Statistics

If someone asks you for the "average" of a set of numbers, they probably mean the mean — but in Grade 12 Statistics you have three different averages to choose from. The mean, the median and the mode each answer a slightly different question about your data. Once you can see what each one is actually doing, choosing the right one for an exam question becomes much simpler.

The mean — sharing the total equally

The mean is the average most learners meet first. You find it by adding every value and dividing by how many values there are. It treats every data point as if it carries the same weight, which makes it the natural choice when the data is fairly evenly spread. If five learners scored 4, 6, 8, 10 and 12 on a short test, the mean is 40 ÷ 5 = 8. The mean "balances" the data set — it is the value every learner would have if all the marks were shared out equally.

Because the mean uses every value in its calculation, it is sensitive to extremes. One very large or very small mark can pull the mean away from where most of the data actually sits. That is a feature, not a bug — it just means you have to think before you use the mean, especially in questions about income, house prices or test marks with one outlier.

The median — the middle value when sorted

The median is the value that sits in the middle once you have written the numbers in order. For an odd count of values, there is a single middle number: in 4, 6, 8, 10, 12 the median is 8. For an even count, you take the average of the two middle numbers: in 4, 6, 8, 10 the median is (6 + 8) ÷ 2 = 7.

The median does not care how large the biggest value is — it only cares about position. That makes it very useful when one or two extreme values would pull the mean away from the rest of the data. If nine learners earn roughly R3 000 a month and one earns R120 000, the median stays close to R3 000 while the mean jumps to nearly R15 000. The median tells the more honest story.

The mode — the value that appears most often

The mode is the value that occurs more times than any other. In the list 3, 4, 4, 5, 5, 5, 6, the mode is 5 because it appears three times. A data set can have one mode, more than one mode (bimodal or multimodal), or no mode at all if every value is different.

The mode is the only average that works for non-number data. You can talk about the most common shoe size, the most common car colour, or the most common province represented in a class. In an exam, the mode often appears in questions about survey data, where categories matter more than the size of any single value.

A worked example with all three

Let us calculate all three averages for one small data set: 5, 7, 7, 9, 12.

Which average should you name in an exam?

In an exam answer, never just write "the average is 8". Always say which average you mean: "the mean is 8", "the median is 7" or "the mode is 7". Examiners give marks for using the correct word, not just the correct number.

A useful rule of thumb: use the mean when the data is fairly even with no extreme values, use the median when one or two values sit far away from the rest, and use the mode when you need the most common value, especially for non-numerical data. Once you can compute all three and name which one you are using, you have the foundation for every other Statistics question in Grade 12.

Many Grade 12 learners can calculate the mean, median and mode but freeze when an exam asks them to choose the most appropriate one for a given data set. The truth is that no single average is "best" — each one answers a different question. The skill is matching the average to the story the data is actually telling.

The mean loves symmetry

The mean uses every value in the data set, so it works best when the values are spread fairly evenly around a central point. Symmetric data — where the left side of the distribution mirrors the right side — is the natural home of the mean. Think of the heights of Grade 12 learners in a school, the masses of apples from one tree, or the times of athletes running 100 m. In each case, most values cluster near the middle and the few extreme values on either side roughly cancel each other out.

Because the mean uses every value, it is also the average you should reach for when you need to do further calculations. Totals, expected values and many probability formulas all rely on the mean rather than the median or mode. If the question asks for an average that can be used in further work, the mean is almost always the one.

The median loves skewed data

The median only looks at the middle position. It is not pulled around by extreme values, so it is the right average to use when the data is skewed — that is, when a few values sit far away from the rest. Income, house prices and wealth are the classic examples. A small number of very high earners can drag the mean well above what most people actually earn, while the median stays close to the typical value.

When you read a news report that says "the average South African earns R27 000 per month", check whether that is the mean or the median. If it is the mean, it is probably higher than what most people actually earn, because a few very high salaries are pulling it up. The median gives a more honest picture of the "typical" person.

The mode loves categories

The mode is the value that appears most often. It is the only average that works for non-number data. If a question asks about the most common shoe size sold, the most popular choice in a survey, or the most frequent colour of car in a parking lot, the mode is your answer. The mean and median simply do not exist for this kind of data.

For numerical data, the mode is useful when the data has a clear "peak" — a value that repeats often. Exam marks that cluster at 60, sizes that cluster at 32, or ages that cluster at 17 all have an obvious mode that summarises the typical member of the group.

A quick guide for exam day

When you sit with a question and have to choose an average, ask yourself three things. First, is the data symmetric or skewed? Symmetric data favours the mean; skewed data favours the median. Second, is the data numerical or categorical? Categorical data forces you to use the mode. Third, will the answer be used in a further calculation? If yes, the mean is almost always the right choice.

A useful final check: if your chosen average is very different from the other two, ask why. A mean that sits well above the median is a strong hint that the data is skewed — and your written answer should mention this. Examiners reward learners who can explain why a particular average is appropriate, not just learners who can compute it.

Practise by arguing both sides

The fastest way to build this skill is to take one data set and argue why each of the three averages could be the "best" answer. Take the marks 42, 55, 58, 60, 62, 65, 71. Every value is distinct, so there is no mode — but the mean and median are both close to 59, which tells you the data is roughly symmetric and the mean is a fair choice. Now replace 71 with 99 and recalculate. The mean jumps to about 63 while the median stays at 60. The median is now the better summary. Doing this kind of comparison a few times makes the choice feel automatic.

Before you read any further, try the question below. Work it out on paper, write your final answer, and only then scroll down to check. The point is not to get it right on the first try — it is to find out where your method breaks down before the exam does it for you.

Here is the question

Eight learners scored the following marks, out of 50, on a Mathematics test: 28, 34, 41, 36, 47, 22, 39, 33. Calculate the mean mark for this group, rounded to one decimal place. Before you start, write down the formula you are going to use. Many learners lose marks not because they cannot add, but because they cannot remember whether to divide by the count of values or by something else.

