What fraction of the music students were girls?' makes the music total the denominator. Same cell, different whole.

Part 2 of 7

From one value to an inference

There are three levels here. Reading is taking one value off the table or chart. Interpreting is saying what that value means in context. Inferring is drawing a conclusion about the whole group that the data supports.
Take the stem-and-leaf diagram above. Reading: the longest time is $36$ minutes. Interpreting: one person took $36$ minutes. Inferring: $7$ of the $10$ times are $20$ minutes or more, so most of this group took at least $20$ minutes.
An inference must point to its evidence. 'Most took at least $20$ minutes' is only worth writing with ' $7$ out of $10$ ' beside it. A claim with no number behind it is an opinion.
On a pie chart, the angle is the share of the total. A sector's fraction of the whole is its angle divided by $360 °$ . To turn that into a number of people, multiply by the total.
$number in sector = \frac{angle}{360 °} \times total$
On a line graph over time, describe the trend first: the overall direction across the whole period. Then mention any clear exception, such as one month that breaks the pattern. Do not list every point.
Keep your inference inside what was measured. If the data are for one class, your conclusion is about that class. Stretching it to 'all students' is a restriction you will meet in a later section.

Part 3 of 7

Averages and spread answer two different questions

To compare two sets of data you need two kinds of measure. An average tells you what a typical value is. A measure of spread tells you how much the values vary.
The averages: the mean is the total divided by how many values there are. The median is the middle value once the data are in order. The mode is the value that occurs most often.
The measures of spread: the range is the largest value minus the smallest. The interquartile range (IQR) is the upper quartile minus the lower quartile, which is the spread of the middle half of the data.
$range = largest - smallest IQR = upper quartile - lower quartile$
A small spread means the values are close together, so the data are consistent. A large spread means they are more varied. Spread says nothing about whether the values are high or low.
Extreme values matter. In $4, 5, 5, 6, 30$ the mean is $50 \div 5 = 10$ , but the median is $5$ . Four of the five values are $6$ or less, so the median describes this set far better than the mean does.
The same is true of spread. One extreme value stretches the range, because the range uses only the two end values. The IQR ignores the top and bottom quarters, so it is not affected.
So when a data set has an extreme value, the median and the IQR give a fairer picture than the mean and the range. You will calculate all of these in full in 10.4; here the skill is knowing what each one tells you.

Part 4 of 7

How to write a comparison that earns the marks

When you are asked to compare two data sets, write two separate sentences: one comparing an average, one comparing a spread. Each sentence earns its own mark.
Each sentence must actually compare. 'Group A has a median of $44$ . Group B has a median of $51$ .' is two facts, not a comparison. Join them with a comparing word: higher, lower, larger, smaller, more consistent.
Each sentence must say what the difference means in context. A lower median time means the group was generally faster. A lower median mark means the group generally did worse. The same word 'lower' means opposite things, so think about what is being measured.
Compare like with like: median with median, IQR with IQR, mean with mean, range with range. Comparing one group's mean with the other group's median tells you nothing.
Here is a pattern you can follow.
$\begin{matrix} A’s median (44 s) is lower than B’s (51 s), so A were generally faster. \\ A’s range (36 s) is larger than B’s (20 s), so A’s times were less consistent. \end{matrix}$
The trap is the word 'better'. A larger spread is not better or worse in itself; it means less consistent. Only say which group did better when the question tells you what counts as better, and base it on the average.
Quote the values you are comparing. They show where your conclusion came from, and if your values came from an earlier part with an error, a correct comparison of them can still earn the follow-through mark.

Part 5 of 7

Comparing groups of different sizes

If two groups are different sizes, comparing raw counts can mislead you. $30$ girls is more than $25$ boys, but if there are $70$ girls and only $50$ boys, $30$ girls is the smaller share.
So convert each count into a fraction or percentage of its own group, then compare those.
$boys: \frac{25}{50} = 50 % girls: \frac{30}{70} = 42.9 %$
Pie charts show shares, not numbers. A bigger sector means a bigger share of that chart's total. If two pie charts have different totals, the bigger sector can stand for fewer people.
graph diagram — not drawn yet
Two pie charts side by side, each split into sectors labelled Walk, Bus and Car. Left chart titled 'School A (200 students)': Walk 90°, Bus 144°, Car 126°. Right chart titled 'School B (600 students)': Walk 60°, Bus 210°, Car 90°. Angles marked at the centre of each sector.
School A's Walk sector is larger, yet 50 walk to School A and 100 walk to School B.
In that example, School A: $\frac{90}{360} \times 200 = 50$ walk. School B: $\frac{60}{360} \times 600 = 100$ walk. A bigger share at School A, but twice as many walkers at School B.
Bar charts give a choice. A dual bar chart puts the groups' bars side by side, which compares counts category by category. A percentage composite bar chart makes each group's bar $100 %$ , which compares shares even when the groups differ in size.

