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Central tendency • grouped data • quartiles • box & whisker
Grade 10 CAPS: measures of central tendency for ungrouped and grouped data, quartiles, the five-number summary, and the box-and-whisker diagram — the toolkit every later grade builds on.
How to summarise a set of data with a single typical value, and how spread-out it really is. Every worked example below moves from a direct calculation up to a genuine application, so work through them in order.
Grade 10 builds the summary toolkit every later grade extends.
Central tendency • grouped data • quartiles • box & whisker
Histograms • ogives • variance & standard deviation • outliers
Bivariate data • scatter plots • regression & correlation
Two weeks, five connected skills, all about describing ONE set of data.
Revise measures of central tendency (mean, median, mode) in ungrouped data.
Measures of central tendency in grouped data: calculate an estimate of the mean, and identify the modal interval and the interval in which the median lies.
Revise range as a measure of dispersion, extended to include percentiles, quartiles, the interquartile range and the semi-interquartile range.
Five-number summary (minimum, quartiles, maximum) and the box-and-whisker diagram.
Use these statistical summaries and graphs to analyse and make meaningful comments on the context of the data.
Before summarising any data set, know what TYPE of data you're holding.
Described in words, not numbers — e.g. favourite subjects, eye colours. Cannot be added, averaged, or put on a number line. Splits further into CATEGORICAL (a limited, fixed set of options, e.g. sandwich fillings chosen from a list) and ANECDOTAL (open-ended, story-form answers with no fixed set of options, e.g. "describe your dream job").
Numerical and COUNTED, so it only takes whole-number values — e.g. number of siblings, number of cars in a parking lot.
Numerical and MEASURED, so it can take any value in a range, including decimals — e.g. height, mass, time.
Classify each data set: (a) \(\{23; 25; 22; 26; 27\}\), sweets counted in a packet. (b) \(\{\text{"cheese"; "jam"; "honey"; "cheese"}\}\), sandwich fillings learners chose. (c) \(\{1{,}70\text{ m}; 1{,}41\text{ m}; 1{,}60\text{ m}; 1{,}32\text{ m}\}\), heights of learners.
Three different "typical" values, each with its own use.
Add every value, divide by how many there are. Uses every single value — but one extreme outlier can drag it a long way from the "typical" value.
Arrange the data in order; the middle value (or the average of the two middle values, if there is an even number of values). Barely affected by an outlier.
The value that occurs most often. The only measure that also makes sense for non-numerical (categorical) data.
Straight from a raw list of values.
Find the mean, median and mode of: \(7,\ 9,\ 4,\ 15,\ 7,\ 12,\ 10\)
Find the mean and median of: \(20,\ 15,\ 30,\ 12,\ 25,\ 18,\ 22,\ 28\)
The mean of 5 numbers is 12. Four of the numbers are \(8,\ 10,\ 14,\ 16\). Find the fifth number.
12 learners' resting heart rates (beats per minute) are: \(71,\ 65,\ 82,\ 74,\ 62,\ 89,\ 78,\ 68,\ 75,\ 80,\ 73,\ 85\). (a) Organise this data using a stem-and-leaf diagram. (b) Use it to find the median. (c) Find the range.
| Stem | Leaves |
|---|---|
| 6 | 2 5 8 |
| 7 | 1 3 4 5 8 |
| 8 | 0 2 5 9 |
A class of 9 learners scores: 45, 52, 58, 60, 62, 65, 70, 75, 98 on a test. Which measure of central tendency is most affected by the score of 98?
The mean of 4 numbers is 20. Three of the numbers are 15, 18, 25. What is the fourth number?
When there's too much raw data to list, group it into intervals first.
Multiply each interval's midpoint by its frequency, add these products together, then divide by the total frequency \(n\).
The interval with the HIGHEST frequency — you can no longer name one exact modal value, only the interval it falls in.
Build a running (cumulative) total of the frequencies; the median interval is the one where the running total first reaches or passes \(\dfrac{n}{2}\).
A small example first, then the exact style CAPS itself uses.
Estimate the mean age, and state the modal and median intervals, for this grouped data:
| Age (years) | Frequency |
|---|---|
| \(10\le x<20\) | 3 |
| \(20\le x<30\) | 7 |
| \(30\le x<40\) | 6 |
| \(40\le x<50\) | 4 |
The percentage marks of 200 Grade 10 learners are summarised below. (a) Calculate the approximate mean mark. (b) Identify the interval in which the median lies. (c) Identify the interval containing the lower quartile. (d) Identify the interval containing the upper quartile. (e) Identify the interval containing the 30th percentile.
| Percentage | Number of candidates |
|---|---|
| \(0\le x<20\) | 4 |
| \(20\le x<30\) | 10 |
| \(30\le x<40\) | 37 |
| \(40\le x<50\) | 43 |
| \(50\le x<60\) | 36 |
| \(60\le x<70\) | 26 |
| \(70\le x<80\) | 24 |
| \(80\le x<100\) | 20 |
For the interval \(30\le x<40\), what value is used as its "typical" value when estimating the mean?
A grouped data set has cumulative frequencies 5, 18, 40, 55 across four intervals, with n = 55. Which interval contains the median?
Central tendency alone can hide very different data sets.
\(\text{Range}=\text{max}-\text{min}\). Simple, but only uses two values — one extreme outlier at either end distorts it completely.