The formula in plain words

The mean is the sum of all the values divided by the number of values. In symbols, we write this as mean = Σx ÷ n, where Σx means "add up all the values" and n is the count. The Greek letter sigma (Σ) is just shorthand for "add everything up". Once you know that, the formula stops looking scary.

Two things catch learners out with this formula. The first is forgetting to count the values — some learners divide by 10 when there are only 8 numbers, usually because they assume a "round" number. The second is including the same value twice when adding. Both mistakes are easy to make under exam pressure, which is why a neat list of values and a clear method matter more than speed.

The worked solution

Here is the full calculation. Add the eight values: 28 + 34 + 41 + 36 + 47 + 22 + 39 + 33 = 280. Then divide by the count of values, which is 8.

So the mean mark is 35. In this case the answer is a whole number, so rounding to one decimal place gives 35.0. Always show the rounded value clearly so the marker can see what your final answer is.

How to check your answer in the exam

Once you have a mean, sanity-check it. The mean must sit somewhere between the smallest and largest value in your data set. Here the lowest mark is 22 and the highest is 47, so a mean of 35 sits comfortably in the middle. If your calculation gave you a mean of 60, you would know immediately that something went wrong.

A second check is to look at the median. The sorted data is 22, 28, 33, 34, 36, 39, 41, 47. The median is the average of the 4th and 5th values: (34 + 36) ÷ 2 = 35. The mean and median agree, which tells you the data is roughly symmetric and your answer is consistent.

Three variations to expect in tests

Exam questions rarely stop at "find the mean". Three common variations are worth practising. First, you might be given the mean and asked to find a missing value — write the formula as an equation and solve. Second, you might be asked to find a new mean after one more value is added — recalculate the sum and the count, then divide again. Third, you might be given a frequency table — multiply each value by its frequency, add those products, and divide by the total frequency.

If you can do all three of these, you have covered most of the mean questions that come up in the Grade 12 Statistics exam. The arithmetic is rarely the hard part; the structure is. Practise the structure, and the marks will follow.

The mean is the most familiar average, but it has one big weakness: a single extreme value can drag it far away from where most of the data actually sits. In Grade 12 this idea comes up in almost every Statistics question on outliers, skewness and the choice between mean and median. Understanding why the mean is so sensitive is the key to using it well.

Why the mean listens to every value

The mean is calculated by adding every value in the data set and dividing by the number of values. Because every value takes part in the sum, every value also takes part in the answer. If one value is much larger or much smaller than the rest, that single value contributes a large amount to the total — and the mean has to absorb it.

Think of the mean as a balance point on a seesaw. Every value sits at its own position, and the mean is the fulcrum that keeps the seesaw level. Move one value far to one side and the fulcrum has to move with it to keep things balanced. That is exactly what an outlier does to the mean — it shifts the balance point away from the bulk of the data.

Why the median does not move

The median is calculated differently. You sort the values and pick the middle one (or the average of the two middle ones). The position of the middle value does not depend on how far the extreme values are — it only depends on how many values there are. So an outlier can be enormous or tiny, and the median stays in the same place as long as the order of the middle values does not change.

This is the reason the median is often used to report incomes and house prices. A few very large values would push the mean well above what most people earn, while the median stays close to the typical earner. The next time a news report says "median household income", remember: that choice was made precisely because the mean would have told a misleading story.

A worked example to see the effect

Consider the salaries, in thousands of rand, of nine employees: 18, 20, 22, 24, 26, 28, 30, 32, 34. The mean is 26 and the median is 26 — they agree because the data is symmetric. Now replace the largest value with an outlier.

One outlier pushed the mean from 26 to 44.4 — an increase of about 71% — while the median stayed exactly where it was. That is the whole point. If you used the mean to describe the "typical" salary in this company, you would describe a person who earns roughly 1.7 times what most employees actually earn.

What this means for the exam

In an exam question, the words "outlier", "extreme value" or "skewed" should make you reach for the median rather than the mean. If the question asks you to identify the better average for a data set with an outlier, the answer is almost always the median, because it is resistant to extreme values. The mean is "non-resistant" — it gets dragged around.

Be ready to explain this in words, not just numbers. A common follow-up question asks why the median is more appropriate. A good answer says: "Because the mean uses every value in its calculation, it is sensitive to outliers. The median only uses the middle position, so it is not affected by extreme values. The median therefore gives a better summary of the typical value when the data is skewed."

The bigger picture

This single idea — that the mean listens to every value while the median does not — explains almost every choice you will be asked to make in the central-tendency part of the syllabus. Whenever a question gives you a data set with one value that stands out, pause and ask: "Would the mean and the median tell different stories here?" If yes, the median is usually the more honest average.

Once you can spot that pattern, the rest of the Statistics chapter becomes much easier — because the same logic explains why we use the interquartile range instead of the range, and why we draw box plots in the first place.

An outlier is a value that sits far away from the rest of the data — so far that it makes you wonder whether it belongs at all. Outliers show up in almost every real data set, and the way you handle them can change your final answer. In Grade 12 you need to be able to spot one, explain why it matters, and know the formal test used to decide whether a value really is an outlier.

The informal definition

Informally, an outlier is a data point that is much smaller or much larger than most of the other values. If you have a list of test marks between 40 and 60 and one learner scored 98, that 98 is clearly an outlier — it sits well above the rest. Equally, if every other salary in a survey is between R10 000 and R40 000 and one person earns R250 000, that value is an outlier on the high side.

The informal definition is a good starting point, but it is not enough for the exam. Two different learners can look at the same data set and disagree about whether a value is "much larger" than the rest. To avoid that, Statistics gives us a formal test based on the quartiles of the data set.