Part 6 of 7

What the data cannot tell you

Data describe the items that were measured. Every conclusion beyond them carries a restriction, and the objective asks you to recognise those restrictions.
Sample size. A conclusion from a few people may not hold for the whole population. Asking $10$ people out of a town of $20 000$ gives very little evidence about the town.
Who was asked. A sample is biased if the way it was chosen favours some answers. Asking people at a gym how often they exercise is likely to give higher answers than the town as a whole. A larger sample chosen the same way is still biased.
When and where. Data collected on one day, in one place or in one season may not describe other days, places or seasons.
Beyond the range. If data cover ages $10$ to $16$ , a pattern in them says nothing reliable about age $30$ . Predicting outside the data is guesswork.
An average is about the group. A class mean of $62$ does not mean every student scored $62$ , and it cannot tell you how one particular student did.
Small differences. If two averages differ by a little and both data sets are widely spread, a different week or a different sample could easily reverse the order. Say so before you crown a winner.
Grouped data lose detail. In a grouped frequency table you know only which class each value is in, not the value itself. So the mean is an estimate, and the exact range cannot be found.
Linked is not caused. Two quantities can rise and fall together because both depend on something else. Ice-cream sales and cases of sunburn both rise in sunny weather; neither causes the other.

Part 7 of 7

When the diagram itself misleads

A chart can be accurate in every value and still give the wrong impression. Reading the frame, from the first section, is your defence.
The first feature to check is a vertical axis that does not start at zero. The bars then show only the part above the starting value, so small differences look large.
graph diagram — not drawn yet
Two bar charts of the same data side by side: weekly visitors to Museum A (820) and Museum B (860). Left chart has a vertical axis from 0 to 900; the two bars look almost equal. Right chart has a vertical axis from 800 to 880; Museum B's bar is three times the height of Museum A's.
Same data. Starting the axis at 800 makes a difference of under 5% look like three times as many.
With the axis starting at $800$ , Museum A's bar is $20$ units tall and Museum B's is $60$ . B's bar looks three times as tall, yet $\frac{860 - 820}{820} \times 100 = 4.9 %$ , so B had less than $5 %$ more visitors.
Other things to check: pictograms where symbols are different sizes, a missing scale, a gap in the time axis, and percentages given without the totals they are percentages of.
When you are asked why a diagram is misleading, name the feature and say what impression it creates, for example: 'the axis starts at $800$ , so the difference between the bars looks much bigger than it is'.

Worked examples

Example 1. The two-way table shows the after-school activity chosen by $120$ students. $\begin{matrix} Sport & Music & Drama & Total \\ Boys & 25 & 12 & 50 \\ Girls & 22 & 18 \\ Total & 55 & 120 \end{matrix}$ (a) Complete the table. (b) Sam says, 'Sport is more popular with girls than with boys, because $30$ girls chose sport and only $25$ boys did.' Comment on Sam's claim.

Show the working

Start with the gap that has everything else in its row. Boys who chose drama: the boys' total minus the other boys.
$50 - 25 - 12 = 13$
The girls' total is the grand total minus the boys' total.
$120 - 50 = 70$
Girls who chose sport: the sport column total minus the boys in that column.
$55 - 25 = 30$
Check the girls' row by adding across. It must give the girls' total.
$30 + 22 + 18 = 70 ✓$
Now the remaining column totals, and a final check that they add to $120$ .
$12 + 22 = 34, 13 + 18 = 31, 55 + 34 + 31 = 120 ✓$
For (b), notice the groups are different sizes: $50$ boys and $70$ girls. Comparing $30$ with $25$ is unfair, so find the share of each group that chose sport.
$boys: \frac{25}{50} \times 100 = 50 % girls: \frac{30}{70} \times 100 = 42.9 %$
A greater proportion of boys chose sport. Sam's claim is not supported: he compared counts from groups of different sizes.

