Common standard deviation mistakes on exams

The dataset and the correct answer

Seven quiz scores: 4, 7, 7, 9, 10, 12, 15. The question says they are a sample. The full method is on the how to calculate standard deviation page; here is just the result to check against.

x̄ = 64 / 7 = 9.142857… Σ(x − x̄)² = 78.857 s² = 78.857 / 6 = 13.143 s = √13.143 = 3.63

Every mistake below starts from these numbers.

Try it: sample standard deviation calculator

The calculator below is loaded with the seven quiz scores; switch between sample and population to see mistake 1 happen, or type in a practice question of your own.

Open the full sample standard deviation calculator for more examples and the step-by-step working.

1. Dividing by n when the question wants n − 1

Wrong: √(78.857 / 7) = √11.265 = 3.36 Right: √(78.857 / 6) = √13.143 = 3.63

The two formulas answer different questions. Divide by n when the values are the entire population you care about; divide by n − 1 when they are a sample used to estimate a larger group's spread. The reasoning is covered in sample vs population standard deviation. On an exam, the fastest check is the wording: "a sample of", "randomly selected" and "drawn from" all mean n − 1. With seven values the correct answer is 8% larger, which is enough to lose the mark.

2. Stopping at the variance

Wrong: s = 13.14 Right: s = √13.14 = 3.63

The variance is the last number you write before the answer, so it is easy to box it and move on. Two checks catch this. First, units: if the scores are in points, the variance is in points squared, and a spread of 13 "points squared" is not something a quiz score can have. Second, size: a standard deviation of 13.14 would be larger than the whole range from 4 to 15, which is impossible for this data.

3. Rounding the mean too early

With the definition formula, rounding x̄ to 9.1 gives s = 3.6256, which still rounds to 3.63. The damage comes with the shortcut formula, where the rounded mean is squared and multiplied by n before a subtraction of two large numbers:

Σx² = 664 Wrong (x̄ ≈ 9): s² = (664 − 7 × 9²) / 6 = 97 / 6 = 16.17 → s = 4.02 Wrong (x̄ ≈ 9.1): s² = (664 − 7 × 9.1²) / 6 = 84.33 / 6 = 14.06 → s = 3.75 Right (x̄ = 64/7): s² = (664 − 64²/7) / 6 = 78.86 / 6 = 13.14 → s = 3.63

Rounding the mean by 0.14 moved the answer by 0.39. The safe habit is to write the shortcut as Σx² − (Σx)²/n, which uses only whole-number sums and never needs the mean at all.

4. Forgetting to square the deviations

Wrong: Σ(x − x̄) = 0, so s = 0 Right: Σ(x − x̄)² = 78.857

The raw deviations from the mean always add up to zero; that is a property of the mean, not a sign that the data has no spread. Students who notice this sometimes take absolute values instead, which gives 2.73. That is the mean absolute deviation, a legitimate measure of spread, but not the one the question asked for. A useful self-check: the deviations column should sum to zero (or very nearly, after rounding), and the squared column must be all positive.

5. Using the range, or the range rule, as the answer

Range: 15 − 4 = 11 Range rule: 11 / 4 = 2.75 Actual: s = 3.63

The range is a different measure of spread and depends on only two values. The "range ÷ 4" rule is a rough estimate for bell-shaped data with a moderate sample size, useful for checking an answer is in the right neighbourhood, never as the answer itself. Here it is 24% too low. If a question asks you to estimate the SD from the range it will say so.

6. Giving the standard error when the question asks for the SD

Standard error: 3.63 / √7 = 1.37 Standard deviation: 3.63

The standard error is the standard deviation of the sample mean, not of the scores. It is smaller by a factor of √n, and it answers "how precisely do we know the average?" rather than "how spread out are the scores?". Mixing them up is most common in multi-part questions where part (a) wants the SD and part (b) a confidence interval. The standard error calculator shows both side by side.

7. Reading σx instead of Sx on the calculator

Graphing calculators show both standard deviations after a one-variable statistics run. On a TI-84 the screen for these seven scores reads Sx = 3.625307869 and σx = 3.356382893. Sx divides by n − 1 and σx by n, so this is mistake 1 in another form, made by pressing the right buttons and copying the wrong line. The TI-84 guide walks through the screen. Recent Casio models show the same pair as sx and σx.

All seven at a glance

The "off by" column is measured against the correct 3.63. Only the n versus n − 1 error is small enough to look plausible, which is why it is the one that survives to the final answer most often.

A thirty-second check before you move on

Three quick tests catch almost every error above. The SD must be positive and smaller than the range. For most datasets it lands between about a sixth and a half of the range (here 3.63 against 11, about a third). And the units should match the data: points, not points squared. If you have a spare minute, run the numbers through the sample standard deviation calculator while revising, so that you know what a correct answer to a practice question looks like before the exam.