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Central tendency • grouped data • quartiles • box & whisker
Grade 11 CAPS: histograms, frequency polygons, ogives, variance and standard deviation, symmetric vs skewed data, and identifying outliers — the tools that turn Grade 10's summaries into real graphical and numerical analysis.
Grade 10 gave you the five-number summary. Grade 11 gives you the graphs and the calculations that go deeper: histograms, ogives, variance, standard deviation, skewness and outliers. Work through every example in order.
You are here: the graphs and calculations year.
Central tendency • grouped data • quartiles • box & whisker
Histograms • ogives • variance & standard deviation • outliers
Bivariate data • scatter plots • regression & correlation
Three weeks, five connected skills, all about SEEING and MEASURING spread.
Draw histograms and frequency polygons from grouped data.
Draw an ogive (cumulative frequency curve) and use it to estimate the median and quartiles.
Calculate variance and standard deviation of ungrouped data — by hand for a small data set, and using a calculator for a larger one.
Identify whether data is symmetric or skewed, using the relationship between the mean and the median.
Identify outliers using a scatter plot AND a box-and-whisker diagram.
Bars whose HEIGHT shows the frequency of each interval.
Bars sit directly next to each other with NO gaps (unlike a bar graph for categorical data) — the data is continuous, so the intervals flow into each other.
The tallest bar shows the modal interval at a glance. The overall shape tells you immediately whether the data is roughly symmetric or lopsided.
One data set, used again for the frequency polygon and the ogive that follow.
Draw a histogram for this grouped frequency table of test scores (out of 100):
| Score | Frequency |
|---|---|
| \(40\le x<50\) | 3 |
| \(50\le x<60\) | 8 |
| \(60\le x<70\) | 14 |
| \(70\le x<80\) | 10 |
| \(80\le x<90\) | 4 |
| \(90\le x<100\) | 1 |
Using the histogram drawn above: (a) How many learners scored at least 80? (b) What percentage of learners scored below 60? (c) Which interval has the fewest learners?
The same data, as connected points instead of bars.
Plot a point at each interval's MIDPOINT, at a height equal to that interval's frequency, then join the points with straight lines.
To bring the line back down to the axis at both ends, add one extra point at each end: the midpoint of the (empty, frequency-0) interval immediately before the first, and immediately after the last.
Same 40 test scores as the histogram above.
Draw a frequency polygon for the same grouped data used in the histogram above.
Why do a histogram's bars touch, with no gaps between them?
A frequency polygon is plotted using each interval's...
A running total, turned into a graph — and a way to read off the median and quartiles.
Plot each point at the UPPER boundary of an interval, at a height equal to the CUMULATIVE (running-total) frequency up to and including that interval. Start with one extra point at the lower boundary of the first interval, at a height of 0.
Find the target position on the vertical axis (\(n/2\) for the median, \(n/4\) for \(Q_1\), \(3n/4\) for \(Q_3\)), draw a horizontal line to the curve, then drop straight down to read the value off the horizontal axis.
The same 40 test scores, one more time.
Draw an ogive for the same grouped data, then use it to estimate the median, \(Q_1\) and \(Q_3\).
Using the SAME ogive above, estimate the mean test score without being given the original frequency table.
An ogive is plotted using each interval's...
For a data set of n = 60, at what cumulative frequency position would you read off Q3 on an ogive?
A single number that measures spread around the MEAN — more powerful than the range or IQR.
For every value, find its distance from the mean (its "deviation"), square it (so negatives don't cancel positives), average all the squared deviations — that average IS the variance.
\(\sigma^2=\dfrac{\sum(x-\bar{x})^2}{n}\)
\(\sigma=\sqrt{\sigma^2}\) — the square root brings the units back to the SAME units as the original data (variance's units are "squared," which is hard to interpret directly).
A SMALL standard deviation means the data is tightly clustered around the mean. A LARGE standard deviation means the data is spread widely.
Every deviation and squared deviation shown — the manual method CAPS requires for a small data set.