The formal definition using quartiles

In Grade 12 you use a rule based on the interquartile range, or IQR. The IQR is the difference between the upper quartile (Q3) and the lower quartile (Q1). A value is an outlier if it is more than 1.5 × IQR above Q3 or more than 1.5 × IQR below Q1. In symbols, the outlier boundaries are:

Lower boundary = Q1 − 1.5 × IQR. Upper boundary = Q3 + 1.5 × IQR. Any value outside these boundaries is an outlier.

How to test a value step by step

Once you know the rule, the method is the same every time. Find Q1 and Q3 from the data. Calculate the IQR. Multiply the IQR by 1.5 to get the outlier "step". Add that step to Q3 to find the upper boundary, and subtract it from Q1 to find the lower boundary. Finally, scan the data for any values outside those boundaries — those are your outliers.

Notice that the method matters: you must exclude the median value itself when you split the data into the lower and upper halves, exactly as we do in Post 166. If you include the median in both halves you will get different quartiles and a different IQR, and your outlier test will be wrong. Always write down the sorted data and the lower/upper halves explicitly so the marker can see your method.

Why outliers matter so much

Outliers matter because they change the answer you get. As we saw in the previous post, a single large outlier can drag the mean far away from the median. The range is even more vulnerable — one extreme value can make the range look enormous even when the rest of the data is tightly clustered. That is why Statisticians often prefer the IQR as a measure of spread: it ignores the extremes and tells you about the middle 50% of the data.

Outliers also matter because they make you ask why they exist. An outlier could be a data-capture error (someone typed an extra zero), a real but unusual event (one learner studied far more than the others), or a sign that two different populations have been mixed together (mixing test marks from two grades). Spotting an outlier is the start of an investigation, not the end of one.

How to write about outliers in an exam

When a question asks you to identify and comment on an outlier, your answer should do three things. First, name the outlier using the formal test ("the value 92 is an outlier because it is above Q3 + 1.5 × IQR"). Second, say which averages are affected ("the mean is pulled upwards by this value; the median is unaffected"). Third, give a possible reason or interpretation ("this could indicate an exceptional learner, or a data-capture error that should be checked").

That three-part answer — identify, quantify, interpret — is what examiners are looking for. Once it becomes a habit, almost every outlier question in the Statistics paper turns into free marks.

Standard deviation sounds intimidating, but the calculation is just a sequence of small, repeatable steps. Once you have done it two or three times, the formula stops being a mystery and starts being a checklist. The trick is to write each step out neatly — sloppy working is the number one reason learners lose marks on this topic.

What standard deviation measures

Standard deviation measures how spread out the values in a data set are around the mean. A small standard deviation means the values are clustered tightly around the mean. A large standard deviation means the values are spread out over a wider range. The symbol for population standard deviation is the Greek letter sigma (σ), and for a sample we use the letter s.

Grade 12 exams usually ask for the sample standard deviation, calculated from a list of values. Your calculator almost certainly has a "statistics mode" that can produce the answer in seconds, but examiners often want to see the full method. That means you need to know the formula and how to apply it.

The formula in plain words

The sample standard deviation formula is: s = square root of [ Σ(x − mean)² ÷ (n − 1) ]. In words: subtract the mean from every value, square each difference, add up all the squares, divide by one less than the count of values, and take the square root. The squared term inside the sum is essential — without it, the positive and negative differences would cancel each other out and the formula would be useless. The "(n − 1)" in the denominator is called the Bessel correction — it makes the sample standard deviation an unbiased estimate of the population standard deviation.

The reason we square the differences is to remove the negative signs. Some values are above the mean and some are below; if we simply added up the differences, they would cancel out. Squaring makes every difference positive so that spread in either direction adds to the total.

The step-by-step method

Set out your working in a table with three columns: the value x, the difference (x − mean), and the squared difference (x − mean)^2. Fill in the table for every value, then add up the third column. Divide that sum by (n − 1) to get the variance, and take the square root to get the standard deviation. A neat table is worth more than fast arithmetic — markers give method marks for each column.

A worked example with five values

So the sample standard deviation is approximately 6.32. Notice that this value sits comfortably between the smallest gap from the mean (4) and the largest gap (8) — that is what you should expect, since the standard deviation is a kind of "average" distance from the mean.

Common mistakes to avoid

The first mistake is dividing by n instead of (n − 1). For a sample, the correct denominator is always (n − 1). Using n gives a slightly smaller answer and will cost you a mark even if everything else is correct. The second mistake is forgetting to square the differences before adding — the squares must be added, not the signed differences. The third mistake is taking the square root at the wrong step: variance is the value before the square root, standard deviation is the value after.

A useful self-check: the standard deviation must be positive (or zero, if all the values are identical). It must also be smaller than the range of the data. If your answer is negative, larger than the range, or zero for a non-identical data set, something has gone wrong.

Using your calculator to check

Once you have done the method by hand, use your calculator's statistics mode to check the answer. Enter the values, set the mode to "single variable" or "1-Var Stats", and read off the value labelled Sx or s. If it matches your hand calculation to two decimal places, you can be confident in your method. The calculator is a checking tool, not a replacement for showing the working — examiners want the table, the formula, and the substitution.

Practise this calculation with three or four different data sets before the exam. The arithmetic never changes; only the numbers do. Once your hand can produce the table without thinking, you have mastered one of the most marks-rich topics in the Statistics paper.

You can calculate a standard deviation perfectly and still not understand what the number means. The calculation is a procedure; the interpretation is the skill the exam is really testing. Once you can read a standard deviation the way you read a temperature — instantly, almost instinctively — the rest of the spread topic falls into place.

Standard deviation is a distance from the mean

The clearest way to think about standard deviation is as a typical distance from the mean. It answers the question: "On average, how far away is each value from the centre?" If the standard deviation is small, most values sit close to the mean. If it is large, the values are spread out over a wide range.

Imagine two classes writing the same test. Both classes have a mean of 60. Class A has a standard deviation of 5, while Class B has a standard deviation of 15. In Class A, most learners scored between 55 and 65 — they are very similar to one another. In Class B, the marks range widely, from below 45 to above 75 — the class is far more varied. The mean hides this difference; the standard deviation reveals it.

