Statistics
Cambridge O Level Mathematics (Syllabus D) 4024 Topic 9 revision chapter covering the whole of Statistics for the 2025-2027 syllabus, version 2. It teaches all seven official subtopics in order, each one taught, demonstrated with a fully worked example and then tested. Classifying and tabulating data establishes the three data types Cambridge assesses: categorical data made of labels or groups, discrete numerical data taking separate countable values, and continuous numerical data measured across an interval; it then builds tally tables, frequency tables, grouped frequency tables with non-overlapping class intervals, and two-way tables whose row totals, column totals and grand total must be checked from both directions before any conclusion is drawn. Interpreting statistical data teaches reading a table or chart properly by finding the title, the key, the units and the axis scale before extracting a single value, then drawing supported inferences, comparing two data sets with one measure of centre and one measure of spread together, and stating the interpretation in the context of the question rather than as a bare number. It teaches the restrictions that weaken a conclusion: a small or unrepresentative sample, missing values, inconsistent measurement, a leading survey question, a truncated or misleading vertical scale, extrapolation beyond the observed range, and the rule that correlation shows association and never proves causation. Averages and measures of spread covers the mean as the sum of the values divided by how many there are, the mean from a frequency table as sigma fx divided by sigma f, the median found only after ordering the data and averaged across the two central values when the count is even, the mode as the most frequent value with the possibility of no mode or several modes, and the range as largest minus smallest. It sets out what each measure is good for and where each one fails, then handles grouped data honestly: the class midpoint stands in for unknown values, the resulting mean is an estimate and must be labelled as one, and the answer to a grouped question is a modal class interval rather than an invented exact mode. Statistical charts and diagrams covers bar charts with separated bars, equal widths and a labelled uniform scale, composite and dual bar charts with a legend and a shared baseline, pie charts using sector angle equals frequency divided by total frequency multiplied by 360 degrees together with the reverse calculation from an angle back to a frequency and the check that the angles total 360 degrees, pictograms with a key applied consistently to partial as well as whole symbols, and simple frequency distributions. Scatter diagrams covers plotting paired data as small crosses, describing correlation as positive, negative or zero and as strong or weak, identifying an outlier as something to investigate rather than delete, and drawing one ruled straight line of best fit that follows the trend across the whole data set with a rough balance of points above and below, is not forced through the origin and is not forced through every point. It separates interpolation inside the observed range from extrapolation outside it, and shows a concrete case where extrapolating a sensible line produces an impossible test mark above 100. Cumulative frequency covers the running total, the rule that the diagram starts at the lower boundary of the first class with cumulative frequency zero and that every other point is plotted at its upper class boundary, joining the points with a smooth increasing curve, and reading the median at N over 2, the lower quartile at N over 4, the upper quartile at three N over 4 and any pth percentile at p over 100 of N, then computing the interquartile range as Q3 minus Q1 as a measure of the middle fifty per cent that ignores extremes. It also reads in the opposite direction to estimate how many or what percentage of the data lie above or below a stated value, and compares two distributions using median and interquartile range together. Histograms covers grouped continuous data with unequal class widths, frequency density as frequency divided by class width, the recovery of a frequency as frequency density multiplied by class width, the fact that bar area rather than bar height represents frequency, why the tallest bar need not have the greatest frequency, why adjacent bars touch, the compulsory frequency density label on the vertical axis, and true class boundaries such as 9.5 to 19.5 for values recorded as whole numbers 10 to 19. A representation decision map, a bar chart against histogram comparison, a histogram clinic, a mistake clinic, fully worked examples, interpretation sentence frames, retrieval practice with answers held back until the end of the set, a mixed challenge, a mastery checklist and a spaced-review plan support both first-pass learning and last-week revision.Show moreShow less
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A summary of this Mathematics (Syllabus D) chapter — open a section to read it. The full notes, worked examples and practice questions are in the study modules above.
What is Statistics about?
