Deciles and percentiles Revision Notes for AQA GCSE Statistics

Deciles and Percentiles

Understanding deciles and percentiles is essential for analysing how data is distributed and finding specific positions within a dataset. These measures help us divide data into equal parts and locate particular values within grouped data.

What are deciles and percentiles?

Deciles are values that split your data into 10 equal parts. Think of it like dividing a group of people into 10 equally-sized queues. Each decile represents 10% of your data.

Percentiles work similarly but divide your data into 100 equal parts instead. Each percentile represents 1% of your data. You've probably heard of percentiles before - exam results are often given as percentiles to show what percentage of students scored below a particular mark.

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Both deciles and percentiles are particularly useful when working with continuous data that has been grouped into intervals, and they can be found using cumulative frequency diagrams.

Finding deciles using cumulative frequency diagrams

To find any decile, you need to use this key formula:

Position of dth decile = $\frac{d}{10} \times n$

Where:

$d$ = the decile number you want (1st, 2nd, 3rd, etc.)
$n$ = total number of data values

Here's how to find a decile step-by-step:

Calculate the position using the formula above
Find this position on the vertical (cumulative frequency) axis of your graph
Draw a horizontal line across to meet the curve
Draw a vertical line down to read the value on the horizontal axis

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The graph above demonstrates finding the 4th decile. The calculation shows: $\frac{4}{10} \times 50 = 20$ , so we look for the 20th value on the cumulative frequency axis, which corresponds to 10 minutes.

Finding percentiles using cumulative frequency diagrams

The method for percentiles is almost identical to deciles, but uses a different formula:

Position of pth percentile = $\frac{p}{100} \times n$

Where:

$p$ = the percentile number you want (1st, 2nd, 3rd, etc.)
$n$ = total number of data values

The same four-step process applies:

Calculate the position using $\frac{p}{100} \times n$
Find this position on the cumulative frequency axis
Draw horizontal line to the curve
Draw vertical line down to read the value

Worked example: distances travelled to work

Let's work through a complete example using a cumulative frequency diagram showing distances 150 people travelled to work.

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Worked Example: Finding Deciles and Percentiles from Cumulative Frequency

Part (a): Find the 6th decile

Step 1: Calculate position = $\frac{6}{10} \times 150 = 90$ th value Step 2: Find 90 on the cumulative frequency axis Step 3: Read across to the curve, then down to the distance axis Step 4: The 6th decile ≈ 11km

Part (b): Find the 36th percentile

Step 1: Calculate position = $\frac{36}{100} \times 150 = 54$ th value
Step 2: Find 54 on the cumulative frequency axis Step 3: Read across to the curve, then down to the distance axis Step 4: The 36th percentile ≈ 7km

Part (c): Find the 20th to 80th interpercentile range

First find the 80th percentile: $\frac{80}{100} \times 150 = 120$ th value ≈ 18km Then find the 20th percentile: $\frac{20}{100} \times 150 = 30$ th value ≈ 5km Interpercentile range = 18 - 5 = 13km

Working with frequency tables

Sometimes you'll need to find deciles and percentiles from frequency tables rather than graphs.

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Alternative Method: Using Frequency Tables

When working with frequency tables, you would:

Calculate the cumulative frequencies
Find the total ( $n$ = 5 + 9 + 10 + 14 + 20 + 2 = 60)
Use the same position formulas
Identify which class interval contains your required position
Use interpolation within that interval if needed

For example, to find the 20th percentile: Position = $\frac{20}{100} \times 60 = 12$ th value

Looking at cumulative frequencies: this falls in the second interval (10 < t ≤ 20), since the cumulative frequency reaches 14 by the end of this interval.

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Key Formulas to Remember

Decile position: $\frac{d}{10} \times n$
Percentile position: $\frac{p}{100} \times n$
Interdecile range: Higher decile value - Lower decile value
Interpercentile range: Higher percentile value - Lower percentile value

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Common Exam Tips and Traps

Always check you're reading the correct axis when using cumulative frequency graphs
Remember that deciles go up to the 9th decile (not 10th) and percentiles go up to the 99th percentile (not 100th)
When asked for a range, subtract the smaller value from the larger value
Round your final answers appropriately based on the context of the question
Make sure you understand whether you're finding a position or an actual value

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Key Points to Remember

Deciles split data into 10 equal parts - use formula $\frac{d}{10} \times n$ to find the position
Percentiles split data into 100 equal parts - use formula $\frac{p}{100} \times n$ to find the position
Always read across then down when using cumulative frequency graphs to find actual values
Ranges are found by subtracting the lower value from the higher value
Check your calculations twice - position formulas are easy to mix up in exam pressure

Deciles and percentiles (AQA GCSE Statistics): Revision Notes