Comparing two data sets

Standard deviation is most useful when you compare it across data sets. The two data sets need to be in the same units and have similar means for the comparison to be fair. If one class writes a test out of 50 and another writes out of 100, comparing standard deviations directly is meaningless — the second one will look bigger just because the scale is bigger.

When the units or means are very different, Statisticians use the coefficient of variation, which is the standard deviation divided by the mean. That gives a unit-free percentage that can be compared across data sets. In Grade 12 you will rarely need this, but knowing it exists helps you understand why standard deviation alone is not always the whole story.

The shape that goes with the number

A small standard deviation produces a tall, narrow curve; a large standard deviation produces a short, wide curve. This is most easily seen on a normal distribution curve, where about 68% of values lie within one standard deviation of the mean, about 95% within two, and almost all within three. These percentages are the "empirical rule" or the "68-95-99.7 rule", and they are very common in Grade 12 exam questions.

Knowing the rule lets you estimate quickly. If a data set has mean 50 and standard deviation 10, then roughly 68% of values lie between 40 and 60, roughly 95% lie between 30 and 70, and almost all values lie between 20 and 80. That single sentence summarises the whole shape of the data without you having to draw it.

What to write in an exam answer

When a question asks you to interpret a standard deviation, your answer should do three things. First, state the size of the standard deviation in plain language ("the standard deviation is 8, which is small relative to the mean of 50"). Second, describe what this means about the spread of the data ("the marks are clustered closely around the mean"). Third, compare it to another data set if one is given ("Class A has a smaller standard deviation than Class B, so Class A's marks are less spread out").

This three-part structure — size, meaning, comparison — is what turns a number into a complete exam answer. Practise it on a few past paper questions until the structure becomes automatic.

The link to other measures of spread

Standard deviation is not the only measure of spread. The range (max − min) is simpler but ignores everything in between. The interquartile range (Q3 − Q1) measures the spread of the middle 50% and is resistant to outliers. Standard deviation uses every value and is therefore sensitive to outliers, but it is also the measure that fits with the normal distribution and with most of the probability work in the second half of the syllabus.

Knowing which measure to use is part of the skill. For symmetric data without outliers, standard deviation is usually best. For skewed data or data with outliers, the IQR tells a more honest story. When you can explain that choice in an exam answer, you are thinking like a Statistician, not just calculating like a student.

If two classes both have a mean test mark of 60, you might think they did about the same. But one class could have every learner between 58 and 62, while the other has marks spread from 20 to 100. The mean tells you where the centre sits — it tells you nothing about how the data is spread. This is one of the most important ideas in the whole Statistics syllabus.

The mean only describes the centre

The mean is a measure of central tendency. It tells you where the balance point of the data sits, but it does not tell you whether the values are clustered tightly around that point or scattered widely across a range. To describe the spread, you need a second number — usually the standard deviation or the interquartile range.

Thinking about a single number as a complete summary of a data set is one of the most common mistakes in Grade 12 Statistics. A complete description of a data set always uses two numbers together: one for centre (mean, median or mode) and one for spread (standard deviation, IQR or range). Without both, you cannot picture the data.

Three data sets, one mean

Here is a simple way to see the problem. Consider three small data sets, each with five values, each with the same mean of 10. Set A is 10, 10, 10, 10, 10. Set B is 8, 9, 10, 11, 12. Set C is 2, 6, 10, 14, 18. Each of these has a mean of 10, but they could not look more different.

Set A has no spread at all — every value is identical. Set B has a small, even spread. Set C is spread across a much wider range, from 2 to 18. The mean hides all of this. The standard deviation reveals it: s = 0 for Set A, s = 1.58 for Set B, and s = 6.32 for Set C. The means agree; the standard deviations tell you the real story.

The worked example in detail

The conclusion: although both classes have a mean of 35, Class A is tightly clustered while Class B is widely spread. A teacher looking only at the mean would miss this entirely. A teacher looking at both mean and standard deviation would immediately see that Class B needs very different teaching — some learners are far behind, others are far ahead.

Why this matters in real life

The same idea explains why averages can mislead. A town where everyone earns R20 000 a month has the same mean income as a town where one person earns R100 000 and nine people earn nothing — but those two towns are very different places. The mean hides the spread; the standard deviation or the range reveals it. Whenever you read a single average in a news report, ask yourself: what would the spread look like?

This is also why box plots exist. A box plot shows the median, the quartiles, the minimum and the maximum — five numbers that together describe both the centre and the spread of the data. Looking at a box plot gives you in two seconds what a single mean could never give you.

How to write the comparison in an exam

When an exam asks you to compare two data sets, your answer should always include both a measure of centre and a measure of spread. A good sentence is: "Class A and Class B have the same mean (35), but Class B has a much larger standard deviation (15.81 compared to 1.58), which means Class B's marks are far more spread out around the mean." That single sentence contains both numbers and an interpretation — exactly what examiners want.

If you remember nothing else from this post, remember this: never describe a data set with only one number. Pair a measure of centre with a measure of spread, and you will be doing what the exam is actually testing.

A box-and-whisker plot packs five different numbers about a data set into a single picture. Once you know what each part of the plot is showing you, you can read off the centre, the spread and the skewness of the data in a single glance. It is one of the most efficient summaries in Statistics, and it appears in almost every Grade 12 exam paper.

The five numbers in the plot

A box-and-whisker plot (also called a box plot) is built from the five-number summary: the minimum, the lower quartile (Q1), the median (Q2), the upper quartile (Q3), and the maximum. The box itself stretches from Q1 to Q3 and contains the middle 50% of the data — this is the interquartile range, or IQR. A vertical line inside the box marks the median. The two "whiskers" stretch out from the box to the minimum and maximum values.

