Key Stage 4 maths · Statistics

GCSE statistics: sampling, time series and scatter graphs

GCSE statistics is about drawing sensible conclusions from data and knowing how far to trust them. Children infer things about a whole population from a sample, read and draw time series graphs, compare data sets using graphs, averages and spread, and use scatter graphs to spot correlation, draw lines of best fit and make predictions. The theme throughout is that a statistic is only as good as the way the data was collected.

Sampling, trends and correlation, explained

A population is everything you want to know about, such as every student in a school, and a sample is the part that you actually collect data from. Children learn that a sample should be random and large enough to represent the population, and that a sample which is easy to collect can be biased, such as asking only the people on one bus. Some schools also teach stratified sampling, where the sample includes each group in proportion to its size in the population. Whatever the method, an estimate for the population always comes with limitations: another sample would give a slightly different answer.

A time series is data recorded at regular times, such as monthly sales or daily temperature. Children construct tables and line graphs for time series and interpret them by describing the overall trend (rising, falling or level) and any repeating pattern, such as a rise every summer. The points are usually joined with straight lines to make the pattern over time easier to see, although values between the plotted points are not actual readings. Some schools also teach moving averages to smooth out short-term ups and downs so that the trend is easier to see.

To compare distributions, children draw and read suitable graphs for discrete, continuous and grouped data, then summarise each data set with an average and a measure of spread. The mean, median and mode are averages, the modal class is the most common group in grouped data, and the range measures spread. When data is grouped, the exact values are lost, so the mean can only be estimated by using the midpoint of each class. A good comparison uses one average and one measure of spread, and says what they mean in context, for example “the median time for Class A is lower, so students there were generally quicker, but the range is bigger, so they were less consistent”. Children also apply these statistics to describe a population, for example using a sample’s median and range to say what is typical of the whole year group and how much it varies.

A scatter graph plots two measurements for each person or object, such as height and arm span. Children describe correlation as positive (as one rises, so does the other), negative or none, and strong or weak depending on how closely the points follow a line. They know that correlation does not show causation. A line of best fit is drawn by eye to follow the overall trend with roughly equal numbers of points either side, and it is used to make predictions. Reading values inside the range of the data is interpolation and is fairly safe. Going outside the range is extrapolation, which can be badly wrong because the pattern may not continue.

Words your child will hear

Population: The whole group you want to find out about.
Sample: A part of the population that data is actually collected from.
Bias: A built-in unfairness in how data is collected that makes a sample unrepresentative.
Time series: Data collected at regular intervals over time, shown on a line graph.
Modal class: The group with the highest frequency in a grouped table.
Correlation: A link between two variables: positive, negative or none, strong or weak.
Interpolation and extrapolation: Predicting inside the range of the data (reasonably safe) or outside it (risky).

What the National Curriculum says

In Key Stage 4, pupils should be taught to:

Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling
Interpret and construct tables and line graphs for time series data
Interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate graphical representation involving discrete, continuous and grouped data
Interpret, analyse and compare the distributions of data sets from univariate empirical distributions through appropriate measures of central tendency (including modal class) and spread
Apply statistics to describe a population
Use and interpret scatter graphs of bivariate data; recognise correlation and know that it does not indicate causation; draw estimated lines of best fit; make predictions; interpolate and extrapolate apparent trends whilst knowing the dangers of so doing

Objectives from the National Curriculum in England, © Crown copyright, used under the Open Government Licence v3.0.

A worked example

25 students recorded how long they spent on homework one evening. Find the modal class, the class containing the median, and an estimate of the mean.

Set out the grouped table. Add a column for the midpoint of each class (the number halfway between its ends) and another for frequency × midpoint.

Time t (min)    f   Mid   f × Mid
0 ≤ t < 20      4    10        40
20 ≤ t < 40    10    30       300
40 ≤ t < 60     8    50       400
60 ≤ t < 80     3    70       210
Total          25             950

Modal class: the highest frequency is 10, so the modal class is 20 ≤ t < 40.
Median class: with 25 values, the median is the 13th. The first class holds 4 values and the first two together hold 14, so the 13th lies in 20 ≤ t < 40.
Estimated mean: add up the f × midpoint column and divide by the total frequency. 950 ÷ 25 = 38.
Say why it is only an estimate. We assumed every student in a class was exactly at the midpoint, but the real times are not known.

Modal class 20 ≤ t < 40; median in the class 20 ≤ t < 40; estimated mean 38 minutes.

Common mistakes and how to help

Believing that correlation means one thing causes the other

When two graphs rise together, a cause feels obvious. Ice cream sales and sunburn cases rise together, but both are driven by hot weather.

What helps: Ask “what else could explain this?” A third factor, or coincidence, can produce correlation without cause. Strong correlation is evidence that two things are linked, not that one causes the other.

Drawing the line of best fit from the first point to the last, or trusting it everywhere

Joining the end points seems tidy. A line of best fit should follow the trend of all the points, with a similar number above and below it. Extending it far beyond the data assumes the pattern never changes, which is rarely true: a child’s height can’t be predicted at age 40 from a line drawn from ages 5 to 15.

What helps: Use a clear ruler and look at where the bulk of the points go. Predictions within the data are interpolation. For predictions outside the data, say that it may be unreliable.

Treating the estimated mean of grouped data as exact, or using the class boundaries instead of the midpoint

Having a single number from a calculation feels like having a precise answer. In grouped data, the original values are lost, so the answer depends on the assumption that the values sit at the middle.

What helps: Always label the answer “estimate”. Write the midpoint column first, since using the lower or upper boundary would push the mean consistently too low or too high.

Thinking that a bigger sample is always a good sample

A large survey sounds convincing. But if everyone surveyed came from the same place, such as a gym or a school gate, the results can still be biased however many people are asked.

What helps: Ask who was left out, and how people were chosen. A smaller random sample can be more trustworthy than a huge sample of volunteers.

How to help at home

Make a scatter graph from the family. Measure the hand span and the height of everyone in the house, plot the pairs, and draw a line of best fit. Does it predict a visiting friend’s height? Discuss why predicting for a toddler or a giant would be unreliable.
Track something for two weeks. Record the daily temperature, steps walked or minutes of reading, and plot a line graph. Describe the trend and any pattern across the days of the week. This is the time series skill in practice.
Question a headline. When a news story quotes a survey, ask how many people were asked, how they were chosen and who was left out. Spotting a biased sample is a skill GCSE questions reward and daily life needs.
Compare two sets of times. Time how long each of you takes to do something simple, such as tying shoelaces or typing a sentence, ten times each. Work out the median and range for each person and write one sentence comparing the two in context.

Builds on

Key Stage 3: Averages, charts and scatter graphs