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Averages & Spread0%

Statistics & Probability · Topic 1 of 4

Averages & Spread

Video coming soon3 worked examples

Theory

The Golden Rule: Before finding the median or quartiles, you must always rewrite your list of numbers in ascending order. When comparing the interquartile range or standard deviation, you must never use the phrase "on average," as these are measures of spread, not averages.

1. Mean, Mode, Median and Range

Three of these are measures of average; the range is a measure of spread.

Mean — add all the values and divide by how many there are.
Mode — the value that appears most often (there can be more than one, or none).
Median — the middle value once the data is in order.
Range — the largest value minus the smallest.

\text{Mean} = \frac{\text{total of the values}}{\text{number of values}} \qquad \text{Range} = \text{Highest} - \text{Lowest}

2. The Five-Figure Summary

A five-figure summary is used to describe the spread of a dataset and consists of five specific values: the Lowest value (L), the Lower Quartile (Q1), the Median (Q2), the Upper Quartile (Q3), and the Highest value (H).

Step 1: Rewrite the raw list of numbers in order from lowest to highest.
Step 2: Draw an arrow to show the exact middle of the list; this is your median. If the middle falls between two numbers, add them together and divide by two.
Step 3: Find the middle of the lower half of the data to find the lower quartile, and the middle of the upper half to find the upper quartile.

3. Interquartile Range (IQR) & Semi-Interquartile Range (SIQR)

The Interquartile Range is calculated by subtracting the lower quartile from the upper quartile:

\text{IQR} = Q_3 - Q_1

The Semi-Interquartile Range is simply the IQR divided by 2:

\text{SIQR} = \frac{Q_3 - Q_1}{2}

where $Q_1$ is the lower quartile and $Q_3$ is the upper quartile.

4. Box Plots

A box plot is a visual drawing of the five-figure summary.

The lowest and highest values form the ends of the "whiskers".
A box is drawn starting at the lower quartile and ending at the upper quartile.
A vertical line is drawn inside the box to represent the median.

5. Standard Deviation

The standard deviation of a list of numbers is a measure of how spread out the numbers are from the mean.

A lower standard deviation indicates that the numbers are more consistent.
A higher standard deviation indicates that the numbers are more varied (or more spread out).

The formula is given in the exam booklet in two equivalent forms, so you do not need to memorise it — but you must be able to apply it:

s = \sqrt{\frac{\sum(x - \bar{x})^2}{n - 1}}

or, the version that avoids working out every difference from the mean:

s = \sqrt{\frac{\sum x^2 - \dfrac{(\sum x)^2}{n}}{n - 1}}

Here $x$ is each value, $\bar{x}$ is the mean, and $n$ is how many values there are. Either form gives the same answer — the first is easier to follow, the second is quicker when the mean is not a whole number.

⚠️ Common Examiner Traps

The "On Average Spread" Trap: a very common lost mark is writing "on average" when comparing interquartile ranges or standard deviations. You must completely avoid the phrase "on average" when making your second comment about standard deviation or the IQR, because these measure spread, not average.
The "Unordered Data" Trap: Candidates frequently rush to find the middle number of the list exactly as it is printed on the exam paper. If you do not rearrange the raw data into ascending order first, your median and quartiles will be completely wrong.
The "Forgotten Square Root" Trap: When calculating the standard deviation, the final step of the formula requires you to take the square root of your total. Candidates often do the complex division and then stop, losing the final process mark.

Worked examples

Example 1

The number of films Megan downloaded each month for a year is shown below:

34, 19, 22, 10, 13, 38, 9, 12, 26, 7, 19, 21

For this data, calculate the median, the lower quartile, the upper quartile, and the interquartile range. (3 marks)

Step 1: Rewrite the 12 numbers in ascending order.

7, 9, 10, 12, 13, 19, | 19, 21, 22, 26, 34, 38

Step 2: Find the median (the middle). The middle falls exactly between the two 19s. $\text{Median} = 19$

Step 3: Find the lower quartile (the middle of the bottom 6 numbers: 7, 9, 10, 12, 13, 19). The middle is between 10 and 12.

(10 + 12) \div 2 = 11 \text{ (Lower Quartile)}

Step 4: Find the upper quartile (the middle of the top 6 numbers: 19, 21, 22, 26, 34, 38). The middle is between 22 and 26.

(22 + 26) \div 2 = 24 \text{ (Upper Quartile)}

Step 5: Calculate the Interquartile Range (Q3 - Q1).

24 - 11 = 13 \text{ (IQR)}

Example 2

The prices of lambs sold in September were recorded. A sample of the prices, in pounds, is shown:

72, 75, 73, 68, 65, 70

For these prices, calculate the mean and the standard deviation. (4 marks)

Step 1: Calculate the mean ( $\bar{x}$ ). Add all the numbers and divide by $n$ (which is 6).

72 + 75 + 73 + 68 + 65 + 70 = 423

423 \div 6 = £70.50 \text{ (Mean)}

Step 2: Set up a table to find $(x - \bar{x})^2$ for the formula.

$65 - 70.5 = -5.5 \rightarrow (-5.5)^2 = 30.25$
$68 - 70.5 = -2.5 \rightarrow (-2.5)^2 = 6.25$
$70 - 70.5 = -0.5 \rightarrow (-0.5)^2 = 0.25$
$72 - 70.5 = 1.5 \rightarrow (1.5)^2 = 2.25$
$73 - 70.5 = 2.5 \rightarrow (2.5)^2 = 6.25$
$75 - 70.5 = 4.5 \rightarrow (4.5)^2 = 20.25$

Step 3: Sum the squared column ( $\sum(x - \bar{x})^2$ ).

30.25 + 6.25 + 0.25 + 2.25 + 6.25 + 20.25 = 65.5

Step 4: Divide by $n - 1$ (which is $6 - 1 = 5$ ) and then take the square root.

65.5 \div 5 = 13.1

s = \sqrt{13.1} = 3.61939...

Final Answer: The standard deviation is £3.62.

Example 3

Tommy also recorded the number of films he downloaded each month. The interquartile range for the number of films Tommy downloaded is 10.

Make one valid comment comparing the number of films Megan and Tommy downloaded. (1 mark)

Step 1: Identify that IQR is a measure of spread. Tommy's IQR (10) is lower than Megan's IQR (13, calculated in Example 1).

Step 2: Link a lower spread to the word "consistent" and completely avoid using the word "average".

Final Answer: "The number of films Tommy downloaded was more consistent than Megan because his interquartile range is lower."

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9 Averages and Spread questions with answers and video solutions