Reading a box plot is therefore a matter of reading five numbers off the diagram: where the left whisker starts (minimum), where the box starts (Q1), where the line inside the box sits (median), where the box ends (Q3), and where the right whisker ends (maximum). Practise naming these five parts out loud whenever you see a box plot, and you will never confuse them again.

What the shape of the box tells you

The shape of the box gives you a quick read on the spread and symmetry of the data. If the median line sits in the middle of the box and the two whiskers are about the same length, the data is roughly symmetric. If the median is closer to Q1 and the right whisker is much longer, the data is positively skewed — a few large values are pulling the upper tail out. If the median is closer to Q3 and the left whisker is longer, the data is negatively skewed — a few small values are pulling the lower tail out.

The width of the box tells you about the spread of the middle 50% of the data. A narrow box means the middle half of the data is tightly clustered; a wide box means it is spread out. Comparing the box width to the total length of the whiskers tells you whether the data is mostly packed in the middle or stretched across a wide range.

Outliers on a box plot

If a value lies more than 1.5 × IQR above Q3 or below Q1, it is shown as a separate dot or star outside the whisker, instead of being part of the whisker. The whisker itself is then drawn only to the most extreme value that is not an outlier. This makes box plots one of the easiest ways to spot outliers — they appear as isolated points on the diagram.

When you describe outliers on a box plot in an exam, say which side they are on (upper or lower) and what they tell you about the data. For example: "There is one outlier on the upper side, indicating one value much larger than the rest of the data."

Reading a worked example

The data has a centre of 28, an interquartile range of 20 and a total range of 58. Because the right whisker is much longer than the left whisker, and the median sits closer to Q1 than to Q3, the data is positively skewed. A few large values are stretching the upper tail out, which is why the maximum (70) is so much further from the median than the minimum (12) is.

Comparing two box plots side by side

Box plots are especially powerful when you compare two of them drawn on the same scale. Look at three things in order. First, compare the medians — which data set has the higher centre? Second, compare the IQRs (the widths of the boxes) — which data set is more spread out in the middle? Third, compare the overall ranges and the whiskers — which data set has the wider extremes, and is either of them skewed?

This three-step comparison is what examiners want when they ask you to "compare the two distributions shown in the box plots below". Mention the median, the IQR, and the skewness in your answer, and you have all the comparison marks in the bag.

What to write in an exam answer

A complete description of a box plot should include the five-number summary (or the relevant values from it), a sentence about the centre, a sentence about the spread, and a sentence about the shape (symmetric, positively skewed or negatively skewed). If outliers are shown, mention them by name. Practise writing this four-sentence summary for any box plot you come across, and you will never lose marks on this kind of question again.

Here is a short challenge to check whether you can read a box plot quickly and accurately. Try the question below before you scroll down to the worked answer. The point is not just to find the right number — it is to check that you know exactly which part of the plot to look at, because in an exam you will often have only a few seconds to find the median.

Here is the box plot

Imagine a box-and-whisker plot drawn on a scale from 0 to 100. The left whisker starts at 20. The box starts at 35 (this is Q1). A vertical line inside the box sits at 52. The box ends at 68 (this is Q3). The right whisker ends at 95. There is one small dot drawn above the scale at 99, just past the right whisker.

Your task: identify the median, the interquartile range, the range, and any outliers shown on the plot. Write your answers down before you read on.

The answer, step by step

Here is how to read the plot. The median is the vertical line inside the box, which sits at 52. The interquartile range is the width of the box: Q3 − Q1 = 68 − 35 = 33. The total range is the distance from the start of the left whisker to the end of the right whisker: 95 − 20 = 75. There is also a small dot above 99 drawn separately from the upper whisker — in a Grade 12 box plot, a dot drawn like that is meant to represent an outlier, and the question will expect you to identify it as one and explain why.

That last step highlights an important exam point. In a real Grade 12 box plot, the upper whisker ends at the largest data value that is NOT an outlier, and any value beyond the 1.5 × IQR boundary is drawn as a separate dot. If a dot is drawn at 99, then by convention 99 must be above the upper boundary for it to appear that way. In this illustrative example the numbers do not quite work out (99 would fall inside the formal boundary), which is a reminder to always compute the boundary yourself and check what the question is testing — never assume a dot is an outlier without verifying it against the 1.5 × IQR rule.

What to do when the box is uneven

Notice that the median in this box plot sits slightly to the right of the centre of the box. The left half of the box (from Q1 to the median) is 17 units wide, while the right half (from the median to Q3) is 16 units wide. They are very close, which means the middle 50% of the data is almost symmetric. However, the right whisker is much longer than the left whisker (27 units versus 15 units), which suggests a slight positive skew in the tails.

When you describe a box plot in an exam, always comment on both the box and the whiskers. The box tells you about the middle 50% of the data; the whiskers tell you about the extremes. Together, they give a full picture.

Three questions you should always answer

Every box plot question can be answered by responding to three prompts. First, what is the centre? (Read the median from the line inside the box.) Second, what is the spread? (Read the IQR from the width of the box, and the range from the total length of the whiskers.) Third, what is the shape? (Compare the lengths of the two whiskers and the position of the median inside the box.) Add a fourth if outliers are shown: which values are outliers, and on which side?

If you can answer those three or four prompts for any box plot, you have covered everything an examiner can ask. Practise it on the next five box-plot questions you find in past papers, and the method will become automatic.

Where learners lose marks

Two mistakes are common. The first is reading the median off the wrong line — learners sometimes give Q1 or Q3 as the median because the labels look similar. Slow down and check that you are looking at the vertical line inside the box, not the edges of the box. The second is forgetting to mention outliers when they are shown. If a dot appears on a box plot, the examiner put it there to test whether you would comment on it. Always do so.

Once you can read off all five numbers in seconds and comment on the shape, box plot questions stop being scary. They become one of the most predictable sources of marks in the Statistics paper.