The daily rainfall (in mm) recorded over one week in a small town was: \(4,\ 7,\ 9,\ 10,\ 12,\ 13,\ 15\). Calculate the mean, variance and standard deviation.
| \(x\) | \(x-\bar{x}\) | \((x-\bar{x})^2\) |
|---|---|---|
| 4 | \(-6\) | 36 |
| 7 | \(-3\) | 9 |
| 9 | \(-1\) | 1 |
| 10 | 0 | 0 |
| 12 | 2 | 4 |
| 13 | 3 | 9 |
| 15 | 5 | 25 |
A larger data set — CAPS expects the calculator's statistical mode here, not a full hand table.
Fifteen learners' test scores are: \(62,\ 58,\ 71,\ 65,\ 69,\ 73,\ 60,\ 66,\ 68,\ 64,\ 72,\ 59,\ 63,\ 70,\ 67\). Use your calculator's statistical mode to find the mean and standard deviation.
The hardest version of this skill: two unknowns, two equations, one genuine algebra problem.
Eight community health clinics each recorded the number of patients seen in one morning. The data set has a mean of 14 and a variance of \(18{,}5\): \(10,\ 14,\ 9,\ 17,\ 12,\ 11,\ a,\ b\). Determine the values of \(a\) and \(b\).
Six employees' monthly bonuses (R) are: \(800,\ 950,\ 1000,\ 1100,\ 1200,\ 1350\), with mean \(R1066{,}67\) and standard deviation \(\approx R177{,}17\). (a) If every employee's bonus is increased by a flat R200, what are the new mean and standard deviation? (b) If every employee's bonus is increased by 15% instead, what are the new mean and standard deviation?
Why are deviations squared before being averaged, instead of just averaging them directly?
A data set has variance 25. What is its standard deviation?
Two independent tests: compare the mean to the median, AND compare the median's distance to each quartile.
Mean \(\approx\) median. The data is evenly balanced on both sides of the centre.
Mean \(>\) median. A few unusually HIGH values pull the mean up above the median, stretching a "tail" to the right.
Mean \(<\) median. A few unusually LOW values pull the mean down below the median, stretching a "tail" to the left.
Four data sets, moving from a direct given comparison to full classification from raw data.
Classify: \(8,\ 9,\ 10,\ 11,\ 12\)
Nine stalls at a heritage festival recorded the following number of visitors (in tens): \(12,\ 14,\ 15,\ 16,\ 17,\ 18,\ 19,\ 20,\ 45\). Classify the shape of this data.
Nine voters' waiting times (in minutes) at a polling station on election day were: \(5,\ 30,\ 31,\ 32,\ 33,\ 34,\ 35,\ 36,\ 38\). Classify the shape of this data.
Classify: \(30,\ 32,\ 35,\ 37,\ 39,\ 43,\ 46,\ 47,\ 48,\ 50,\ 52,\ 58,\ 61,\ 67\). Confirm your answer using both the mean-vs-median test AND the quartile-distance test.
Classify: \(9,\ 12,\ 15,\ 17,\ 19,\ 20,\ 22,\ 24,\ 25,\ 27\)
A value far removed from the rest — found visually AND numerically.
Plot every value on a single number line. An outlier is a point sitting visibly far away from the main cluster of dots — usually obvious just by looking.
A value is an outlier if it lies below \(Q_1-1{,}5\times\text{IQR}\) (the lower fence) or above \(Q_3+1{,}5\times\text{IQR}\) (the upper fence). The whiskers then stop at the most extreme NON-outlier value, and any true outlier is plotted as a separate dot.
A short, clean lead-in, then the same data set checked both by eye and by the fence rule.
Seven community volunteers logged the following number of volunteering hours in one month: \(5,\ 6,\ 7,\ 8,\ 9,\ 10,\ 25\). Identify any outlier(s).
Eleven informal traders at a local market recorded the following weekly income (in hundreds of rand): \(22,\ 24,\ 25,\ 27,\ 28,\ 29,\ 30,\ 31,\ 33,\ 35,\ 58\). Identify any outlier(s).