If you remember nothing else: continuous data goes into intervals, not categories; the median needs ordered data; a grouped mean is an estimate; cumulative frequency is plotted at upper class boundaries; and in a histogram the area of a bar is its frequency.
The classification is not a labelling exercise for its own sake — it decides what is legal downstream. Categorical data can have a mode but never a mean. Discrete data can be listed value by value. Continuous data is collected into intervals as soon as the values are many and varied, and it is that grouped continuous data which leads to cumulative-frequency curves and histograms. A short list of measurements can still be handled value by value, with an exact mean and median — see WE 9.2a in 9.2. Get this line wrong and every later step inherits the error.
A statistical conclusion is a sentence with evidence attached, not a number. “Class A did better” states no evidence at all. “Class A has the higher median (\(6\) against \(5\)), so its typical mark is higher” states all of it. The pattern to write, every single time, is: which measure, which way round, what that means here.
The four measures are not interchangeable. The mean uses every value, which is its strength and its weakness — one extreme observation drags it. The median ignores everything except position, so extremes cannot move it. The mode is the only average available for categorical data. The range uses nothing but the two extremes, so it is the measure most easily distorted by a single unusual value. Choosing the right one is the statistics.
Charts differ in what they make easy to see, not in what they contain. A bar chart makes it easy to compare categories against each other. A pie chart makes it easy to see each category's share of the whole — and hard to compare two similar slices. A pictogram makes the count approachable and the precision poor. Choosing the chart is choosing which question the reader will find easy to answer.
Direction and strength are two separate statements. “Positive” says which way the trend runs; “strong” says how closely the points follow it. The syllabus names the direction explicitly — positive, negative or zero — so that is the part it asks for by name; the strength costs one extra word and makes the description sharper, so give both. A relationship can be weak and positive, or strong and negative, and the two words are chosen independently. Then — every time — add the third sentence: association is not causation.
Key ideas to remember
- DATA — Define the variable and its type · Arrange it in a table or diagram · Take the measure asked for · Answer in context, saying whether the figure is exact or an estimate.
What you need to be able to do
- I can decide whether a variable is categorical, discrete or continuous, and justify the choice.
- I can build a tally table from raw data and check the frequency total against the number of observations.
- I can choose grouped class intervals that are equal in width where sensible and that never overlap.
- I can complete a two-way table and verify the grand total from both the row totals and the column totals.
- I read the title, key, units and axis scale of a chart before extracting any value from it.
- I can compare two data sets using one measure of centre and one measure of spread, and then interpret both in context.
- I can name the specific limitation that weakens a stated conclusion — sample size, bias, missing data, measurement, wording, scale or extrapolation.
- I can explain why correlation does not establish causation, using the data in front of me.
- I can find the mean, median, mode and range of a list, ordering the list first.
- I can find the mean from a frequency table using \(\dfrac{\sum fx}{\sum f}\).
- I can estimate the mean of grouped data from class midpoints and state that the answer is an estimate.
- I can identify the modal class and explain why I do not quote an exact mode from grouped data.
- I can say what each average is good for and how each one fails.
- I can draw and read bar charts, including composite and dual bar charts with a legend and a shared scale.
- I can calculate a pie-chart sector angle, do the reverse calculation from angle to frequency, and check the angles total \(360^\circ\).
- I can apply a pictogram key to partial symbols as well as whole ones.
- I can explain why a category bar chart is not a histogram.
- I can plot paired data accurately as crosses and describe the correlation by direction and by strength.
- I can draw one ruled line of best fit across the whole data set with a rough balance of points either side.
- I can use the line to interpolate, and can explain why extrapolating it is unreliable.
- I can identify an outlier and say what should be done about it.
- I can build a cumulative-frequency column as a running total and plot it at upper class boundaries.
- I start the curve at the lower boundary of the first class with cumulative frequency \(0\) and join with a smooth increasing curve.
- I can read the median, \(Q_1\), \(Q_3\) and any percentile, and calculate the interquartile range.