Quartiles are the three values that split a sorted data set into four equal parts. They are the foundation of the box plot and the interquartile range, and almost every spread question in the Grade 12 Statistics paper starts with finding them. Once you know the rule for odd and even counts of values, the calculation is the same every time.

The three quartiles in plain words

The first quartile (Q1) is the value below which 25% of the data lies. The second quartile (Q2) is the median — 50% of the data lies below it. The third quartile (Q3) is the value below which 75% of the data lies. Together with the minimum and maximum, these three values make up the five-number summary that drives most of the box plot work in the syllabus.

The key fact about quartiles is that they are positional — they describe where values sit in the ordered list, not how large they are. This is what makes the IQR resistant to outliers, since the position of Q1 and Q3 does not change when an extreme value moves further away.

The basic method for an odd count

To find the quartiles of a data set, first sort the values from smallest to largest. Then find the median — this is Q2, the middle value. The lower half of the data is everything below the median (do not include the median itself). Q1 is the median of that lower half. The upper half is everything above the median. Q3 is the median of that upper half.

This method is clean and consistent. The only thing to remember is to exclude the median value itself when you split the data into the lower and upper halves. If you include it, you will get the wrong quartiles.

The basic method for an even count

When the count of values is even, the median is the average of the two middle values. Split the data into a lower half (the first half of the values) and an upper half (the second half). Q1 is the median of the lower half, and Q3 is the median of the upper half. Because the count is even, there is no excluded middle value — every value goes into one half or the other.

Some textbooks use a slightly different rule that gives quartile values which can differ from this method by a fraction. In Grade 12 the difference is almost always less than one unit and should not change the answer to a "describe the data" question. Stick with the method your teacher uses, and use the same method throughout the exam so your answers stay internally consistent.

A worked example with nine values

Notice that the lower and upper halves each have four values, so their medians are the average of the two middle values. If the original count had been seven instead of nine, the lower and upper halves would each have three values, and the quartiles would be single middle values rather than averages.

A worked example with an even count

For comparison, consider the data: 4, 8, 12, 16, 20, 24, 28, 32. There are eight values, so the median is the average of the 4th and 5th values: (16 + 20) ÷ 2 = 18. The lower half is 4, 8, 12, 16; Q1 is the median of these four, which is (8 + 12) ÷ 2 = 10. The upper half is 20, 24, 28, 32; Q3 is the median of these, which is (24 + 28) ÷ 2 = 26. So Q1 = 10, Q2 = 18, Q3 = 26.

Checking with your calculator

Most calculators with a statistics mode will give you Q1, Q2 and Q3 directly once you have entered the data. Use this as a check on your hand calculation, but be aware that some calculators use slightly different rules for quartiles and may give a value that differs by a small fraction. If your calculator's value disagrees with your hand calculation, write down the method you used and the value it produced. Examiners give method marks, so a clear method matters more than a single "magic" answer.

Practise this procedure with five or six small data sets before the exam. Once the steps become automatic, finding quartiles takes about thirty seconds per data set — and that is thirty seconds that can earn you several marks further on in the question.

The interquartile range, or IQR, is one of the most useful measures of spread in Grade 12 Statistics. It tells you how wide the middle 50% of your data is, and it does so without being fooled by a few extreme values. If you understand what the IQR is doing, you understand why box plots are drawn the way they are.

What the IQR actually measures

The IQR is the difference between the upper quartile and the lower quartile: IQR = Q3 − Q1. It measures the spread of the middle 50% of the data — the values that sit between the 25th percentile and the 75th percentile. Half of all the data values fall inside this range, so the IQR is a measure of how packed that middle half is.

A small IQR means the middle 50% of the data is tightly clustered — most values are close together. A large IQR means the middle 50% is spread out — values are more varied. Because the IQR only looks at the middle half, it ignores the smallest 25% and the largest 25% of the data, which is where outliers usually sit.

Why we use IQR instead of the range

The simplest measure of spread is the range, calculated as maximum − minimum. The range is easy to find, but it has one big weakness: it depends on just two values, and any single extreme value can make it enormous. A data set with values between 20 and 30 plus one value at 200 has a range of 180, even though most of the data sits in a 10-unit window.

The IQR solves this problem by ignoring the extremes. The smallest 25% and the largest 25% of values are simply not part of the calculation, so an outlier far from the bulk of the data has no effect on the IQR. This makes the IQR a "resistant" or "robust" measure of spread, in the same way that the median is a resistant measure of centre.

Calculating the IQR step by step

To calculate the IQR, first sort your data from smallest to largest. Find the median (Q2), then find the median of the lower half (Q1) and the median of the upper half (Q3). Subtract Q1 from Q3 to get the IQR. The arithmetic is short; the work is in finding Q1 and Q3 correctly.

The IQR of 20 means that the middle 50% of the data spans 20 units — from 12 up to 32. Half of all the values fall inside this window. If you compare that to the range of 32 (from 6 to 38), you can see that the middle half of the data is much more tightly packed than the range would suggest.

Using the IQR to identify outliers

The IQR has a second important use: it defines the formal rule for outliers. A value is an outlier if it is more than 1.5 × IQR above Q3 or more than 1.5 × IQR below Q1. The boundaries are calculated as Q3 + 1.5 × IQR on the upper side and Q1 − 1.5 × IQR on the lower side. Any value outside those boundaries is flagged as an outlier.

Using the example above: IQR = 20, so 1.5 × IQR = 30. The upper boundary is Q3 + 30 = 32 + 30 = 62, and the lower boundary is Q1 − 30 = 12 − 30 = −18. In this data set, no value lies outside those boundaries, so there are no outliers. If the value 80 had been in the data set, it would be above 62 and therefore an outlier.

What to write in an exam answer

When a question asks you to interpret an IQR, your answer should do three things. First, state the value of the IQR clearly. Second, say what it means about the spread of the middle 50% of the data — for example, "the IQR is 20, which means the middle 50% of the marks lie within a 20-mark range". Third, compare it to another data set if one is given — for example, "Class A has a smaller IQR than Class B, so the middle 50% of Class A's marks are less spread out".

