Statistics

A population is everyone or everything being studied; a sample is the part actually surveyed. Samples are used because studying a whole population is usually too slow and too expensive.

Inferring properties of a population from a sample only works if the sample is representative, and the specification asks you to know the limitations.

• A larger sample gives more reliable results — but costs more and takes longer • A biased sample gives conclusions that do not hold for the population. Surveying people outside a gym about exercise habits is the standard example • A random sample gives every member an equal chance of selection, which is what keeps it unbiased • However good the sample, a conclusion drawn from one is an estimate rather than a certainty

Questions often ask you to criticise a sampling method. Say what makes it unrepresentative and who has been left out or over-represented — a general comment that it is "too small" earns less than naming the bias.

For grouped data the individual values are unknown, so the mean is estimated: multiply each class midpoint by its frequency, add those products, and divide by the total frequency. The answer is an estimate because the midpoint stands in for values that could be anywhere in the class — and saying that is often worth a mark.

Comparing two data sets is a standard question, and it wants two things: a comparison of an average and a comparison of a measure of spread, both written in the context of the question. "The medians are 6 and 8" is not a comparison. "Group B's median is higher, so B generally scored more, and B's interquartile range is smaller, so B's scores were more consistent" is.

Each chart type suits a particular kind of data, and choosing correctly is examined:

• Bar chart — categorical data, with gaps between the bars • Pie chart — proportions of a whole. Each category's angle is its frequency divided by the total, times 360° • Pictogram — categorical data with a key giving the value of each symbol • Vertical line chart — ungrouped discrete numerical data • Line graph — time series data, showing change over time • Histogram — grouped continuous data, with no gaps between the bars

A cumulative frequency graph plots running totals against the upper bound of each class, giving an S-shaped curve. Read the median at half the total frequency, the lower quartile at a quarter and the upper quartile at three quarters — across from the vertical axis, then down. It also answers "how many were less than…" questions directly.

A box plot displays five values: minimum, lower quartile, median, upper quartile and maximum. Two box plots side by side are the quickest comparison there is, and the comparison should still name an average and a spread.

A histogram is for grouped continuous data with unequal class widths, and it is the diagram students most often get wrong. The vertical axis is frequency density, not frequency:

frequency density = frequency ÷ class width

which rearranges to frequency = frequency density × class width. So the frequency is the area of the bar, and reading it off the vertical axis gives the wrong answer every time.

A scatter graph plots two variables against each other to show whether they are related.

Correlation describes the pattern. Positive correlation means that as one goes up so does the other; negative means one rises as the other falls; and no correlation means the points show no pattern. Strong correlation has points close to a line, weak correlation has them scattered around it.

A line of best fit is drawn by eye, following the trend with roughly as many points either side. Use it to predict: read across from a known value and down to the other axis.

Two warnings the specification asks for explicitly.

Interpolation — predicting within the range of the data — is reasonably reliable. Extrapolation — predicting beyond the data — is not, because there is no evidence the trend continues outside the range measured. A question asking whether a prediction is reliable is nearly always asking you to spot an extrapolation.

Correlation does not imply causation. Two things can rise together because both depend on a third factor, or by coincidence. Ice cream sales and drowning incidents correlate strongly, because both rise in hot weather — neither causes the other. Saying that a correlation shows one thing causes another is the error the mark scheme is looking for.