Range, quartiles and interquartile range Revision Notes for Edexcel GCSE Statistics

Range, quartiles and interquartile range

When working with grouped data, you cannot determine the exact individual values. This means you can only calculate estimates of the range, quartiles, and interquartile range. Understanding these measures of spread is crucial for analysing how scattered your data is.

Understanding the range for grouped data

The range tells us about the spread of data from the smallest to the largest value. For grouped data, we need to think about the possible minimum and maximum values within each class interval.

Formula:

$\text{Range} = \text{largest possible value} - \text{smallest possible value}$

Method for calculating range

Identify the smallest possible value: Look at the lowest class interval and find the minimum value that could occur
Identify the largest possible value: Look at the highest class interval and find the maximum value that could occur
Calculate the difference: Subtract the smallest from the largest

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Worked Example: Newspaper reading habits

Consider this frequency table showing numbers of newspapers people read:

Number of newspapers	Frequency
0-7	5
8-14	16
15-21	22
22-50	12

Step 1: The smallest possible value is 0 (from the 0-7 class) Step 2: The largest possible value is 50 (from the 22-50 class)
Step 3: Range = 50 - 0 = 50 newspapers

chatImportant

When data is rounded (like weights to the nearest kg), be extra careful about minimum and maximum possible values. For example, if weight $w$ satisfies $45 ≤ w ≤ 50$ , the actual minimum could be 45.5kg and maximum could be 65.5kg.

Finding quartiles for grouped data

Quartiles divide your data into four equal parts. Since we're working with grouped data, we need to use cumulative frequency and interpolation techniques.

The three quartiles

$Q_1$ (first quartile): The value below which 25% of the data falls
$Q_2$ (second quartile/median): The value below which 50% of the data falls
$Q_3$ (third quartile): The value below which 75% of the data falls

Step-by-step method for finding quartiles

Create a cumulative frequency column: Add up frequencies as you go down the table
Find the positions:
- $Q_1$ position = $\frac{n}{4}$ (where $n$ is the total frequency)
- $Q_2$ position = $\frac{n}{2}$
- $Q_3$ position = $\frac{3n}{4}$
Locate the class intervals: Find which class contains each quartile position
Use linear interpolation: Calculate the exact value within the class

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Worked Example: Travel distances

Distance, x (km)	Frequency	Cumulative frequency
0 < x ≤ 5	34	34
5 < x ≤ 10	38	72
10 < x ≤ 15	22	94
15 < x ≤ 20	16	110
20 < x ≤ 25	10	120

Total frequency ( $n$ ) = 120

Finding $Q_1$ :

$Q_1$ position = $\frac{120}{4} = 30$ th value
The 30th value falls in the 0 < x ≤ 5 class (cumulative frequency goes up to 34)
Since we need the 30th value out of 34 in this class: $Q_1 = 0 + \frac{30}{34} \times 5 = :success[4.41km]$

Finding $Q_2$ :

$Q_2$ position = $\frac{120}{2} = 60$ th value
The 60th value falls in the 5 < x ≤ 10 class (cumulative frequency goes from 34 to 72)
$Q_2 = 5 + \frac{(60-34)}{38} \times 5 ≈ :success[8.4km]$

Finding $Q_3$ :

$Q_3$ position = $\frac{3 \times 120}{4} = 90$ th value
The 90th value falls in the 10 < x ≤ 15 class
$Q_3 = 10 + \frac{(90-72)}{22} \times 5 ≈ :success[14.1km]$

Linear interpolation formula

infoNote

When using linear interpolation to find quartiles, use this formula:

$Q = L + \frac{(\text{position} - CF_{\text{before}})}{f} \times c$

Where:

$L$ = lower boundary of the class containing the quartile
position = the quartile position ( $\frac{n}{4}$ , $\frac{n}{2}$ , or $\frac{3n}{4}$ )
$CF_{\text{before}}$ = cumulative frequency before the quartile class
$f$ = frequency of the quartile class
$c$ = class width

Calculating the interquartile range

The interquartile range (IQR) measures the spread of the middle 50% of your data.

Formula:

$\text{IQR} = Q_3 - Q_1$

This is particularly useful because it's not affected by extreme values (outliers), making it a robust measure of spread.

Using our travel distance example: $\text{IQR} = Q_3 - Q_1 = 14.1 - 4.41 = \text{:success[9.69km]}$

chatImportant

Common exam traps and tips

Trap 1: Forgetting that grouped data gives estimates, not exact values Solution: Always use words like "estimate" in your answers

Trap 2: Using the wrong quartile positions Solution: Remember - $Q_1$ is $\frac{n}{4}$ , $Q_2$ is $\frac{n}{2}$ , $Q_3$ is $\frac{3n}{4}$

Trap 3: Incorrect interpolation when the quartile position falls exactly on a cumulative frequency boundary Solution: Check your cumulative frequency column carefully and ensure you're interpolating within the correct class

Trap 4: Mixing up class boundaries with class values Solution: Pay close attention to whether inequalities are strict (< or >) or inclusive (≤ or ≥)

Tip: Always show your working clearly, especially for the interpolation steps - this often earns method marks even if your final answer isn't perfect.

bookmarkSummary

Key Points to Remember:

Range for grouped data = largest possible value - smallest possible value (these are estimates)
Quartiles divide data into quarters - $Q_1$ (25%), $Q_2$ (50%), $Q_3$ (75%)
Use cumulative frequency to locate which class contains each quartile
Linear interpolation helps you find the exact quartile value within its class
IQR = $Q_3 - Q_1$ gives you the spread of the middle 50% of data

Range, quartiles and interquartile range (Edexcel GCSE Statistics): Revision Notes