This three-part structure — value, meaning, comparison — is what turns a number into a full answer. Practise it on past paper questions until it becomes automatic. Once you can describe the IQR confidently, you have the language you need to talk about spread throughout the rest of the Statistics paper, from box plot questions to outlier identification to comparing distributions.

A scatter plot is the simplest way to see a relationship between two numerical variables. It puts one variable on each axis and plots one dot for every pair of values. Reading a scatter plot well is one of the most useful skills in Grade 12 Statistics, because almost every question about bivariate data starts with one. Once you know what to look for, you can describe the relationship in two sentences.

What a scatter plot shows you

Each dot on a scatter plot represents one observation, with its horizontal position given by the value of the first variable and its vertical position given by the value of the second. The pattern of the dots tells you how the two variables move together. If the dots rise from bottom-left to top-right, the relationship is positive. If they fall from top-left to bottom-right, the relationship is negative. If the dots are scattered randomly with no clear direction, there is no relationship.

A scatter plot does not prove that one variable causes the other. It only shows association. We will come back to that important distinction in the next post — for now, focus on reading what the plot actually shows.

Direction, strength and shape

Three things describe any scatter plot: the direction of the relationship (positive, negative or none), the strength of the relationship (strong, moderate or weak), and the shape of the relationship (linear or non-linear). Direction tells you which way the dots move. Strength tells you how tightly the dots cluster around a single line. Shape tells you whether that line is straight or curved.

A strong positive linear relationship has dots that rise clearly along a straight line. A weak positive linear relationship has dots that rise in general but spread widely around the line. A strong negative linear relationship has dots that fall clearly along a straight line. If the dots form a clear curve — for example a U-shape — the relationship is non-linear, and you should not fit a straight line to it.

The correlation coefficient

The correlation coefficient, written as r, is a number between −1 and +1 that summarises the strength and direction of a linear relationship. A value of +1 means a perfect positive linear relationship; −1 means a perfect negative linear relationship; 0 means no linear relationship. The closer r is to ±1, the stronger the linear relationship; the closer r is to 0, the weaker it is.

A rough guide: |r| above 0.7 is a strong linear relationship, |r| between 0.4 and 0.7 is moderate, and |r| below 0.4 is weak. The sign of r matches the direction of the relationship. Always pair the value of r with a description of the scatter plot — the number alone is not enough.

A worked example

Notice that the description includes all three elements: direction (positive), strength (strong), and shape (linear). A complete exam answer always includes those three words. Saying just "positive correlation" or "r = 0.86" without describing the strength and shape will lose marks.

What scatter plots do not show

A scatter plot shows association, not causation. Even a very strong linear relationship between two variables does not mean that one variable causes the other. There may be a third variable driving both, or the relationship may be a coincidence. We will look at this in detail in the next post, but keep it in mind whenever you describe a scatter plot in an exam — use words like "association" or "relationship", not "causes" or "leads to", unless the question gives you a reason to do so.

Scatter plots also do not show categorical data well. If one of your variables is a category (like province or gender), a scatter plot is the wrong graph to use. Box plots, bar charts or grouped frequency tables are usually better for that kind of data.

How to write a complete answer

When a question asks you to interpret a scatter plot, your answer should do four things. First, name the two variables being plotted. Second, state the direction of the relationship. Third, state the strength. Fourth, state the shape (linear or non-linear). If you are given a value of r, mention it. A good sentence is: "There is a strong positive linear relationship (r = 0.86) between hours studied and test mark, with higher study times associated with higher marks."

Notice the word "associated". That is the safe word — it describes what the plot shows without claiming anything the plot cannot prove. Practise using it whenever you describe a scatter plot, and your answers will be both complete and accurate.

"Correlation does not imply causation" is one of the most quoted sentences in Statistics, and one of the most misunderstood. Many Grade 12 learners lose marks because they write "X causes Y" when the data only shows that X and Y move together. Once you can draw the line between correlation and causation clearly, you will never make this mistake again.

What correlation means

Correlation is a statistical relationship between two variables. It says that when one variable changes, the other tends to change in a predictable way. A positive correlation means the two variables move in the same direction — when one goes up, the other tends to go up. A negative correlation means they move in opposite directions — when one goes up, the other tends to go down.

Correlation is measured by the correlation coefficient, r, which ranges from −1 to +1. The closer r is to ±1, the stronger the linear relationship. But correlation only describes the pattern in the data — it does not say anything about why the pattern exists. Two variables can be strongly correlated without either one having any effect on the other.

What causation means

Causation is a much stronger claim. It says that changing one variable directly causes the other to change. To establish causation, you need more than a scatter plot — you usually need a controlled experiment in which one variable is deliberately changed while everything else is held constant. Only then can you reasonably conclude that the change in the first variable caused the change in the second.

Causation requires three things: a correlation between the variables, a logical mechanism by which one could cause the other, and evidence that other possible explanations (such as a third hidden variable) can be ruled out. Correlation alone gives you only the first of these.

Why correlation does not imply causation

There are two main reasons why a correlation can exist without causation. The first is a confounding variable — a third variable that causes both of the others, making them appear to be linked when they are actually both being driven by the same thing. The second is pure coincidence — two variables can move together over a short period simply by chance, especially when only a small number of data points are involved.

Both of these can produce strong correlations that look just like causal relationships. The scatter plot alone cannot tell you which is which. That is why Statisticians are so careful with the language they use, and why your exam answers should be too.

A side-by-side comparison

Here is the contrast in a single view.

Correlation: Two variables move together in a predictable way. It can be measured by the correlation coefficient r. It can be shown on a scatter plot. It does not require any particular explanation — only a pattern. It is a statement about what the data shows, not why it shows it.