In a data set, the mean is 45 and the median is 52. What does this tell you?
A data set has Q1 = 20 and Q3 = 32. Using the 1,5×IQR rule, what is the upper fence?
A full multi-part question combining every skill from this topic — skewness, variance and standard deviation, and identifying an outlier, all on one data set.
A small business recorded the number of items sold by each of its 12 sales agents in one week: \(18,\ 20,\ 21,\ 22,\ 23,\ 24,\ 25,\ 26,\ 27,\ 28,\ 29,\ 52\). (a) Classify the skewness of this data set. (b) Use your calculator's statistical mode to determine the variance and standard deviation. (c) Identify any outlier(s) using the \(1{,}5\times\text{IQR}\) rule. (d) Comment on what this outlier might mean for the business.
Work first. Open one answer only when your own line of working is complete.
Bars with frequencies \(5,9,16,8,2\). Answer: the 3rd interval (frequency 16, the tallest bar).
\(n=50\), ogive reads 25 at \(x=48\). Answer: the estimated median is 48 (position \(n/2=25\)).
Squared deviations \(4,1,0,1,4\), \(n=5\). Answer: variance \(=\dfrac{10}{5}=2\), \(\sigma=\sqrt2\approx1{,}41\).
Mean \(=80\), median \(=74\). Answer: mean \(>\) median, so positively (right) skewed.
Plotting an ogive point at an interval's midpoint instead of its upper boundary.
Reading your calculator's sample standard deviation (\(s_x\)) when CAPS wants the population one (\(\sigma_x\)).
Calling data "skewed" without actually comparing the mean and median numbers.
Drawing histogram bars with unequal widths — every bar's width must match its own interval exactly.
Assuming an ogive must start at \((0,0)\) — it starts at the lower boundary of the FIRST interval, whatever that value is.
Double-check: does your ogive only ever go up, never down?
State both the mean AND standard deviation together — one without the other is an incomplete answer.
Always finish an outlier question by naming the actual outlier value, not just "yes there is one."
If every value gets the same amount ADDED, only the mean moves — the standard deviation stays the same.
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.
Drawing and reading bar-height frequency histograms.
Khan Academy · How to interpret a histogram
Building the running total and drawing the curve.
Khan Academy India · Graph of Cumulative Frequency Distribution - Ogive
Calculating both step by step from a small data set.
StatQuest with Josh Starmer · Calculating the Mean, Variance and Standard Deviation, Clearly Explained
Using the interquartile range to flag unusual values.
Khan Academy · Judging outliers in a dataset
The core teaching is above. These are the next steps, not a replacement for it.
20 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 11 Mathematics.
This topic continues from Grade 10 into Grade 12.
Short answers for the checks learners make while preparing for the Grade 11 CAPS exam.
Draw and interpret histograms and frequency polygons from grouped data; draw an ogive (cumulative frequency curve) and use it to read off the median and quartiles; calculate variance and standard deviation of ungrouped data, both by hand for a small data set and using a calculator for a larger one; identify whether a data set is symmetric or skewed; and identify outliers using a scatter plot and a box-and-whisker diagram.
A histogram uses bars whose height shows the frequency of each interval. A frequency polygon plots a point at the midpoint of each interval (at its frequency) and joins the points with straight lines — it can be drawn directly from a histogram, or on its own.
Find n/2 on the cumulative frequency (vertical) axis, draw a horizontal line across to the ogive curve, then drop straight down to the horizontal axis — that value is the estimated median. The same method with n/4 gives Q1 and 3n/4 gives Q3.
CAPS expects both: you must be able to calculate the mean, variance and standard deviation manually for a small ungrouped data set (showing every deviation and squared deviation), and you must also be able to use your calculator's statistical mode for a larger data set.
Compare the mean and median. If they are equal (or very close), the data is symmetric. If the mean is greater than the median, the data is skewed to the right (positively skewed) — a few unusually high values pull the mean up. If the mean is less than the median, the data is skewed to the left (negatively skewed).
Finish the interactive slides, open the Mastery Bank, then take the short self-test without notes.