- I can read the curve backwards to estimate how many observations lie above or below a stated value.
- I can compare two distributions using median and IQR together.
- I can calculate frequency density and use it as the bar height.
- I can recover a frequency from a histogram using \(\text{frequency}=\text{frequency density}\times\text{class width}\).
- I can explain why bar area represents frequency and why the tallest bar need not be the largest class.
- I label the vertical axis “frequency density” and draw adjacent bars touching.
- I can find true class boundaries for values recorded to the nearest whole unit.
Key terms in Statistics
- Classifying Statistical Data
- Deciding what kind of variable has been recorded before any statistics is done with it. Categorical data consists of labels or groups with no numerical size, discrete data consists of separate countable values with nothing in between them, and continuous data is measured and can take any value across an interval. The classification determines which table, which chart and which average are legitimate, so it is the first step of every statistics question rather than a preliminary to it.
- Histograms
- A diagram for grouped continuous data in which the horizontal width of each bar is the class width and the vertical height is the frequency density, defined as the frequency divided by the class width. Because height multiplied by width returns the frequency, it is the area of a bar rather than its height that represents how many observations the class contains, so the tallest bar need not be the class with the greatest frequency. Adjacent bars touch because the classes are continuous and share boundaries, the vertical axis must be labelled frequency density rather than frequency, and values recorded to the nearest whole unit must first be converted to true class boundaries such as 9.5 to 19.5 for a class recorded as 10 to 19.
- Scatter Diagrams
- A diagram in which each individual contributes one point, plotting two variables measured on that same individual against each other, so that a relationship between the variables becomes visible. The pattern is described by direction — positive when y tends to rise with x, negative when y tends to fall, and zero when no trend is present — and by strength, meaning how tightly the points cluster around that trend. A single ruled straight line of best fit may be drawn through the overall trend with a rough balance of points either side; it supports interpolation inside the observed range of x but is unreliable outside it, and a correlation of any strength shows association only and never establishes cause.
- Interpreting Statistical Data
- Turning a table or diagram into a statement that the data actually supports. It requires reading the title, key, units and axis scale before extracting any value, comparing data sets using a measure of centre together with a measure of spread, expressing the result in the context of the question rather than as a bare number, and naming the restriction — sample size, bias, missing data, measurement, question wording, misleading scale or extrapolation — that limits how far the conclusion can go.
- Statistical Charts and Diagrams
- The family of diagrams used to display categorical and discrete data in Cambridge O Level Mathematics 4024: bar charts with separated bars of equal width and a labelled frequency scale, composite and dual bar charts with a legend and a shared baseline, pie charts whose sector angle is the frequency divided by the total frequency multiplied by 360 degrees and whose angles must sum to 360 degrees, pictograms whose key applies to partial symbols as strictly as to whole ones, and simple frequency diagrams. Every one of them is a scale drawing of a frequency table, so every height, angle and symbol count is computed from that table rather than judged by eye.
- Cumulative Frequency Diagrams
- A diagram built from the running totals of a grouped frequency table. Each running total is plotted against the upper boundary of its class, the curve starts at the lower boundary of the first class with cumulative frequency zero, and the points are joined with a smooth increasing curve. Because the vertical axis counts how many observations fall at or below each value, the curve can be read in both directions: horizontally at N over 4, N over 2 and three N over 4 to estimate the lower quartile, median and upper quartile, at p over 100 of N for any percentile, and vertically from a data value to find how many observations lie below or above it. The interquartile range, Q3 minus Q1, measures the spread of the middle fifty per cent and is unaffected by extreme values.
- Averages and Measures of Spread
- The four summary statistics of Cambridge O Level Mathematics 4024. The mean is the sum of the values divided by how many there are, and for a frequency table it is the sum of f times x divided by the sum of f. The median is the middle value of the ordered data, averaged across the two central values when the count is even. The mode is the most frequent value or category. The range is the largest value minus the smallest. The first three describe where the data sits and the fourth describes how spread out it is, so a complete summary quotes one of each. For grouped data the class midpoint replaces the unknown individual values, which makes the resulting mean an estimate and makes the answer to a mode question a modal class interval.