Causation: One variable directly produces a change in another. It cannot be proved by a scatter plot alone. It requires a logical mechanism, controlled evidence, and the ruling out of other explanations. It is a statement about why the data shows what it shows.

The key contrast is this: correlation is about what we see; causation is about why it happens. A correlation is a necessary but not sufficient condition for causation. To move from "these variables are correlated" to "this variable causes that one", you need much more than a graph.

A classic example

Consider this: ice cream sales and drownings both rise sharply in summer and fall in winter. If you plot ice cream sales against drownings, you will see a strong positive correlation. But eating ice cream does not cause drowning. The hidden variable is hot weather, which causes both more ice cream sales and more swimming, which leads to more drownings. The correlation is real; the causation is not.

This example shows why exam answers should be so careful. If a question shows a scatter plot of ice cream sales and drownings and asks what the plot shows, the correct answer is "a positive correlation between ice cream sales and drownings". The wrong answer is "ice cream causes drownings". The first answer is what the data supports; the second goes far beyond it.

How to argue causation responsibly

If you want to argue that a correlation also reflects causation, you need to do three things. First, show that the correlation is strong and consistent. Second, propose a sensible mechanism by which the first variable could affect the second. Third, rule out likely confounding variables — the third factors that could be driving both. Only when all three are in place can you reasonably use the word "causes" in your answer.

In a Grade 12 exam, you will rarely be asked to prove causation. You will much more often be asked to interpret a scatter plot or a correlation coefficient, and the safe answer is to describe the association, mention the strength and direction, and stop there. Use words like "associated", "related to", or "tends to move with". Leave "causes" for situations where you have actual evidence.

What to write in the exam

If a question shows a strong correlation and asks what conclusion you can draw, a safe complete answer is: "There is a strong positive linear correlation between X and Y, which means that higher values of X tend to go with higher values of Y. This does not prove that X causes Y, because the relationship could also be explained by a third variable or by chance." That answer describes what the data shows and refuses to claim more than the data supports — exactly what examiners want.

Once you can make this distinction in writing without thinking, you have one of the most marks-saving habits in the Statistics paper. The phrase "correlation does not imply causation" is not just a slogan — it is the difference between a wrong answer and a right one.

A statistical graph question can feel overwhelming because so much information is packed into one picture. But there is a reliable four-step routine that turns any graph question into a series of small, predictable tasks. Once you have practised this routine a few times, you will start every graph question with a clear plan instead of staring at the diagram.

Step 1: Read the title and the axes first

Before you look at the shape of the graph, read every word around it. The title tells you what the graph is about — which variable, which group, which time period. The axis labels tell you what each side is measuring and in what units. Many learners lose marks simply because they did not notice the units on the y-axis were in thousands, or because they misread a percentage as a count.

Write down the variables in words. If the x-axis shows "time studied (hours)" and the y-axis shows "test mark (out of 50)", say so out loud. That tiny act forces you to see the graph as a story about two specific things, rather than an abstract shape.

Step 2: Identify the type of graph

Different graph types answer different questions. A bar chart compares categories. A histogram shows the distribution of a single numerical variable. A box plot summarises five numbers about a data set. A scatter plot shows the relationship between two numerical variables. A pie chart shows how a total is divided into proportions. A line graph shows how a variable changes over time.

Identifying the type of graph tells you what kind of description the examiner is looking for. For a box plot, you describe the centre, the spread and the shape. For a scatter plot, you describe the direction, the strength and the shape of the relationship. For a histogram, you describe the centre, the spread, the shape and any unusual features. Each graph type has its own checklist of things to mention.

Step 3: Look for centre, spread and shape

For any graph that shows a distribution, three things should always be in your answer. The centre — what is a typical value, and is it best described by the mean, the median or the mode? The spread — are the values tightly clustered or widely scattered, and how does the IQR or standard deviation compare? The shape — is the distribution symmetric, positively skewed or negatively skewed?

If the graph shows two distributions, compare each of these three things across the two. Start with the centre, then the spread, then the shape. Examiners give marks in that order, so writing your answer in that order makes it easy for them to find each mark.

Step 4: Comment on outliers and trends

Once you have described the centre, the spread and the shape, add anything unusual. Are there outliers? If so, on which side, and what might they tell you? Is there a clear trend over time? If so, in which direction, and is it linear or curved? Is there a gap in the data that suggests a hidden group of values? Examiners reward learners who notice these features and put them into words.

A useful sentence frame is: "There is one outlier on the upper side, indicating one value much larger than the rest of the data, which could be due to ...". The blank at the end forces you to suggest a reason, which is often worth a mark on its own.

A worked example of the routine

Suppose a question shows a box plot of test marks for Class A next to a box plot for Class B, on the same scale. Apply the four steps. First, read the title: "Test marks out of 50 for two Grade 12 classes". Second, identify the type: two box plots on the same scale, which means a comparison is expected. Third, look at centre, spread and shape: Class A has a higher median, Class B has a wider IQR, and Class B has a long upper whisker (positive skew). Fourth, comment on outliers: Class B shows one outlier on the upper side.

A complete exam answer based on this routine would be: "Class A has a higher median than Class B, so Class A's typical mark is higher. Class B has a wider IQR than Class A, so Class B's middle 50% of marks are more spread out. Class B is positively skewed, with a long upper whisker, and shows one outlier on the upper side, indicating one learner who scored much higher than the rest of Class B." That answer covers every mark the question is likely to carry.

Practise the routine until it is automatic

The fastest way to make this four-step routine automatic is to apply it to every graph you see in past papers, even when the question only asks for one specific thing. Describe the title, the type, the centre, the spread, the shape and the outliers in your head before you write anything. Once that mental checklist is second nature, every graph question becomes a translation exercise — you already know what to look for, you just have to put it into words.

With a clear routine, the graph questions in the Statistics paper stop being the scariest part of the exam. They become some of the most predictable marks available — and in a paper where every mark matters, that is exactly what you want.