Running a data investigation properly — samples and bias, grouping data into class intervals, drawing and reading histograms, choosing between the mean, median and mode when the data has outliers, and then interpreting the result: writing a conclusion, questioning a graph before you trust it, and making a prediction.
Also known as: histogram · frequency table · grouped data · data handling
Grade 9 data handling is less about calculating and more about judgement: is this sample fair, are these intervals sensible, and is the mean actually the right summary to use?
1Samples, populations and bias
The population is everyone you want to know about; the sample is the part you actually ask. A sample is only useful if it is representative.
- Random sample — everyone has an equal chance of being picked. Usually the fairest.
- Biased sample — some group is over-represented, so the results are misleading.
Surveying shoppers outside a sports shop about "how often do you exercise?" gives a badly biased answer — the location itself selects sporty people. Always ask where and who before trusting a statistic.
2Grouping data into class intervals
With many different values, group them into equal, non-overlapping class intervals such as 0–9, 10–19, 20–29. You lose the individual values but the pattern becomes visible.
Intervals must not overlap, and they should be equal in width. Writing 0–10, 10–20 is wrong — 10 belongs to both.
3Histograms
A histogram shows grouped numerical data. Its bars touch, because the classes are continuous. A bar graph shows separate categories, so its bars have gaps between them.
- 1The tallest bar is 20–29 with a frequency of 9.The modal class is the interval with the highest frequency.
- 2Total = 3 + 7 + 9 + 5 + 2.Add every frequency.
- 3Total = 26 values.
4Pie charts
A pie chart shows how a whole splits into parts. Each slice’s angle is that category’s share of the total, out of 360°.
Always check that your slice angles add up to 360° — it catches arithmetic slips instantly.
5Choosing the right average
- Mean — uses every value, but a single extreme value drags it badly.
- Median — the middle value; barely affected by outliers, so it is safer for skewed data.
- Mode — the most common value or category; of the three, the only one that still works for categories with no natural order, like favourite colour.
- 1Mean = 116 500 ÷ 5 = R23 300.But nobody earns near that — the R80 000 distorts it.
- 2Median = R9 500.The middle value once ordered.
- 3The median is the fairer summary here.One outlier has pulled the mean far above almost everyone.
If a data set has an obvious outlier, expect the exam to ask which average is more appropriate — the answer is nearly always the median, and you must say why.
6Double bar graphs: comparing two groups
The ATP lists bar graphs and double bar graphs together. A double bar graph carries two data sets on one pair of axes, which is what makes a comparison honest — both groups are read off the same scale.
- Always include a key; it carries its own mark.
- Compare within a pair to see the difference between the groups, and across pairs to see the trend.
- A double bar graph is for categorical data. For grouped continuous data use a histogram, whose bars touch throughout.
- 1Netball: 14 girls against 4 boys — a difference of 10, the largest.Compare the two bars in each pair.
- 2Soccer: 12 boys against 5 girls — a difference of 7.
- 3The graph does not tell you the total number of learners surveyed, or whether anyone plays two sports.Reading beyond what a graph shows is the classic data-handling error.
Comparing bar heights from two separate graphs that use different scales. That is how misleading statistics are made — a double bar graph exists precisely so both groups share one scale.
7Interpreting and reporting your data
Collecting the data and drawing the graph is only half of a data investigation. The ATP also asks you to analyse the data critically, draw a conclusion and make a prediction — and those are the marks most learners walk past.
8Writing a conclusion that earns the mark
A conclusion answers the question you started with and quotes a number from the data as evidence. “The boys did better” earns nothing. “The boys’ mean was 62% against the girls’ 55%, so the boys scored higher on this test” earns the mark.
- Answer the original question — not some other question the data also happens to answer.
- Quote a value from your table or graph as the evidence.
- Limit the claim to the data you actually have. Thirty learners in one class do not speak for the whole school.
- 1Compare with the numbers: 8,1 − 6,4 = 1,7 hours.The evidence is the difference, not an impression.
- 2Answer the question: the Grade 9s in this survey slept about 1,7 hours a night more than the Grade 12s.It answers what was asked.
- 3Limit it: this is based on 60 learners at one school, so it cannot be claimed for all Grade 9s and 12s.The sample is the limit of the claim — this is the mark most learners miss.
