GCSE Statistics common mistakes
GCSE Statistics · Understand the traps and how to avoid them.
When asked to design a questionnaire item, the candidate writes a leading or loaded question (e.g. 'Don't you agree that homework is too heavy?') and supplies response boxes that are not mutually exclusive or not exhaustive (e.g. '0-5 hours', '5-10 hours', 'more than 10 hours'), so a respondent doing exactly 5 hours has two valid boxes.
Attention goes entirely into making the topic sound plausible; the candidate never tests the item against the different answers a real respondent could give, and everyday phrasing ('don't you agree') reads as natural rather than biased.
What to do instead
Before finalising any question, run two explicit checks and write them in the answer: (i) neutrality - remove any wording that suggests a preferred answer and avoid emotive words; (ii) coverage - list three extreme or awkward possible answers and confirm each fits exactly one, non-overlapping box (use 'under 5', '5 to under 10', etc.). Mention that the questionnaire would be piloted on a small group to catch problems.
Draws a histogram for grouped data with unequal class widths by making each bar's height equal to its frequency, so a class covering 20 units looks twice as important as a class covering 10 units with the same frequency.
The candidate transfers the bar-chart habit of 'height = count' straight across and never asks what the vertical axis of a histogram measures; the unequal widths are treated as decoration rather than as information that must be compensated for.
What to do instead
Add an extra column headed 'frequency density' to the table for every class before drawing anything, computed as frequency divided by class width. Label the vertical axis 'frequency density' (never 'frequency') and check the finished diagram by confirming that the area of each bar equals its class frequency.
Plots a cumulative frequency curve at class midpoints rather than upper class boundaries, and/or reads the median by going from n/2 on the vertical axis across to the curve but then taking the value from the wrong axis or forgetting to read down.
Midpoints have just been used for estimating the mean from a grouped table, so the candidate reuses them mechanically; plotting at boundaries also requires starting with a cumulative frequency of zero at the lower boundary of the first class, which is a step candidates skip.
What to do instead
Write the cumulative frequency table with a first row of 'lower boundary of first class, cf = 0', then accumulate at each upper class boundary and plot only those points. For the median, trace n/2 horizontally to the curve, drop vertically to the horizontal axis, and state the answer with units in the context of the data; use the same route for quartiles at n/4 and 3n/4 before finding the interquartile range.
Compares two distributions by quoting a single average ('the mean of A is 12.4 and the mean of B is 11.9, so A is higher') and concludes one set is better or more consistent without quoting any measure of spread or checking for outliers.
'Average' has been learned as the summary of a data set, and consistency is a vague everyday idea rather than something the candidate knows must be quantified; the question's instruction to 'compare' is read as 'find a difference'.
What to do instead
Treat every comparison as a required pair: an average (mean or median, with a reason for the choice) plus a measure of spread (range, interquartile range or standard deviation). Write the comparison as two sentences - one about the typical value, one about the spread - and finish with the interpretation in context (e.g. which shop is more consistent and why that matters).
Computes the variance correctly but reports that number as the standard deviation, so a data set measured in centimetres is summarised by a value in squared centimetres (e.g. reporting 36 instead of 6).
The calculation is a long chain of steps - mean, deviations, squares, sum, divide - and the final square root is a separate line that gets dropped once the candidate feels the arithmetic is finished; the units give no warning because the intermediate values are never labelled.
What to do instead
Write the definition on the answer line before substituting: standard deviation = square root of (sum of squared deviations divided by n). Check the units of every quantity as you go - squared deviations carry squared units, so the answer must be square-rooted to return to the original units - and sanity-check that the standard deviation is a sensible fraction of the range rather than a number comparable to the variance.
Describes a scatter diagram only as 'positive correlation' or 'strong correlation' with no reference to the variables, or asserts that changing one variable will cause the other to change (e.g. 'revision causes better marks').
The vocabulary of correlation is memorised as a label to be attached to the graph, and the everyday assumption that a tidy pattern implies cause and effect fills the gap; the candidate does not realise the description must name the variables and comment on the form of the pattern.
What to do instead
Describe every scatter diagram with a fixed sentence pattern that names the variables and covers three features: 'as [x variable] increases, [y variable] tends to increase/decrease, the relationship is strong/moderate/weak, and it is approximately linear/curved'. State separately that correlation does not establish causation and give a plausible lurking variable that could explain the link.
Draws the line of best fit by joining the first and last plotted points (or forcing it through the origin because the axes meet there), then uses it to predict a value well outside the range of the plotted data.
The candidate treats the line as a way of connecting dots, which works on a distance-time graph, and does not appreciate that a line of best fit summarises the whole cloud of points; the origin is visually compelling and the axis scales are not checked.
What to do instead
Before drawing, compute the mean of the x values and the mean of the y values, mark that point, and draw a straight line through it with roughly equal numbers of points above and below along its whole length. Then state that predictions are only reliable between the smallest and largest plotted x values - interpolate, and explicitly refuse (in words) to extrapolate beyond the data.
Finds a four-point moving average by averaging each block of four consecutive values and plotting the result against the first (or last) quarter of the block, rather than placing it halfway between the two middle periods.
With an odd number of points the moving average genuinely sits on a data point, so the even case looks like the same procedure; the need to centre arises only because there is no middle period, and that is a conceptual point rather than a memorised step.
What to do instead
For an even period, average the moving averages of two consecutive blocks and plot the result at the midpoint of the periods covered (for a block of quarters 1-4 the first plotted point is at 2.5). Label the plotted points and remember the purpose: the moving average smooths out seasonal variation so the underlying trend is revealed, so assess the trend separately from the seasonal effect.
On a tree diagram for drawing items without replacement, keeps the same denominator on the second set of branches (e.g. 4/10 followed by 4/10) or keeps the same numerator for the 'success' branch even after one success has been taken.
The first pair of branches is set up carefully and then the diagram is treated as a template to copy; the phrase 'without replacement' is read as a detail rather than as an instruction that the composition of the remaining population has changed.
What to do instead
Above each pair of branches write the composition in words - 'first pick: 6 red, 4 blue out of 10' then 'second pick: 5 red, 4 blue out of 9'. Then check two things on the finished diagram: the fractions on each pair of branches sum to 1, and the denominators decrease by one at every stage when there is no replacement.
Computes the probability that a randomly selected person has property A or property B by adding the two probabilities without subtracting the overlap, even though the same person can have both properties (e.g. is a girl and studies French).
'Or means add' is a rule from mutually exclusive examples such as dice scores, and the candidate does not test whether the events being combined can occur together; the assumption of independence is likewise made silently when multiplicating.
What to do instead
Before adding, test for overlap by listing the outcomes, drawing a Venn diagram or building a two-way table. Use P(A or B) = P(A) + P(B) - P(A and B), and state explicitly that the subtraction is needed because the overlap has been counted twice; note that it vanishes only when the events are mutually exclusive. Never multiply to combine events unless independence is stated in the question or can be justified.