Answer: (a) Boys drama $13$ ; girls sport $30$ ; girls total $70$ ; music total $34$ ; drama total $31$ . (b) Sam is wrong: $50 %$ of boys chose sport compared with $42.9 %$ of girls, so sport is more popular with boys. He compared counts from groups of different sizes.

Example 2. Two basketball players record the points they score in six games. Amira: $12, 15, 14, 18, 11, 14$ . Bella: $6, 25, 9, 22, 10, 18$ . (a) Calculate the mean and the range for each player. (b) Compare the two players' scores. (c) Comment on whether these data show that Bella is the better scorer.

Show the working

Amira's mean: add the scores and divide by $6$ .
$\frac{12 + 15 + 14 + 18 + 11 + 14}{6} = \frac{84}{6} = 14$
Amira's range: largest minus smallest.
$18 - 11 = 7$
Bella's mean and range, the same way.
$\frac{6 + 25 + 9 + 22 + 10 + 18}{6} = \frac{90}{6} = 15, 25 - 6 = 19$
For (b), write one sentence on the average and one on the spread, each comparing and each in context: 'Bella's mean ( $15$ points) is higher than Amira's ( $14$ points), so on average Bella scores more.'
Then the spread: 'Amira's range ( $7$ points) is much smaller than Bella's ( $19$ points), so Amira's scores are more consistent.'
For (c), look at the size of the difference against the spread. The means differ by only $1$ point, Bella's scores swing by $19$ points, and there are only six games. A single extra game of $6$ points would bring Bella's mean down to $96 \div 7 = 13.7$ , below Amira's $14$ .
So the data do not show that Bella is the better scorer. They show a higher average and much less consistency.

Answer: (a) Amira: mean $14$ , range $7$ . Bella: mean $15$ , range $19$ . (b) Bella's mean is higher, so on average she scores more; Amira's range is smaller, so her scores are more consistent. (c) No: the means differ by only $1$ point over six games, and Bella's scores vary widely, so the difference could easily reverse.

Example 3. A pie chart for School A ( $200$ students) shows a $90 °$ sector for students who walk to school. A pie chart for School B ( $600$ students) shows a $60 °$ sector for students who walk. Jo says, 'More students walk to School A, because its sector is bigger.' Determine whether Jo is right.

Show the working

A sector's angle out of $360 °$ is the fraction of that school's students. Multiply by the school's total to get a number of students.
School A.
$\frac{90}{360} \times 200 = 50$
School B.
$\frac{60}{360} \times 600 = 100$
School A's sector is bigger because a larger share of its students walk: $25 %$ against about $16.7 %$ . But School B is three times the size, so more students walk there.
Jo has read a share as if it were a number. A pie chart can compare shares between schools, but not numbers, unless you use each chart's total.

Answer: Jo is wrong: $50$ students walk to School A and $100$ walk to School B. The bigger sector means a bigger share ( $25 %$ against $16.7 %$ ), not a bigger number.

Example 4. A cycling magazine's website asks visitors to vote for their favourite sport. Of the $400$ votes, $85 %$ are for cycling. The editor writes, ' $85 %$ of people in the country prefer cycling to any other sport.' Calculate the number of votes for cycling, and give two reasons why the editor's conclusion is not reliable.

Show the working

The number of votes for cycling.
$0.85 \times 400 = 340$
Now ask who gave the data. Visitors to a cycling magazine's website are likely to be interested in cycling, so the sample does not represent the people of the country. It is biased.
Then ask how they were chosen. The voters chose themselves: only people who visited the site and wanted to vote took part. Nobody selected a fair cross-section.
Notice that more votes would not fix this. $4000$ votes from the same website would be biased in the same way. Size and bias are separate problems.

Answer: $340$ votes. The conclusion is not reliable because (1) visitors to a cycling website are likely to favour cycling, so the sample is biased and not representative of the country; (2) the voters selected themselves rather than being chosen to represent the whole population.

Where people go wrong

Mistake: Comparing raw counts from two groups of different sizes.

Instead: Turn each count into a fraction or percentage of its own group's total, then compare.

A count depends on how big the group is. $30$ out of $70$ is a smaller share than $25$ out of $50$ .

Mistake: Writing two separate facts and calling it a comparison: 'A's median is $44$ . B's median is $51$ .'

Instead: Use a comparing word and say what it means: 'A's median is lower than B's, so A were generally faster.'

A comparison states how the two sets differ and what that difference means in context. Listing values leaves that work undone.

Mistake: Saying a larger range or IQR means a group did better, or scored higher.

