Statistics

Statistics involves collecting, organising, representing, analysing and interpreting data. In examinations, learners must calculate accurately, choose suitable graphs and explain what the results mean in context.

Measures of central tendency

Measures of central tendency describe the centre or typical value of a data set.

• Mean: the sum of all values divided by the number of values.
• Median: the middle value after arranging the data in ascending order.
• Mode: the value occurring most frequently.

The mean formula is:

\overline{x}=\frac{\sum x}{n}

For a frequency table:

\overline{x}=\frac{\sum fx}{\sum f}

Example

Question: Calculate the mean, median and mode of the marks:

4,\ 7,\ 7,\ 8,\ 9

Given information:

n=5

Mean formula:

\overline{x}=\frac{\sum x}{n}

Substitute:

\overline{x}=\frac{4+7+7+8+9}{5}

Simplify:

\overline{x}=\frac{35}{5} \overline{x}=7

The third value is the median:

\text{Median}=7

The most frequent value is:

\text{Mode}=7

Common Mistake: Do not find the median before arranging the data in ascending order.

Measures of dispersion

Measures of dispersion describe how spread out the data values are.

• Range measures the total spread.
• Interquartile range measures the spread of the middle half of the data.
• Semi interquartile range is half the interquartile range.
• Standard deviation measures the average spread around the mean.

The formulas are:

\text{Range}=\text{Maximum}-\text{Minimum} \text{IQR}=Q_3-Q_1 \text{Semi-interquartile range}=\frac{Q_3-Q_1}{2}

Example

Given:

2,\ 4,\ 5,\ 7,\ 8,\ 10,\ 12

Calculate the range:

\text{Range}=12-2 \text{Range}=10

The lower half is:

2,\ 4,\ 5

Therefore:

Q_1=4

The upper half is:

8,\ 10,\ 12

Therefore:

Q_3=10

Calculate the interquartile range:

\text{IQR}=10-4 \text{IQR}=6

Remember: The interquartile range is less affected by extreme values than the range.

Standard deviation

Standard deviation shows how far data values generally lie from the mean. A small standard deviation indicates that the values are clustered near the mean. A large standard deviation indicates greater variation.

The population standard deviation is:

\sigma=\sqrt{\frac{\sum(x-\overline{x})^2}{n}}

Example

Question: Calculate the standard deviation of:

2,\ 4,\ 4,\ 6

Step 1: Calculate the mean.

\overline{x}=\frac{2+4+4+6}{4} \overline{x}=\frac{16}{4} \overline{x}=4

Step 2: Write the formula.

\sigma=\sqrt{\frac{\sum(x-\overline{x})^2}{n}}

Step 3: Substitute.

\sigma=\sqrt{\frac{(2-4)^2+(4-4)^2+(4-4)^2+(6-4)^2}{4}}

Step 4: Simplify.

\sigma=\sqrt{\frac{4+0+0+4}{4}} \sigma=\sqrt{2}

Step 5: Final answer.

\sigma\approx1{,}41

Exam Tip: Use the population standard deviation function on your calculator unless the question instructs otherwise. Do not round intermediate answers too early.

Five number summary

The five number summary provides the main values needed to describe the distribution and draw a box and whisker plot.

It consists of:

• Minimum value
• Lower quartile
• Median
• Upper quartile
• Maximum value

Example

Given:

3,\ 5,\ 6,\ 8,\ 9,\ 11,\ 14,\ 16

There are eight values. The median is the mean of the two middle values:

Q_2=\frac{8+9}{2} Q_2=8{,}5

The lower half is:

3,\ 5,\ 6,\ 8 Q_1=\frac{5+6}{2} Q_1=5{,}5

The upper half is:

9,\ 11,\ 14,\ 16 Q_3=\frac{11+14}{2} Q_3=12{,}5

The five number summary is:

3;\ 5{,}5;\ 8{,}5;\ 12{,}5;\ 16

Box and whisker plots

A box and whisker plot displays the five number summary on a number line.

• The box extends from the lower quartile to the upper quartile.
• A line inside the box represents the median.
• The whiskers extend to the minimum and maximum values.
• A longer section indicates greater spread.
• The median’s position helps identify skewness.

Example

Using the five number summary:

3;\ 5{,}5;\ 8{,}5;\ 12{,}5;\ 16

Draw a number line and mark these five values. Draw the box from:

Q_1=5{,}5\quad\text{to}\quad Q_3=12{,}5

Draw the median at:

Q_2=8{,}5

Extend the whiskers to:

\text{Minimum}=3,\quad\text{Maximum}=16

Exam Tip: Box and whisker plots being compared must use the same scale.

Histograms

A histogram represents continuous grouped data. The bars touch because the intervals are continuous.

Important facts:

• The horizontal axis shows class intervals.
• The vertical axis usually shows frequency.
• There are no gaps between bars.
• Class boundaries must be used where necessary.
• A histogram differs from a bar graph because bar graph categories are separate.

