Grouped continuous data Revision Notes for Edexcel GCSE Statistics

Grouped continuous data

Understanding the formula for grouped continuous data

When working with grouped continuous data, calculating the standard deviation requires a specific approach. The most straightforward formula to use is:

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

This formula is designed specifically for grouped data where we don't have individual values, but rather intervals or classes with frequencies.

Why we get estimates with continuous data

An important concept to understand is that when dealing with continuous data that has been grouped into intervals, our standard deviation calculation produces an estimate rather than an exact value.

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This happens because we use the midpoint of each interval to represent all values within that class, just like we do when estimating the mean for continuous data.

Setting up your working columns

To calculate standard deviation for grouped continuous data, you need to add working columns to your frequency table:

Mid (midpoint) - Find the middle value of each class interval
f × Mid - Multiply frequency by midpoint
f × Mid² - Multiply frequency by the square of the midpoint

These working columns allow you to find the three key sums needed for the formula: ∑f, ∑fx, and ∑fx².

Worked example: Time to solve puzzles

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Worked Example: Time to Solve Puzzles

Let's work through a complete example using data about the time students took to solve a puzzle:

Time, T (s)	Frequency, f	Mid T	fT	fT²
0 < T ≤ 20	6	10	60	600
20 < T ≤ 40	11	30	330	9900
40 < T ≤ 60	13	50	650	32500
60 < T ≤ 80	6	70	420	29400
80 < T ≤ 100	4	90	360	32400
Totals	40		1820	104800

Step-by-step calculation process:

Find the midpoints: For each interval, add the boundaries and divide by 2
- For 0 < T ≤ 20: midpoint = (0 + 20) ÷ 2 = 10
Calculate working columns:
- fT: multiply each frequency by its midpoint
- fT²: multiply each frequency by the square of its midpoint
Find the totals:
- ∑f = 40, ∑fT = 1820, ∑fT² = 104800
Calculate the mean:
- Mean = ∑fT ÷ ∑f = 1820 ÷ 40 = 45.5 seconds
Apply the standard deviation formula:
- Standard deviation = $\sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}$
- Standard deviation = $\sqrt{\frac{104800}{40} - 45.5^2}$
- Standard deviation = $\sqrt{2620 - 2070.25}$
- Standard deviation = $\sqrt{549.75}$ = 23.4 seconds (to 3 s.f.)

Another worked example: Bird flight distances

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Worked Example: Bird Flight Distances

Here's another example showing distances birds flew from their nests:

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

Completing the calculation:

Calculate the mean:
- Mean = 2440 ÷ 60 = 40.67 metres (to 2 d.p.)
Calculate standard deviation:
- Standard deviation = $\sqrt{\frac{145200}{60} - 40.67^2}$
- Standard deviation = $\sqrt{2420 - 1654.05}$
- Standard deviation = $\sqrt{765.95}$ = 27.7 metres (to 3 s.f.)

Important exam tips

Understanding the process is crucial, but avoiding common pitfalls will help you achieve accurate results in your calculations.

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Common Mistakes to Avoid:

Remember the square: Don't forget to square the midpoint when calculating the fT² column
Check your arithmetic: Double-check your column totals as errors here will affect your final answer
Round appropriately: Usually give your answer to 3 significant figures unless told otherwise
Show working columns: In most exam questions, you'll need to show your working columns clearly
Label your answer: Always include appropriate units in your final answer

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Key Points to Remember:

The standard deviation formula for grouped continuous data is: $\sqrt{\frac{\sum fx^2}{\sum f} - \left(\frac{\sum fx}{\sum f}\right)^2}$
You need three working columns: midpoint, f × midpoint, and f × midpoint²
The result is an estimate because we use midpoints to represent all values in each class interval
Always check your column arithmetic carefully - small errors multiply through the calculation
Remember to take the square root as your final step, and round to an appropriate number of significant figures

Grouped continuous data (Edexcel GCSE Statistics): Revision Notes