Instead: Spread measures consistency. A larger spread means the values are more varied; use the average to say which is higher.

A group with low marks can have a large spread, and a group with high marks can have a small one. Spread and level are independent.

Mistake: Reading a bigger pie-chart sector as a bigger number when the charts have different totals.

Instead: Multiply each sector's fraction by that chart's own total before comparing numbers.

A sector shows a share of its own chart's total. Different totals mean the same angle stands for different numbers of people.

Mistake: Treating a conclusion from a small or biased sample as true for everyone.

Instead: State the restriction: who was asked, how many, and when or where. Then limit the conclusion to what the sample can support.

A sample only represents a population if it is large enough and chosen fairly. A biased sample stays biased however many people are in it.

Check yourself

In a stem-and-leaf diagram the key is $4 | 2$ means $42$ . The row with stem $3$ has leaves $1 5 8$ . Write down the three values in that row.
Answer
$31$ , $35$ and $38$ .
A pie chart shows the drinks chosen by $72$ people. The sector for tea has an angle of $100 °$ . How many people chose tea?
Answer
$\frac{100}{360} \times 72 = 20$ people.
In Class P, $12$ of the $30$ students wear glasses. In Class Q, $10$ of the $20$ students wear glasses. Which class has the greater proportion of students who wear glasses?
Answer
Class Q: $\frac{10}{20} = 50 %$ against $\frac{12}{30} = 40 %$ for Class P, even though Class P has more students who wear glasses.
Test marks for two classes: Class X has median $45$ and IQR $10$ ; Class Y has median $52$ and IQR $18$ . Write two sentences comparing the classes.
Answer
Class Y's median ( $52$ ) is higher than Class X's ( $45$ ), so Class Y generally scored higher. Class X's IQR ( $10$ ) is smaller than Class Y's ( $18$ ), so Class X's marks were more consistent.
For the data $3, 4, 4, 5, 39$ , find the mean and the median. Which better represents a typical value, and why?
Answer
Mean $= 55 \div 5 = 11$ ; median $= 4$ . The median is better: the extreme value $39$ pulls the mean above four of the five values.
Why can you not find the exact range of data given only in a grouped frequency table?
Answer
The table shows only which class each value falls in, not the values themselves, so the exact largest and smallest values are unknown.
A bar chart's vertical axis starts at $80$ . One bar reaches $85$ and another reaches $90$ . How many times as tall does the second bar look, and what is the true percentage difference?
Answer
The bars are $5$ and $10$ units tall, so the second looks twice as tall. In fact $\frac{90 - 85}{85} \times 100 = 5.9 %$ more.
In one town, ice-cream sales and the number of sunburn cases both rise in the same months. Does this show that eating ice cream causes sunburn?
Answer
No. Both rise in hot, sunny weather. Two quantities that change together are not proof that one causes the other.

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1. A cinema asked $160$ people which type of film they prefer. The results are shown in the two-way table. $\begin{matrix} Comedy & Action & Animation & Total \\ Under 18 & 15 & 21 & 60 \\ 18 and over & 47 & 15 \\ Total & 53 & 160 \end{matrix}$

(a) Complete the table. 2 marks

Mark scheme

B2 all five correct: 24, 38, 100, 68, 39 (B1 for any three correct)

(b) Calculate the percentage of the people who prefer animation that are under 18. 2 marks

Mark scheme

M1 24 / 39 [× 100] oe; FT their table
A1 61.5 awrt (%)

(c) Ravi says, 'Action films are more popular with people aged 18 and over than with people under 18, because 47 people aged 18 and over prefer action but only 21 people under 18 do.' Comment on Ravi's reasoning. 3 marks

Mark scheme

M1 21 / 60 or 47 / 100 seen, as a fraction, decimal or percentage; FT their table
A1 35% and 47% oe
B1 states that the groups are different sizes so the counts alone are not a fair comparison, and that the percentages still show action is more popular with people aged 18 and over

Question 2. Eleven students in Class A each solved a puzzle. Their times, in seconds, are $32, 35, 38, 40, 41, 44, 46, 47, 50, 53, 68$ . For Class B the median time was $51$ seconds and the range was $20$ seconds.