Example

Suppose the following grouped marks are given:

\begin{array}{c|c} \text{Mark interval}&\text{Frequency}\\ 0\leq x<10&2\\ 10\leq x<20&5\\ 20\leq x<30&8\\ 30\leq x<40&4 \end{array} [/katex] <p class="wp-block-paragraph">Draw four touching bars with heights:</p> [katex display="true"] 2,\ 5,\ 8,\ 4

The modal class is the interval with the greatest frequency:

20\leq x<30 [/katex] <p class="wp-block-paragraph">Common Mistake: Do not leave spaces between the bars of a histogram.</p> <div style="height:50px" aria-hidden="true" class="wp-block-spacer"></div> <figure class="wp-block-image size-large"><a href="https://sqooltutors.co.za/sign-up-b/"><img src="https://lmxddlwowqlsyucleuyv.supabase.co/storage/v1/object/public/pdf_ads/batch%201/Doc%20Image%20Aug%2022,%202026,%2004_22_20%20PM%20(1).png" alt=""/></a></figure> <h2 class="wp-block-heading">Ogives</h2> <p class="wp-block-paragraph">An ogive is a cumulative frequency curve. It shows how many observations are less than or equal to particular upper class boundaries.</p> <p class="wp-block-paragraph">To draw an ogive:</p> <ul class="wp-block-list"> <li>Calculate cumulative frequencies.</li> <li>Plot upper class boundaries against cumulative frequencies.</li> <li>Include the lower starting boundary with cumulative frequency zero.</li> <li>Join the points using a smooth increasing curve.</li> </ul> <div style="height:50px" aria-hidden="true" class="wp-block-spacer"></div> <h3 class="wp-block-heading">Example</h3> <p class="wp-block-paragraph">Using frequencies:</p> [katex display="true"] 2,\ 5,\ 8,\ 4

Calculate cumulative frequencies:

2 2+5=7 7+8=15 15+4=19

Plot the points:

(0;0),\ (10;2),\ (20;7),\ (30;15),\ (40;19)

The median corresponds to half the total frequency:

\frac{19}{2}=9{,}5

Locate the cumulative frequency of approximately 9{,}5 on the vertical axis and read the estimated median from the horizontal axis.

Scatter plots

A scatter plot represents paired numerical data as points on the Cartesian plane. It helps identify a relationship between two variables.

• Positive correlation: points generally rise from left to right.
• Negative correlation: points generally fall from left to right.
• No correlation: no clear pattern is visible.
• An outlier lies far from the general pattern.

Example

Consider the points:

(1;9),\ (2;7),\ (3;6),\ (4;4),\ (5;2)

As x increases, y generally decreases. The data therefore shows a strong negative correlation.

Practical Application: A scatter plot can compare the age of a car with its selling price.

Regression line

A least squares regression line models the relationship between two variables and can be used for prediction.

Its equation is commonly written as:

\widehat{y}=a+bx

Here, b is the gradient and a is the y-intercept.

An examination data set relating car age to selling price produced the regression equation:

\widehat{y}=331\,397{,}20-22\,988{,}32x

The negative gradient shows that the predicted selling price decreases as the car becomes older.

Example

Question: Predict the selling price of a five-year-old car.

Given information:

x=5

Write the formula:

\widehat{y}=331\,397{,}20-22\,988{,}32x

Substitute:

\widehat{y}=331\,397{,}20-22\,988{,}32(5)

Simplify:

\widehat{y}=331\,397{,}20-114\,941{,}60

Final answer:

\widehat{y}=\text{R}216\,455{,}60

The gradient means that the predicted selling price decreases by approximately \text{R}22\,988{,}32 per year.

Common Mistake: Do not use the regression equation to predict far outside the range of the original data. This is called extrapolation and may be unreliable.

Correlation coefficient

The correlation coefficient, represented by r, measures the strength and direction of a linear relationship.

-1\leq r\leq1

• A value close to 1 indicates strong positive correlation.
• A value close to -1 indicates strong negative correlation.
• A value close to 0 indicates weak or no linear correlation.

Example

Given:

r=-0{,}95

The negative sign indicates negative correlation. Since the value is close to -1, the relationship is strong. The points should lie close to a downward regression line, making predictions within the observed range reasonably reliable.

Remember: Correlation does not prove that one variable causes the other variable to change.

Interpretation of statistical data

Interpretation requires learners to explain calculations and graphs in the context of the question.

Important comparisons include:

• A larger mean indicates a higher average.
• A smaller standard deviation indicates more consistent data.
• A larger interquartile range indicates greater spread in the middle half.
• A longer box or whisker indicates greater variation.
• An outlier may affect the mean, range, standard deviation and regression line.
• The median and interquartile range are resistant to extreme values.

Example

Class A has:

\overline{x}=65,\quad \sigma=4

Class B has:

\overline{x}=65,\quad \sigma=11

Both classes have the same mean, so their average performance is equal. Class A has the smaller standard deviation, so its marks are more consistent and lie closer to the mean. Class B’s marks are more widely spread.

Exam Tip: Do not merely write “Class A is better”. State whether you are comparing average performance, consistency or spread, and support the conclusion with the relevant statistic.