Common mistakes to avoid
- “The mean of the grouped table is \(25.25\) minutes.” Why it is wrong The exact values inside each class are unknown. The midpoint is a stand-in, so the result is an estimate of the mean, not the mean. Write instead “An estimate of the mean is \(25.25\) minutes.” One word. Section 9.3
- “The modal class is \(14\).” Why it is wrong A modal class is an interval. \(14\) is the frequency of that interval, not the class itself. Write instead “The modal class is \(20<t\le30\).” Section 9.3
- “The median of \(7,3,9,4,6\) is \(9\), because it is the middle one written down.” Why it is wrong The median is the middle of the ordered list. Unordered, the middle position means nothing. Write instead Order first: \(3,4,6,7,9\). Median \(=6\). Section 9.3
- “Plot each cumulative frequency against the midpoint of its class.” Why it is wrong A cumulative frequency counts everything up to and including the class. That total is only reached at the upper boundary, not halfway through. Write instead Plot \((2,6)\), \((4,20)\), \((6,42)\)… and start at \((0,0)\), the lower boundary of the first class. Section 9.6
- “The classes have different widths, but the frequencies are \(20,15,18,12\), so those are the bar heights.” Why it is wrong With unequal widths, using frequency as height makes a wide class look enormous. It is the area that must equal the frequency. Write instead Divide each frequency by its class width to get frequency density: \(2.0,\;3.0,\;1.2,\;0.4\). Section 9.7
- “Class A is better than Class B because its mean is higher.” Why it is wrong One measure of centre is half an answer, and “better” is not a statistical word. Two sets can share a mean and be nothing alike. Write instead “The two classes have the same median (\(6\) each), so their typical mark is the same. Class A has the smaller range (\(4\) against \(10\)), so its marks are more consistent.” Section 9.2
- “There is strong positive correlation, so more revision causes higher marks.” Why it is wrong A scatter diagram shows that two variables move together. It cannot rule out a third factor driving both, or the causation running the other way. Write instead “There is strong positive correlation: students who revised longer tended to score higher. This does not prove that revision caused the higher marks.” Section 9.5
- Pets \(0,1,2,3,4\) with frequencies \(11,14,9,4,2\). “The mode is \(14\).” Why it convinces \(14\) is the largest number in the table, and “mode = most” is half-remembered correctly. The error \(14\) is how often, not what. It lives in the frequency row, which is a count of the data, not the data. Correct The largest frequency is \(14\), which belongs to \(x=1\). The mode is \(1\) pet.
- Same table. “The range is \(14-2=12\).” Why it convinces It is largest minus smallest — of the wrong row. The error Range is the spread of the values, so it is computed from the top row only. Correct Values run \(0\) to \(4\), so the range is \(4-0=4\) pets.
- Times \(9,9,10,11,12,14,15,16\). “\(n=8\), so the median position is \(\frac{8+1}{2}=4.5\). The median is \(4.5\).” Why it convinces The formula was applied correctly. Only the last step is wrong. The error \(\frac{n+1}{2}\) gives a position in the ordered list, not a value. Position \(4.5\) means “between the \(4\)th and \(5\)th”. Correct The \(4\)th and \(5\)th values are \(11\) and \(12\), so the median is \(\frac{11+12}{2}=11.5\) s. A median of \(4.5\) would also fail the sense check: it lies outside the data, which begins at \(9\).
- Grouped homework times: \(\sum fx = 1010\) across five classes. “Estimated mean \(=\frac{1010}{5}=202\) minutes.” Why it convinces There genuinely are five classes, and dividing by “how many things” is the right instinct. The error The divisor is \(\sum f\) — how many students — not how many classes. Correct \(\frac{1010}{40}=\mathbf{25.25}\) minutes. And \(202\) minutes fails the boundary check instantly: the largest class ends at \(50\) minutes, so no mean can exceed \(50\).