\(Q_2\) is the median. \(Q_1\) is the median of the lower half of the data; \(Q_3\) is the median of the upper half (excluding \(Q_2\) itself if \(n\) is odd).
\(\text{IQR}=Q_3-Q_1\) — the spread of the MIDDLE 50% of the data, ignoring extreme values at either end.
\(\dfrac{Q_3-Q_1}{2}\) — half the IQR, sometimes used as a single "typical spread from the median" figure.
Odd and even data sets need slightly different care when splitting into halves.
Find the range, \(Q_1\), \(Q_2\), \(Q_3\), the IQR and the semi-IQR of: \(9,\ 5,\ 13,\ 2,\ 11,\ 15,\ 7,\ 8,\ 4\)
Find the range, \(Q_1\), \(Q_2\), \(Q_3\) and the IQR of: \(31,\ 25,\ 38,\ 47,\ 23,\ 33,\ 44,\ 28,\ 40,\ 35\)
Why is the range considered a "weak" measure of spread compared to the IQR?
For the data set 6, 10, 14, 18, 22, 26 (n = 6, even), what is Q1?
One picture, five numbers, the whole shape of the data.
Minimum, \(Q_1\), median (\(Q_2\)), \(Q_3\), maximum — exactly the values you already know how to find.
Drawing one from data, then reading one that's already drawn.
Find the five-number summary for: \(20,\ 15,\ 33,\ 12,\ 28,\ 40,\ 18,\ 22,\ 25,\ 30,\ 36\)
A box-and-whisker plot for a set of test scores has minimum 40, \(Q_1=55\), median 68, \(Q_3=78\), maximum 92. (a) What percentage of learners scored below 55? (b) What is the IQR? (c) Is the data skewed, and if so, in which direction?
A box-and-whisker plot for household incomes has minimum 20, \(Q_1=28\), median 33, \(Q_3=42\), maximum 75 (in R thousands). Describe the shape of this data.
In a box-and-whisker diagram, what does the BOX itself (not the whiskers) represent?
A box plot has min=10, Q1=20, median=30, Q3=32, max=35. What does the short distance between the median and Q3 suggest?
This is the actual exam skill CAPS names: use the numbers AND the picture to say something true.
Set A: \(12,13,14,15,16\). Set B: \(6,10,14,18,22\). Compare the two sets.
Two Grade 10 classes wrote the same test (out of 100). Class A: \(45,52,58,61,63,65,68,70,72,75,80\). Class B: \(40,48,55,60,64,68,71,74,78,85,92\). Compare the two classes' performance.
Work first. Open one answer only when your own line of working is complete.
Data: \(6,9,9,12,14\). Answer: mean \(=10\), median \(=9\), mode \(=9\).
Frequencies \(5,12,8,3\) across 4 intervals. Answer: the 2nd interval (frequency 12, the highest).
\(Q_1=14,\ Q_3=26\). Answer: \(\text{IQR}=26-14=12\).
Median sits much closer to \(Q_1\) than to \(Q_3\). Answer: the upper half of the data is more spread out.
Forgetting to sort the data before finding the median or quartiles.
Including the median itself in the lower/upper half when \(n\) is odd.
Stopping at the numbers without stating what they mean in context.
Always write the sorted list out first — it's the single most common lost mark.
Double-check: does your \(n\) for the lower/upper half match what you expect?
End every "compare" answer with a sentence in words.
Use the Mastery Bank to build fluency. Then take the test without notes and use the result to choose the exact slide to revisit.
Four free, independent videos — not made by Equation Station SA.
The three measures of central tendency, from scratch.
Khan Academy · Statistics intro: Mean, median, and mode
Using midpoints to estimate the mean of a frequency table.
Khan Academy · Finding mean of the grouped data with frequency
Finding Q1, Q3 and the IQR step by step.
Khan Academy · How to calculate interquartile range IQR
Building the five-number summary into a diagram.
Khan Academy · Box and whisker plot
The core teaching is above. These are the next steps, not a replacement for it.
18 questions by skill, with concise reveal answers and methods.
Use the short exam-style self-check when you want a fast confidence check.
Use its own worked examples for extra explanation and exercises.
Official state-owned learner books and teacher support for Grade 10 Mathematics.
This topic continues into Grade 11 and Grade 12.
Short answers for the checks learners make while preparing for the Grade 10 CAPS exam.
Revise measures of central tendency in ungrouped data; calculate an estimated mean, modal interval and median interval for grouped data; revise range and extend it to percentiles, quartiles, the interquartile range and semi-interquartile range; find the five-number summary and draw a box-and-whisker diagram; and use these summaries with graphs to comment meaningfully on data in context.
Once data is grouped into intervals, the individual values are lost — you only know how many values fell inside each interval, not their exact values. Using the midpoint of each interval as a stand-in for every value inside it gives a good estimate, not the exact original mean.
No. Variance and standard deviation are introduced in Grade 11, alongside histograms, frequency polygons and ogives. Grade 10 Statistics covers central tendency, grouped-data estimates, quartiles and the box-and-whisker diagram.
Arrange the data in order and find the median, which splits it into a lower half and an upper half (excluding the median itself if there is an odd number of values). Q1 is the median of the lower half; Q3 is the median of the upper half.
Finish the interactive slides, open the Mastery Bank, then take the short self-test without notes.