Deciles and percentiles Revision Notes for Edexcel GCSE Statistics

Deciles and percentiles

What are deciles and percentiles?

Deciles and percentiles are measures of position that help us understand how data is spread out and where particular values sit within a dataset.

Deciles divide your data into 10 equal parts. This means each decile represents 10% of your data. The 1st decile contains the lowest 10% of values, the 2nd decile contains the lowest 20%, and so on.

Percentiles divide your data into 100 equal parts. Each percentile represents 1% of your data. The 25th percentile (also called the first quartile) contains the lowest 25% of values.

Both deciles and percentiles work brilliantly with grouped data when you use cumulative frequency diagrams or linear interpolation to find the exact values.

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Deciles and percentiles are particularly useful in statistics because they help us understand the distribution of data and identify where specific values fall within the overall dataset. They're commonly used in standardised testing, income analysis, and performance measurements.

Finding deciles from cumulative frequency diagrams

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

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

Where:

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

Step-by-step method:

Calculate the position using the formula above
Draw a horizontal line from this position on the vertical axis
Where this line meets the cumulative frequency curve, draw a vertical line down
Read the value where the vertical line meets the horizontal axis

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Worked Example: Finding the 4th Decile

To find the 4th decile when $n = 50$ :

Step 1: Apply the formula Position = $\frac{4}{10} \times 50 = 20$

Step 2: Locate the position Find the 20th value on the cumulative frequency curve

Step 3: Read the value Read across to find the corresponding data value

Finding percentiles from cumulative frequency diagrams

The method for percentiles follows exactly the same pattern, but uses a different formula:

$\text{Position of } P\text{th percentile} = \frac{P}{100} \times n$

Where:

$P$ = the percentile number you want
$n$ = total frequency

Step-by-step method:

Calculate the position using $\frac{P}{100} \times n$
Find this position on the vertical axis of your cumulative frequency diagram
Read horizontally across to the curve
Read vertically down to find the data value

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Worked Example: Finding the 76th Percentile

To find the 76th percentile when $n = 50$ :

Step 1: Apply the formula Position = $\frac{76}{100} \times 50 = 38$

Step 2: Locate the position Find the 38th value on the curve

Step 3: Read the value Read the corresponding data value from the horizontal axis

Worked example with step-by-step solution

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 the 6th Decile

Step 1: Calculate position = $\frac{6}{10} \times 150 = 90$ th value

Step 2: Find the 90th value on the vertical axis

Step 3: Read across to the curve, then down to the horizontal axis

Step 4: The 6th decile ≈ 11km

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Worked Example: Finding the 36th Percentile

Step 1: Calculate position = $\frac{36}{100} \times 150 = 54$ th value

Step 2: Find the 54th value on the vertical axis

Step 3: Read across to the curve, then down to the horizontal axis

Step 4: The 36th percentile ≈ 7km

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Worked Example: Finding the Interpercentile Range

The 20th to 80th interpercentile range shows the spread of the middle 60% of the data.

Step 1: Find the 80th percentile: $\frac{80}{100} \times 150 = 120$ th value

Step 2: Find the 20th percentile: $\frac{20}{100} \times 150 = 30$ th value

Step 3: From the diagram: 80th percentile ≈ 18km, 20th percentile ≈ 5km

Step 4: Interpercentile range = 18 - 5 = 13km

Common exam traps and tips

Exam traps to avoid:

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Common Mistakes to Watch Out For:

Forgetting to multiply by n: Always remember to multiply your fraction by the total frequency
Reading the wrong axis: Make sure you're reading from the cumulative frequency axis (vertical) to find your position
Mixing up deciles and percentiles: Deciles use tenths (divide by 10), percentiles use hundredths (divide by 100)
Rounding errors: Be careful with your calculations, especially when dealing with positions that aren't whole numbers

Problem-solving tips:

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Helpful Problem-Solving Strategies:

Always show your working: Write down the formula and substitute the values clearly
Check your answer makes sense: A high percentile should give a high data value
Use estimation: If the 50th percentile is around the median, does your answer seem reasonable?
Practice reading cumulative frequency curves: This skill takes practice to get accurate readings

Key formulas to memorise:

dth decile position: $\frac{d}{10} \times n$
pth percentile position: $\frac{P}{100} \times n$
Interdecile range: higher decile - lower decile
Interpercentile range: higher percentile - lower percentile

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Key Points to Remember:

Deciles split data into 10 equal parts - use the formula $\frac{d}{10} \times n$ to find the position
Percentiles split data into 100 equal parts - use the formula $\frac{P}{100} \times n$ to find the position
Always multiply your fraction by the total frequency (n) to get the correct position
Read carefully from cumulative frequency diagrams - horizontal from the vertical axis to the curve, then vertical down to read the value
Interpercentile and interdecile ranges measure spread - subtract the lower value from the higher value to find the range

Deciles and percentiles (Edexcel GCSE Statistics): Revision Notes