- Egg masses in classes \(40\)–\(50\), \(50\)–\(55\), \(55\)–\(60\), \(60\)–\(80\). “Midpoints \(45, 55, 65, 75\).” Why it convinces The first midpoint is right, and midpoints usually do step evenly — when the widths are equal. The error These widths are \(10,5,5,20\). A pattern was assumed instead of four separate calculations. Correct \(\frac{40+50}{2}=45\); \(\frac{50+55}{2}=52.5\); \(\frac{55+60}{2}=57.5\); \(\frac{60+80}{2}=70\). Compute every midpoint from its own two boundaries.
- \(18\) of \(60\) students choose music. “Sector angle \(=\frac{18}{60}\times100=30^\circ\).” Why it convinces \(30\) is a perfectly plausible-looking angle, and \(\times100\) is the reflex from percentage questions. The error A pie chart is a full turn, so the scaling factor is \(360^\circ\). \(\times100\) produces a percentage wearing a degree symbol. Correct \(\frac{18}{60}\times360=\mathbf{108^\circ}\). The check catches it too: with \(\times100\) the five sectors would total \(100^\circ\), not \(360^\circ\).
- Cumulative frequency curve, \(N=80\). “The median is where the curve reaches \(40\) on the mass axis.” Why it convinces The number \(\frac N2 = 40\) is right. Only the axis is wrong. The error \(\frac N2\) is a count, so it belongs on the cumulative frequency axis. On a mass axis running \(0\) to \(12\) kg, \(40\) does not even exist. Correct Go across from \(40\) on the vertical axis to the curve, then down to the mass axis: median \(\approx5.8\) kg.
- Same curve. “\(Q_1\) is at \(\frac{12}{4}=3\) kg, because the masses run from \(0\) to \(12\).” Why it convinces It uses a quarter of something, and produces a mass, which is the right kind of answer. The error \(Q_1\) is the quarter-way point through the data, not through the range of the axis. Those coincide only if the data is perfectly uniform, which it never is. Correct Read across at \(\frac N4=20\): \(Q_1=4.0\) kg. The two answers differ by a whole kilogram, and the wrong method would give the same \(3\) kg for any distribution on that axis — which is the tell that it cannot be right.
- A class recorded as \(10\)–\(19\) to the nearest unit, frequency \(25\). “Width \(=19-10=9\), so density \(=\frac{25}{9}=2.78\).” Why it convinces Subtracting the two printed numbers is exactly what you do when the boundaries are printed. The error These are rounded values, so the class really occupies \(9.5\le x<19.5\). Correct Width \(=19.5-9.5=10\), so density \(=\frac{25}{10}=\mathbf{2.5}\). Check by counting the labels \(10,11,\ldots,19\): ten of them, so the width is \(10\).
- A class of width \(10\) contains \(20\) patients. “Frequency density \(=\frac{10}{20}=0.5\).” Why it convinces Both numbers are correct and the division looks symmetrical. Nothing in the answer flags an error. The error The fraction is upside down. Density means observations per unit, so frequency is on top. Correct \(\frac{20}{10}=\mathbf{2.0}\). Always reverse-check: \(2.0\times10=20\) ✓, whereas \(0.5\times10=5\), which is not the frequency you were given.
Examiner tips
- Do the arithmetic on paper even on Paper 2. Topic 9 is arithmetically light but bookkeeping-heavy: a mean is a sum of twelve products, a cumulative frequency column is five additions in a chain, a histogram is four divisions. Every one of those is a place to slip a digit. Write the \(fx\) column out; write the running totals out. The calculator is for the final division, not for holding the table in its memory.
- Draw the reading lines, always. On a cumulative-frequency question, rule a horizontal line at \(\frac N2\) across to the curve and drop a vertical line to the axis. The lines show the method behind the number, which a bare answer does not, and they make your own read-off easy to re-check and correct. Leave them on the diagram.
