A-Level · Mathematics · AQA · Mark scheme decoded
AQA A-Level Mathematics: Measures of Central Tendency and Variation: Standard Deviation — mark scheme explained
The short answer
Understanding measures of central tendency and variation is crucial in statistics. These measures help us summarize and interpret data effectively. In this section, we will focus on the standard deviation, a key measure of variation, and how to calculate it using both raw data and summary statistics.
The question
Calculate the standard deviation of the dataset: 3, 5, 7, 9, 11.
[Paraphrased for study — not reproduced from any exam paper.]
Mark scheme, decoded
How the examiner actually awards the marks on this topic.
Gradora's own decode of the marking approach — not the exam board's published mark scheme.
How marks are awarded
For questions involving standard deviation, marks are typically awarded for correct use of formulas, accurate calculations, and appropriate interpretation of results. Ensure you show all steps clearly and round to the required number of decimal places.
What the command words demand
- Calculate
- Perform a mathematical operation to find a specific value.
- Interpret
- Explain the meaning of statistical measures in the context of the data.
- Compare
- Identify and explain differences between two or more sets of data.
Model answer
A full-mark response to the question above, worked through step by step.
Timing: Allocate about 5-7 minutes for a 5-mark question on standard deviation. This allows time for careful calculation and checking your work.
- Calculate the mean (μ): (3 + 5 + 7 + 9 + 11) / 5 = 71 mark
- Calculate the squared differences from the mean: (3 - 7) 2 , (5 - 7) 2 , (7 - 7) 2 , (9 - 7) 2 , (11 - 7) 21 mark
- Sum the squared differences: 16 + 4 + 0 + 4 + 16 = 401 mark
- Divide by the number of values (N): 40 / 5 = 81 mark
- Take the square root: √8 ≈ 2.831 mark
Final answer: The standard deviation is approximately 2.83.
Work through every step correctly and you earn all 5 marks.
Another worked example
Given the summary statistics for a dataset: Σx i = 100, Σx i 2 = 2400, N = 10. Calculate the standard deviation.
- Calculate the mean (μ): μ = Σx i / N = 100 / 10 = 101 mark
- Use the formula for population standard deviation: σ = √((Σx i 2 / N) - μ 2 )1 mark
- Substitute the values: σ = √((2400 / 10) - 10 2 ) = √(240 - 100) = √140 ≈ 11.833 marks
Final answer: The standard deviation is approximately 11.83.
Work through every step correctly and you earn all 5 marks.
Common mistakes
Using the population formula for a sample dataset
Why it happens: Students often confuse the formulas for population and sample standard deviations. The sample standard deviation uses (n - 1) in the denominator to provide an unbiased estimate.
Fix: Always use the correct formula based on whether you are dealing with a population or a sample.
Forgetting to square the differences from the mean
Why it happens: Students sometimes forget to square the differences when calculating the variance, leading to incorrect results.
Fix: Ensure you square each difference before summing them up.
Using the wrong mean in calculations
Why it happens: Students might use the population mean when they should be using the sample mean, or vice versa.
Fix: Double-check which mean you are using and ensure it matches the type of standard deviation you are calculating.
Dividing by N instead of (n - 1) for sample standard deviation
Why it happens: Students often forget to use (n - 1) in the denominator when calculating the sample standard deviation, leading to a biased estimate.
Fix: Always use (n - 1) in the denominator for sample standard deviation.
Forgetting to take the square root at the end
Why it happens: Students sometimes forget to take the final step of taking the square root, resulting in the variance instead of the standard deviation.
Fix: Always remember to take the square root of the variance to get the standard deviation.
Using incorrect summary statistics
Why it happens: Students might use incorrect values for Σx i or Σx i 2 , leading to incorrect calculations.
Fix: Double-check the given summary statistics and ensure they are used correctly in the formula.
Rounding too early
Why it happens: Students often round intermediate results, which can lead to significant errors in the final answer.
Fix: Avoid rounding until the final step and use as many decimal places as necessary during calculations.
Where the marks go
The question types you’ll meet on this topic and the marks each one carries — so you know what to expect and where to focus.
| Question type | What you’re asked to do | Marks |
|---|---|---|
| Calculate Standard Deviation | Find the standard deviation of a small dataset by working through the variance formula. | 5 |
| Calculate Standard Deviation | Find the population standard deviation from summary statistics using the given totals. | 5 |
| Calculate Standard Deviation | Compute the sample standard deviation from a small set of given data values. | 5 |
| Calculate Standard Deviation | Use the summary statistics to compute the sample standard deviation from given sums and sample size. | 5 |
| Total across these question types | 20 | |
Question types and mark tariffs are Gradora’s guidance based on how this topic is typically examined — not the board’s official paper structure.