Grouped continuous data Revision Notes for AQA GCSE Statistics

Grouped continuous data

When working with grouped continuous data, calculating the standard deviation requires a special approach since we don't have individual data values. Instead, we work with class intervals and frequencies, which means our result will be an estimate rather than an exact value.

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Key Concept: Unlike working with individual data points, grouped continuous data calculations always produce estimates because we use mid-point values to represent entire class intervals.

Understanding the formula

For grouped continuous data, the standard deviation formula is:

$\text{Standard deviation} = \sqrt{\frac{\Sigma fx^2}{\Sigma f} - \left(\frac{\Sigma fx}{\Sigma f}\right)^2}$

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This formula might look complex, but it's essentially saying "find the square root of the mean of the squares minus the square of the mean". When dealing with grouped data, we use the mid-point of each class interval to represent all values in that group.

Setting up your calculation table

To calculate standard deviation for grouped continuous data, you need to create a table with these essential columns:

Class intervals: The ranges of your continuous data
Frequency (f): How many values fall in each interval
Mid-point: The middle value of each class interval
fD: Frequency multiplied by the mid-point
fD²: Frequency multiplied by the mid-point squared

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Why Mid-points Matter: The mid-point represents the typical value for all data in that class interval, which is why our final answer is an estimate. We assume all values in each class are concentrated at the mid-point.

Step-by-step calculation process

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Systematic Approach: Here's how to work through the calculation systematically:

Find the mid-point of each class interval
Calculate fD for each row (frequency × mid-point)
Calculate fD² for each row (frequency × mid-point²)
Sum all columns: Find $\Sigma f$ , $\Sigma fD$ , and $\Sigma fD^2$
Calculate the mean: Mean = $\frac{\Sigma fD}{\Sigma f}$
Apply the standard deviation formula

Worked example: Bird flight distances

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Worked Example: Calculating Standard Deviation for Bird Flight Distances

Let's work through a complete example using data about distances that birds flew from their nests.

Given data:

Distance intervals from 0m to 100m in 20m groups
Total of 60 birds observed

Step 1: Complete the table

Distance, D (m)	Frequency, f	Mid D	fD	fD²
0 < D ≤ 20	19	10	190	1900
20 < D ≤ 40	14	30	420	12600
40 < D ≤ 60	10	50	500	25000
60 < D ≤ 80	10	70	700	49000
80 < D ≤ 100	7	90	630	56700
Totals	60		2440	145200

Step 2: Calculate the totals

$\Sigma f = 60$
$\Sigma fD = 2440$
$\Sigma fD^2 = 145200$

Step 3: Find the mean $\text{Mean} = \frac{\Sigma fD}{\Sigma f} = \frac{2440}{60} = 40.67\text{m (to 2 d.p.)}$

Step 4: Calculate standard deviation $\text{Standard deviation} = \sqrt{\frac{\Sigma fD^2}{\Sigma f} - \left(\frac{\Sigma fD}{\Sigma f}\right)^2}$

$= \sqrt{\frac{145200}{60} - (40.67)^2}$

$= \sqrt{2420 - 1654.05}$

$= \sqrt{765.95} = \boxed{27.7\text{m (to 3 significant figures)}}$

Important points to remember

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Why this gives an estimate: Since we're using the mid-point to represent all values in each class interval, we're making an assumption about where the actual data points lie within each group. The real data could be spread differently within each interval.

Accuracy considerations: The estimate will be more accurate when:

Class intervals are relatively narrow
Data is fairly evenly distributed within each interval
You have a reasonable number of data points

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Common exam tips:

Always show your working clearly with the three key totals: $\Sigma f$ , $\Sigma fD$ , $\Sigma fD^2$
Remember to square the mid-point value when calculating $fD^2$
Don't forget the square root in the final step
Round sensibly to an appropriate number of significant figures

Remember!

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Key Points to Remember:

The formula for grouped continuous data standard deviation is $\sqrt{\frac{\Sigma fx^2}{\Sigma f} - \left(\frac{\Sigma fx}{\Sigma f}\right)^2}$
You need three key columns in your table: $f$ , $fD$ , and $fD^2$
Always use the mid-point of each class interval in your calculations
Your answer will be an estimate, not an exact value
Sum your columns carefully and double-check your arithmetic before applying the formula

Grouped continuous data (AQA GCSE Statistics): Revision Notes