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Standard deviation practice problems
Standard deviation practice problems you answer on the page. Pick a topic, work the problem out, type your answer and press Check: you see at once whether it is right, the full worked solution, and the next problem.
Press Check. Leave the box empty and press Check to skip to a new problem.
Show the working, step by step
How to use it
- Choose a topic. A new problem appears straight away; changing the topic always starts a new one.
- Work it out on paper or with a calculator. Keep the mean unrounded until the end.
- Type your answer to 2 decimal places and press Check (or Enter).
- Read the verdict, then open "Show the working, step by step" for the full solution to the problem you just answered.
- The next problem is already showing. Clear the box and go again. To skip a problem, leave the box empty and press Check.
Answers are accepted within 0.01 of the correct value rounded to 2 decimal places. The running score counts every answered problem until you reload the page; nothing is stored.
The seven topics
| Topic | What you are given | What you find |
|---|---|---|
| Sample SD | 5–7 values from a larger group | s, dividing by n − 1 |
| Population SD | 5–7 values that are the whole group | σ, dividing by N |
| Variance | 5–7 values, labelled sample or population | s² or σ², no square root |
| Mean ± k SD | A mean, an SD and k = 1, 2 or 3 | The lower and upper limits |
| Z-score | A mean, an SD and one value | z = (x − μ) ÷ σ |
| Coefficient of variation | 5–7 sample values | CV = s ÷ x̄ × 100% |
| Frequency table | Values with frequencies, sample or population | s or σ using f·(x − x̄)² |
The two formulas behind almost every problem are:
s = √[ Σ(x − x̄)² ÷ (n − 1) ] σ = √[ Σ(x − μ)² ÷ N ]
If you are unsure which one a problem needs, the wording tells you: "a sample drawn from a larger group" means s, and "all the values of interest" means σ. The reasoning is in sample vs population standard deviation.
Three sample problems with solutions
These three show the kind of problem the tool sets and the level of working in its solutions. They are fixed, so you can read them without JavaScript.
Sample problem 1: sample standard deviation
A sample of five daily step counts, in thousands, is 14, 9, 11, 16, 10. Find the
sample standard deviation to 2 decimal places.
Show solution
Sum = 60, n = 5, so x̄ = 12.
| x | x − x̄ | (x − x̄)² |
|---|---|---|
| 14 | 2 | 4 |
| 9 | −3 | 9 |
| 11 | −1 | 1 |
| 16 | 4 | 16 |
| 10 | −2 | 4 |
| Σ 60 | 0 | 34 |
s² = 34 ÷ (5 − 1) = 8.5 s = √8.5 = 2.9155 ≈ 2.92 thousand steps
Dividing by 5 instead would give 2.61, which the checker would mark wrong.
Sample problem 2: z-score
Marks on an exam have a mean of 68 and a standard deviation of 7.5. Find the z-score of a mark of 81, to 2 decimal places.
Show solution
z = (x − μ) ÷ σ = (81 − 68) ÷ 7.5 = 13 ÷ 7.5 = 1.7333 ≈ 1.73
The mark is 1.73 standard deviations above the mean. For a mark below the mean the z-score is negative; type it with a minus sign. The z-score calculator converts a z-score to a percentile.
Sample problem 3: SD from a frequency table
Twenty households were asked how many cars they own. Treating these twenty as the whole population of a small street, find the population standard deviation to 2 decimal places.
| Cars x | Households f |
|---|---|
| 0 | 3 |
| 1 | 5 |
| 2 | 7 |
| 3 | 4 |
| 4 | 1 |
Show solution
N = Σf = 20 and Σf·x = 0 + 5 + 14 + 12 + 4 = 35, so μ = 35 ÷ 20 = 1.75.
| x | f | x − μ | f·(x − μ)² |
|---|---|---|---|
| 0 | 3 | −1.75 | 9.1875 |
| 1 | 5 | −0.75 | 2.8125 |
| 2 | 7 | 0.25 | 0.4375 |
| 3 | 4 | 1.25 | 6.25 |
| 4 | 1 | 2.25 | 5.0625 |
| Σ | 20 | 23.75 |
σ² = 23.75 ÷ 20 = 1.1875 σ = √1.1875 = 1.0897 ≈ 1.09 cars
The common mistake is to forget the f in f·(x − μ)², which treats each row as a single household. The frequency table calculator shows the same working for any table.
Mistakes the checker catches most often
Using the wrong divisor. On five values, dividing by n instead of n − 1 shrinks the SD by about 11% (the factor is √(4/5) = 0.894), which moves the answer well outside the 0.01 margin.
Rounding the mean too early. If the mean is 38.5714 and you use 38.6, every deviation is off by 0.03 and the error builds up in the squares. Keep four or more decimals, or the exact fraction.
Stopping at the variance, or square-rooting it when the question asks for the variance. Check the symbol: s² and σ² are variances, s and σ are standard deviations.
Dividing by the number of rows in a frequency table instead of the total frequency. The table in sample problem 3 has 5 rows but describes 20 households.
Other ways to practise
For paper practice or a class set, the printable standard deviation worksheets have four graded worksheets with answer keys and a generator for fresh sets. To test understanding rather than arithmetic, the standard deviation quiz asks 15 multiple-choice questions and explains each answer. To see the method applied in context first, read the worked standard deviation examples, ten real-world problems from exam scores to process capability.
Common questions
How is my answer marked?
The correct answer is rounded to 2 decimal places, and your answer counts as right if it is within 0.01 of that. So if the answer is 2.9155, then 2.92, 2.91 and 2.93 are all accepted. The margin allows for a slightly different rounding of the mean along the way, but not for using the wrong formula: dividing by n instead of n − 1 usually changes the second decimal by more.
Does the practice keep my score?
Yes, for as long as the page is open. The score counts every problem you answer, across all topics. Skipped problems (an empty box and Check) are not counted. Nothing is saved or sent anywhere, so reloading the page starts again from zero.
How do I type a negative z-score or a percentage?
Type a minus sign in front, such as −1.35 or -1.35; both work. For the coefficient of variation, type the percentage with or without the % sign: 30.65 and 30.65% are marked the same. For an interval, type both limits separated by a comma, such as 61.8, 83.0.
What is the difference between this page, the worksheets and the quiz?
This page is for drilling calculations on screen, one random problem at a time, with instant marking. The worksheets are fixed sets you print, with answer keys for a teacher. The quiz is multiple choice and tests understanding (what the SD means, when to use n − 1) rather than arithmetic.
Can I use a calculator while practising?
Yes, and for anything beyond five values you probably should for the square root. The skill being practised is setting the problem up correctly: the right divisor, the frequencies applied to each row, the mean kept unrounded. To check a data set of your own, paste it into the standard deviation calculator.
Related calculators
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Standard deviation examples
Ten real-world problems worked in full, from exam scores to Cpk.
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Printable worksheets
Four graded worksheets with answer keys, for paper practice.
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Standard deviation quiz
Fifteen multiple-choice questions on the concepts.
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How to calculate standard deviation
The step-by-step method, if you need a refresher first.