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
What each mark is really for — in plain English — and the wording trap that loses it.
- S1
Step 1: Calculate the mean (μ): (3 + 5 + 7 + 9 + 11) / 5 = 7
- S2
Step 2: Calculate the squared differences from the mean: (3 - 7) 2 , (5 - 7) 2 , (7 - 7) 2 , (9 - 7) 2 , (11 - 7) 2
- S3
Step 3: Sum the squared differences: 16 + 4 + 0 + 4 + 16 = 40
- S4
Step 4: Divide by the number of values (N): 40 / 5 = 8
- S5
Step 5: Take the square root: √8 ≈ 2.83
Model answer
Worked through, with each step tagged to the mark it earns.
- S1
Step 1: Calculate the mean (μ): (3 + 5 + 7 + 9 + 11) / 5 = 7
- S2
Step 2: Calculate the squared differences from the mean: (3 - 7) 2 , (5 - 7) 2 , (7 - 7) 2 , (9 - 7) 2 , (11 - 7) 2
- S3
Step 3: Sum the squared differences: 16 + 4 + 0 + 4 + 16 = 40
- S4
Step 4: Divide by the number of values (N): 40 / 5 = 8
- S5
Step 5: Take the square root: √8 ≈ 2.83
Final answer: The standard deviation is approximately 2.83.
Common mistakes
- Using the population formula for a sample dataset — 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 — Ensure you square each difference before summing them up.
- Using the wrong mean in calculations — 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 — Always use (n - 1) in the denominator for sample standard deviation.
- Forgetting to take the square root at the end — Always remember to take the square root of the variance to get the standard deviation.
- Using incorrect summary statistics — Double-check the given summary statistics and ensure they are used correctly in the formula.
- Rounding too early — Avoid rounding until the final step and use as many decimal places as necessary during calculations.
Where the marks go
- Full worked solution (all marking points)5 marks