9Questioning data before you believe it
When you are handed someone else’s graph, the exam wants you to interrogate it before you accept what it says:
- Who was asked, and how many? A small or one-sided sample cannot represent everyone.
- How was the data collected? A questionnaire handed only to the choir tells you about the choir.
- Does the vertical axis start at zero? An axis starting at 40 makes a tiny difference look enormous.
- Are the class intervals equal? Unequal intervals make some bars look more important than they are.
- Who ran the survey, and do they want a particular answer?
Accepting a graph because it looks convincing. A graph can be perfectly correct and still mislead — the two commonest tricks are an axis that does not start at zero and a sample chosen to give the answer the writer wanted.
10Making a prediction
A prediction extends a pattern that is visible in the data. Say what the pattern is, use it, and then say what would make it stop being true.
- 1Find the pattern: 150 − 120 = 30, 180 − 150 = 30, 210 − 180 = 30.A constant difference of R30 a month.
- 2210 + 30 = R240Extend the pattern by one step.
- 3State the assumption: this assumes she keeps saving R30 more each month.A prediction without its assumption is only a guess.
- 4Note the limit: if her income or expenses change, the pattern breaks — and predicting two years ahead from four months is not safe.The further past the data you go, the weaker the prediction.
Every prediction in a data question should carry the words “if the trend continues”. It is often worth a mark on its own, and it is the difference between reading data and guessing.
Practice exercises
Work each one out, then click to reveal the answer.
- 1The mean of 45, 52, 61, 48, 54 and x is 52. Solve for x.
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(45 + 52 + 61 + 48 + 54 + x)/6 = 52 (1)
260 + x = 52 × 6 = 312 (1)
x = 312 − 260 = 52 (1) - 2What is the difference between a population and a sample?
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The population is the whole group; the sample is the smaller part actually surveyed. - 3Why is surveying only your friends about school food biased?
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They are not representative of the whole school. - 4What is wrong with the intervals 0–10, 10–20, 20–30?
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They overlap — 10 and 20 each fit two intervals. - 5How do a histogram and a bar graph differ?
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Histogram bars touch (continuous data); bar-graph bars have gaps (separate categories). - 6In the histogram above, which is the modal class?
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20–29, with a frequency of 9 - 7In the histogram above, how many values are there in total?
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3 + 7 + 9 + 5 + 2 = 26 - 8Find the mean of R8 000, R9 000, R9 500, R10 000 and R80 000.
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116 500 ÷ 5 = R23 300 - 9Find the median of that same salary data.
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R9 500 - 10Which average is fairer for that salary data, and why?
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The median — the R80 000 outlier pulls the mean far above what almost everyone earns. - 11Give TWO reasons why a double bar graph is better than two separate bar graphs for comparing boys and girls.
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Both groups share one scale so the comparison is fair ✓; the pairs sit side by side so differences can be read directly ✓ - 12The double bar graph below shows the sports played by a group of learners. State the difference between the number of girls and the number of boys who play netball, and one thing the graph does NOT tell you.
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Difference = 10 ✓; it does not show the total surveyed / whether learners play more than one sport ✓ - 13Explain how two separate bar graphs could be used to mislead a reader.
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If the two graphs use different vertical scales ✓, a smaller quantity can be drawn taller than a larger one, so the comparison looks the opposite of the truth ✓ - 14In a spelling test the boys’ mean was 71% and the girls’ mean was 78%. Write a conclusion that would earn full marks.
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The girls scored higher, with a mean of 78% against the boys’ 71% ✓ — a difference of 7 percentage points, for the learners in this test only ✓ - 15A shop’s sales were 48, 54, 60 and 66 items over four weeks. Predict week 5 and state the assumption you are making.
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The difference is a constant 6 ✓; 66 + 6 = 72 items ✓; this assumes the trend of +6 a week continues ✓ - 16A learner surveys 20 friends and reports that “most learners at the school play soccer”. Give TWO reasons to question this conclusion.
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The sample is only 20 learners, far too few for the whole school ✓; they are all the learner's friends, so they are not representative — friends tend to share interests ✓ - 17A bar graph’s vertical axis starts at 40 instead of 0. Explain how this could mislead a reader.
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The bars only show the part above 40, so small differences are stretched to look large ✓; a reader glancing at the heights would think one group is far bigger than another when the real gap is small ✓
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