Transforming data Revision Notes for Edexcel GCSE Statistics

Transforming data

What is data transformation?

Data transformation occurs when you change all values in a dataset by the same amount or percentage. This might involve adding or subtracting the same number from each value, or multiplying each value by the same factor.

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The key principle to remember is: when all data values are transformed in the same way, the averages (mean, median, and mode) are also transformed in exactly the same way.

For example, if you add 5 to every data value, the mean, median, and mode will each increase by 5. If you multiply every value by 3, the mean, median, and mode will each be multiplied by 3.

Why transform data first?

Sometimes calculations become much easier when you transform the data before finding the averages. This is particularly useful when dealing with:

Large numbers that are difficult to work with
Decimals with many decimal places
Data that has a consistent pattern

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By simplifying the numbers first, you can perform calculations more quickly and reduce the chance of arithmetic errors. The final result will be exactly the same as if you had worked with the original complex numbers throughout.

Step-by-step method for calculating with transformed data

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Here's the general process:

Transform the data - Apply the same operation to each data value
Calculate the required average using the transformed data
Reverse the transformation - Apply the opposite operation to your answer

Let's see this in action with worked examples.

Worked example 1: Price data

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Worked Example: Finding the Mean with Transformed Data

Question: Find the mean, median and mode for these prices: £45, £28, £36, £57, £28

Original data: 2.05, 2.02, 2.14, 2.01, 2.20, 2.09

Step 1: Transform the data Subtract 2 from each value:

$2.05 - 2 = 0.05$
$2.02 - 2 = 0.02$
$2.14 - 2 = 0.14$
$2.01 - 2 = 0.01$
$2.20 - 2 = 0.20$
$2.09 - 2 = 0.09$

Then multiply each by 100:

$0.05 \times 100 = 5$
$0.02 \times 100 = 2$
$0.14 \times 100 = 14$
$0.01 \times 100 = 1$
$0.20 \times 100 = 20$
$0.09 \times 100 = 9$

Transformed data: 5, 2, 14, 1, 20, 9

Step 2: Calculate using transformed data $\text{Mean} = \frac{5 + 2 + 14 + 1 + 20 + 9}{6} = \frac{51}{6} = 8.5$

Step 3: Reverse the transformation

Divide by 100: $8.5 \div 100 = 0.085$
Add 2: $0.085 + 2 = 2.085$

Therefore, the mean of the original data is 2.085.

Worked example 2: Speed data

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Worked Example: 400m Race Times

Question: Find the mean speed for this 400m race data (in seconds): 59, 52, 56.5, 59.5, 58

Step 1: Transform the data Subtract 50 from each value:

$59 - 50 = 9$
$52 - 50 = 2$
$56.5 - 50 = 6.5$
$59.5 - 50 = 9.5$
$58 - 50 = 8$

Then multiply by 10:

$9 \times 10 = 90$
$2 \times 10 = 20$
$6.5 \times 10 = 65$
$9.5 \times 10 = 95$
$8 \times 10 = 80$

Transformed data: 90, 20, 65, 95, 80

Step 2: Calculate mean $\text{Mean} = \frac{90 + 20 + 65 + 95 + 80}{5} = \frac{350}{5} = 70$

Step 3: Reverse the transformation

Divide by 10: $70 \div 10 = 7$
Add 50: $7 + 50 = 57$

The mean speed is 57 seconds.

Working with percentage increases

When data is increased by a percentage, you multiply by a specific factor:

For a 20% increase: multiply by 1.2 (this gives you 120% of the original value) For a 5% increase: multiply by 1.05 (this gives you 105% of the original value)

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Remember that a percentage increase means you keep 100% of the original value plus the additional percentage. So a 20% increase means 100% + 20% = 120% = 1.2 times the original.

Example with visitor numbers

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Worked Example: Percentage Increase in Visitor Numbers

If the original mean number of visitors per month was £38.80, and visitor numbers increased by 20%:

New mean = £38.80 × 1.2 = £46.56

This same multiplier applies to the median and mode:

New median = £36 × 1.2 = £43.20
New mode = £28 × 1.2 = £33.60

Important tips

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Key Points to Remember:

Always remember to reverse your transformation - this is the most common mistake students make
The same transformation applies to all measures of average - if you multiply the data by 2, multiply the mean, median and mode by 2
Check your answer makes sense - if your original data was around 50-60, your final answer should be in a similar range
Use transformations strategically - they can save time even when the question doesn't explicitly ask you to use them

Common exam traps

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Watch out for these common mistakes:

Forgetting to reverse the transformation - always apply the opposite operation at the end
Only transforming one measure - remember that mean, median AND mode are all affected equally
Getting percentage multipliers wrong - 20% increase means multiply by 1.2, not 0.2

Remember!

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Essential Takeaways:

When all data values are transformed by the same amount, all averages are transformed by that same amount
The process is: transform → calculate → reverse transformation
For percentage increases, use multipliers (20% increase = multiply by 1.2)
Always check your final answer makes sense compared to the original data
This method can save significant time in calculations, even when not explicitly required

Transforming data (Edexcel GCSE Statistics): Revision Notes