- Write the totals row and column even when the question does not print one. A two-way table without totals is a table you cannot check. Adding them costs two additions and converts a guess into a verified answer — and on a “complete the table” question the totals are the working that shows how each entry was found.
- Never write “better”, “worse” or “good”. They are judgements, not statistics, and they name no evidence. Replace them with what you actually measured: higher median, smaller range, more consistent, more spread out, typically larger. And always attach the numbers — “the higher median” without the two values is only half a statement.
- The median position formula gives a position, not a value. For \(n=9\), \(\frac{9+1}{2}=5\) means “the \(5\)th value in the ordered list”, not “the median is \(5\)”. For \(n=8\), \(\frac{8+1}{2}=4.5\) means “halfway between the \(4\)th and \(5\)th values”, so you average those two. Writing \(4.5\) as the answer reports a position where a value belongs — and the sense check catches it at once, since \(4.5\) need not even lie inside the data.
- A grouped mean has a built-in sanity check: it must lie between the smallest lower boundary and the largest upper boundary. In WE 9.3c that is \(40\) to \(80\) g, and \(57.6\) sits comfortably inside. If your answer escapes that interval you have almost certainly divided by \(\sum fx\) instead of \(\sum f\), or lost a digit in one of the products. The check takes two seconds and catches the worst errors.
- When you are asked to draw, draw completely. A bar chart with correct heights but no axis label, no scale label and no title is an incomplete answer — the labelling is part of what was asked for, not decoration. Get into the habit of writing the title first, before you plot anything — it takes four seconds and it also forces you to reread what the data actually is.
- Supporting insight—not required syllabus content: a useful anchor, not a rule. A line of best fit will usually pass close to the point \((\bar x,\bar y)\) — the mean of the \(x\) values against the mean of the \(y\) values. Plotting that point lightly gives you somewhere sensible to pivot the ruler. Cambridge accepts a range of lines, so this is a way of landing inside that range quickly, not a way of finding “the” line. There is no unique correct line of best fit at this level.
- Write the three heights down before you touch the ruler. For \(N=80\): “\(Q_1\) at \(20\), median at \(40\), \(Q_3\) at \(60\)”. Doing this as a separate line of working means that if your read-off drifts, the method is still visibly correct. It also guards against a very easy error — reading the median at \(N/2\) of the mass axis instead of the cumulative frequency axis.
- Two columns, then draw. Whatever the question gives you, add a class width column and a frequency density column to the table before touching the graph paper — and if the data was recorded to the nearest unit, add a true boundaries column first. Those columns are the method. Trying to compute densities in your head while drawing is how the widths \(10, 5, 15, 30\) turn into \(10, 5, 15, 15\).
- If you only remember one sentence from this clinic: densities never add up, areas always do. Any time you find yourself summing a frequency-density column, stop — convert each one back to a frequency first.
- Write the working out, not just the answer. The points below are split into method (M), accuracy (A) and statement (B) lines so you can see which part of an answer each one is testing. A correct \(fx\) column with a slipped final division still shows a correct method; a bare correct number shows none. These are this resource’s own marking points, written to structure your self-marking — they are not an official mark scheme.
Frequently asked questions
Why does the mean sometimes look “wrong”, like \(1.3\) pets or \(2.4\) children?
Because a mean summarises a group, not a member of it. No household owns \(1.3\) pets, and none needs to. The mean is the value each household would own if the total were shared out equally, which is a statement about the total and the count — both of which are whole numbers. Only the mode is guaranteed to be a value that actually occurred.
When should I use the median rather than the mean?
When the data contains an extreme value or is clearly lopsided. The mean uses every value, so a single unusual observation drags it; the median depends only on position, so it does not move. In Figure 9.5, changing one value from \(10\) to \(20\) moved the mean from \(6\) to \(8\) and left the median at \(5\). If a question asks which average “best represents” the data and there is an obvious outlier, the median is almost always the expected answer — and say why: “because it is not affected by the extreme value”.