(a) Find the median and the range of the times for Class A. 2 marks No calculator

Mark scheme

B1 median 44 (s)
B1 range 36 (s)

(b) Compare the times of the two classes. 2 marks No calculator

Mark scheme

B1 Class A's median is lower, so Class A were generally faster oe; FT their median
B1 Class A's range is larger, so Class A's times were less consistent oe; FT their range

(c) Explain why the range may not be a good measure of the spread of Class A's times. 1 mark No calculator

Mark scheme

B1 68 is an extreme value, much larger than the other times, and the range depends on it oe [the interquartile range would not be affected]

Question 3. A town council wants to know whether residents support a new cycle lane. It asks $30$ people at the town's cycling club on a Saturday morning, and $27$ of them say yes. The council reports that $90 %$ of the town's residents support the cycle lane.

(a) Show that $90 %$ of the people asked said yes. 1 mark No calculator

Mark scheme

B1 27 / 30 × 100 = 90 seen, or 27 / 30 = 0.9 seen

(b) Give two reasons why the council's conclusion may not be reliable. 2 marks No calculator

Mark scheme

B1 any one of these reasons, tied to this survey: the sample is small (only 30 people for a whole town); the sample is biased (members of a cycling club are more likely to support a cycle lane); people were asked in only one place at one time (Saturday morning) oe
B1 a second, different reason from the list above oe

(c) Describe how the council could collect data that would give a more reliable conclusion. 2 marks No calculator

Mark scheme

B1 ask a larger number of people
B1 choose them at random from the whole town, or ask in different places and at different times oe

Question 4. A bar chart shows the number of bicycles sold in one year by two shops. The vertical axis starts at $400$ and goes up to $520$ . The bar for Shop X reaches $450$ and the bar for Shop Y reaches $500$ . The manager of Shop Y says, 'We sold twice as many bicycles as Shop X.'

(a) Calculate the percentage by which Shop Y's sales are greater than Shop X's. 2 marks

Mark scheme

M1 (500 − 450) / 450 [× 100] oe
A1 11.1 awrt (%)

(b) Explain why the bar chart makes the manager's statement look true. 1 mark

Mark scheme

B1 the axis starts at 400, so the bars show only 50 and 100, and Shop Y's bar is twice the height of Shop X's oe

How this is examined

Comparing two data sets with one average sentence and one spread sentence, both in context: A 2-mark Compare, after you have found or been given an average and a measure of spread for each group One sentence comparing the averages with a comparing word and its meaning ('A's median time is lower than B's, so A were generally faster'), and one comparing the spreads ('A's range is larger, so A's times were less consistent'). Each sentence is a separate mark; two quoted values with no comparison and no context do not do the job.
Turning counts into proportions before judging a claim about groups of different sizes: A two-way table to complete, then a Comment on someone's claim that compares raw counts A fraction or percentage of each group's own total (method), both values correct (accuracy), and a statement that the groups are different sizes, so the counts alone do not give a fair comparison.
Reading exact values from a table or diagram, using its key or scale: A 1- or 2-mark Write down or Find from a two-way table, stem-and-leaf diagram or pie chart The value read with its key applied ( $3 | 5$ read as $35$ ), missing cells found from row and column totals, or a pie-chart number found as angle $\div 360 \times$ total.
Saying why a conclusion from a survey is not reliable: A 2-mark Give or Explain: reasons why a conclusion may not be valid, which may be followed by how to collect better data Each reason tied to the survey in the question: what is wrong and why it matters ('only $30$ people were asked'; 'they were all at a cycling club, so they are more likely to support a cycle lane'). For an improvement: a larger sample, chosen at random from the whole population.
Explaining why a diagram gives a false impression: A 1-mark Explain, which may follow a calculation of the true difference Name the feature and its effect: 'the vertical axis does not start at zero, so the difference between the bars looks much bigger than it is.'

Words to know

Inference: A conclusion about a group that is drawn from data and supported by the values in it.
Two-way table: A table that sorts each item by two categories at once, with row and column totals.
Average: A single value that represents a typical member of a data set: the mean, the median or the mode.
Spread: How much the values in a data set vary. Measured by the range or the interquartile range.
Interquartile range (IQR): Upper quartile minus lower quartile: the spread of the middle half of the data. It is not affected by extreme values.
Extreme value: A value much larger or much smaller than the rest of the data.
Population and sample: The population is the whole group you want to know about. A sample is the part of it you actually collect data from.
Biased sample: A sample chosen in a way that favours some results, so it does not represent the population fairly.
Proportion: A part given as a fraction or percentage of its own whole.

All Maths lessons Next: 10.3 Discrete and continuous data