Is \(\frac{n+1}{2}\) or \(\frac N2\) the right median rule?
Both, in different situations. For a list of individual values you are counting positions, so the median is at position \(\frac{n+1}{2}\). For a cumulative frequency curve you are reading a continuous model at its halfway height, so you read across at \(\frac N2\). With the sample sizes used in 4024 the graphical difference between \(\frac N2\) and \(\frac{N+1}{2}\) is invisible anyway — but knowing which situation you are in stops the two methods getting crossed.
Why is the modal class not just “the tallest bar”?
On a histogram with unequal class widths, the tallest bar is the one where data is most densely packed, which is not the same as the one containing the most observations. The modal class is defined by the greatest frequency, which on a histogram is the greatest area. In Figure 9.11 the tallest bar holds \(15\) patients while a shorter, wider bar holds \(20\). When the class widths are all equal the two do coincide — which is why the confusion survives.
Why is cumulative frequency plotted at the upper boundary and not the midpoint?
Because a cumulative frequency answers “how many are at or below this value?”. Halfway through a class you have not counted all of it yet, so the running total is not correct there. The total is only reached at the top of the class. Plotting at midpoints shifts the entire curve to the left and makes every median, quartile and percentile too small.
Do I need to know the formula for the median of grouped data?
No. Cambridge 4024 expects the median of grouped data to be estimated from a cumulative frequency curve, by reading across at \(\frac N2\) and down. The algebraic interpolation formula is not required, and a question that supplies a grid is asking you to use it. (Interpolating between two plotted points arithmetically will give a very similar answer and is a legitimate private check — but the graphical reading, with visible reading lines, is what the marks are for.)
How exact does a reading from a graph have to be?
Reading a curve by eye is inherently approximate, so an answer of this kind is judged against a band of acceptable values rather than one exact number. What protects you is showing the construction: rule the horizontal line at \(\frac N2\), rule the vertical line down, and leave both on the diagram. A reading with visible construction lines shows the method behind it; a bare number shows nothing but itself.
Can I draw a line of best fit through the origin?
Only if the trend genuinely goes there. Forcing the line through \((0,0)\) is a common error: it is a decision about the data, not a tidying-up convention. The line should follow the overall trend with a rough balance of points above and below along its whole length — and if that line happens to miss the origin, it misses the origin. Equally, do not force it through the first point, the last point, or any particular point.
Should I delete an outlier?
No — investigate it. An outlier may be a recording error, a genuine unusual case, or a sign that another factor is at work. Removing a point because it spoils the trend is not a statistical decision, it is the opposite of one. Only a reason found outside the data — a known measurement fault, for instance — justifies excluding it, and you would say so.
Is strong correlation ever enough to prove cause?
No, and the strength of the correlation makes no difference. A scatter diagram shows that two variables move together. It cannot rule out a third factor driving both, and it cannot tell you which direction any causation runs. The standard sentence is: “This shows association only; it does not prove that one variable causes the other.”
What is the difference between a bar chart and a histogram, in one line?
A bar chart has separated bars of equal width whose height is the frequency, and it is for categorical or discrete data. A histogram has touching bars whose widths are the class widths, whose height is the frequency density, and whose area is the frequency — and it is for grouped continuous data. The three differences always travel together; see Figure 9.12.
Do I need standard deviation or box plots for 4024?
No. Standard deviation, variance, box-and-whisker plots, correlation coefficients, regression equations, the normal distribution and hypothesis testing are all outside Cambridge O Level Mathematics (Syllabus D) 4024. The required measures of spread are the range and the interquartile range. Using an out-of-syllabus method will not gain marks, and may cost time you needed elsewhere.
Syllabus reference and sources
Written against: Cambridge O Level Mathematics – Syllabus D (4024) 2025–2027 Syllabus (Subject Content, Topic 9: Statistics).
Written by: Academiq Edu Instructor Panel
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