Standard deviation examples

The method is the same in every case: find the mean, subtract it from each value, square the deviations, add them, divide by n − 1 for a sample or N for a population, and take the square root. If any step is unfamiliar, how to calculate standard deviation walks through it slowly. The examples here are about applying it: choosing the right formula, reading the result in context, and following it with the statistic each field actually uses, such as Cpk, RSD or annualised volatility. For a shorter tour of the fields themselves, see where standard deviation is used.

The examples at a glance

Results are rounded at the last step only. Intermediate values such as the mean are kept exact, which is why a hand calculation that rounds early can differ in the final digit.

1. Exam scores for one class

A teacher has the final-exam marks of all ten students in her class: 62, 75, 81, 58, 90, 70, 77, 84, 66, 87.

Which formula. Population, σ. She wants to describe this class and no other; the ten students are the whole group, so nothing is being estimated and the divisor is N = 10.

The marks add to 750, so μ = 750 ÷ 10 = 75.

σ² = 1,054 ÷ 10 = 105.4 σ = √105.4 = 10.27 marks

Interpretation. A typical student's mark sits about 10 marks from the class average of 75. The top mark, 90, is (90 − 75) ÷ 10.27 = 1.46 standard deviations above the mean, and the lowest, 58, is 1.66 below it, so neither is an extreme outlier. A spread of 10 marks on a 100-mark paper suggests the exam separated students reasonably well without leaving part of the class behind. The test score standard deviation calculator gives the same figures, with each student's z-score.

2. Manufacturing tolerances: machined shafts

A shaft is specified at 25.00 ± 0.10 mm, so the lower and upper specification limits are 24.90 and 25.10 mm. An inspector measures eight shafts from a production run: 25.02, 24.98, 25.01, 25.03, 24.99, 25.00, 25.04, 24.97 mm.

Which formula. Sample, s. The eight shafts stand in for every shaft the process makes, so the divisor is n − 1 = 7.

The diameters add to 200.04 mm, so x̄ = 25.005 mm.

s² = 0.0042 ÷ 7 = 0.0006 mm² s = √0.0006 = 0.0245 mm

Quality engineers turn this into capability indices:

Cp = (USL − LSL) ÷ 6s = 0.20 ÷ 0.1470 = 1.36 Cpk = min(USL − x̄, x̄ − LSL) ÷ 3s = min(0.095, 0.105) ÷ 0.0735 = 1.29

Interpretation. The process spread is small compared with the ±0.10 mm tolerance: six standard deviations take up 0.147 mm of the 0.20 mm window. Cpk is lower than Cp because the mean has drifted 0.005 mm towards the upper limit. A Cpk of 1.29 falls just short of the common minimum of 1.33, so a customer asking for 1.33 would want the process re-centred at 25.00 mm (which would lift Cpk to Cp, 1.36) and a larger sample than eight parts before signing off. Try other limits in the Cpk calculator.

3. Lab replicates and relative standard deviation

A chemist runs five replicate assays of the same solution and reads the concentration in mg/L: 10.2, 10.5, 10.1, 10.4, 10.3.

Which formula. Sample, s. The five readings are a sample of the infinitely many readings the method could produce; the goal is the method's precision, so the divisor is 4.

The readings add to 51.5, so x̄ = 10.3 mg/L.

s² = 0.10 ÷ 4 = 0.025 s = √0.025 = 0.158 mg/L RSD = s ÷ x̄ × 100% = 0.158 ÷ 10.3 × 100% = 1.54%

Interpretation. Labs report precision as the relative standard deviation because it can be compared across concentrations. An RSD of 1.54% means repeat measurements agree to about one and a half percent of the value. Whether that passes depends on the method's acceptance criteria; many validated methods require replicate RSDs under 2%, which this does. The relative standard deviation calculator does the division for you.

4. Stock returns and volatility

A stock's monthly returns over six months were 2.1, −1.4, 3.5, 0.8, −2.6, 1.8 percent.

Which formula. Sample, s. Six months are a small sample of the stock's return behaviour, and the reason for measuring it is to judge future risk, so the divisor is 5.

The returns add to 4.2%, so the mean monthly return is 0.7%.

s² = 26.32 ÷ 5 = 5.264 (%²) s = √5.264 = 2.29% a month Annualised volatility ≈ 2.29% × √12 = 7.95%

Interpretation. In a typical month the return lands a couple of percentage points either side of the 0.7% average, and a losing month is not unusual: two of the six were negative. Scaling by √12 assumes months are independent, which is the market convention rather than a law. Six observations give a rough estimate at best; analysts usually want 36 to 60 months. To combine this stock with others, use the portfolio standard deviation calculator, which accounts for correlations.

5. Sports: two players with the same average

Two basketball players each averaged 22 points over their last six games. Player A scored 20, 22, 23, 21, 24, 22; player B scored 12, 30, 25, 15, 31, 19.

Which formula. Sample, s. The coach cares about how each player will score in future games, and six games are a sample of that, so the divisor is 5.

A: s² = 10 ÷ 5 = 2 s = √2 = 1.41 points B: s² = 312 ÷ 5 = 62.4 s = √62.4 = 7.90 points

Interpretation. The averages are identical, and the averages alone would hide the one thing a coach needs to know. Player A is almost metronomic: every game within two points of 22. Player B swings between off nights and big nights, with a coefficient of variation of 7.90 ÷ 22 = 35.9% against A's 6.4%. Which is better depends on the situation: a team that needs a steady contribution picks A; a team that needs a chance of a 30-point game might pick B. The coefficient of variation calculator makes that comparison fair when means differ.

6. Weather: annual rainfall

A weather station recorded these annual rainfall totals, in millimetres, over ten years: 812, 745, 903, 688, 790, 856, 721, 934, 767, 804.

Which formula. Sample, s. The decade is used to describe the local climate, which includes years not yet observed, so the divisor is 9.

The totals add to 8,020 mm, so x̄ = 802 mm.

s² = 54,820 ÷ 9 = 6,091.11 mm² s = √6,091.11 = 78.0 mm

Interpretation. A normal year falls between about 724 and 880 mm (802 ± 78). Six of the ten years did; 688, 721, 903 and 934 mm fell outside. For roughly bell-shaped data about 68% of values sit within one standard deviation, and 6 of 10 is close to that. A year below 646 mm (two standard deviations under the mean) would be a genuinely dry year for this station. The empirical rule calculator lays out the ±1, 2 and 3 SD bands.

7. Psychology: reaction times

In a simple reaction-time task, one participant's ten trials took 312, 287, 345, 298, 330, 276, 301, 355, 289, 317 milliseconds.

Which formula. Sample, s. The ten trials are a sample of the participant's reaction times; the researcher wants that person's typical speed and variability, so the divisor is 9.

The times add to 3,110 ms, so x̄ = 311 ms.

s² = 6,044 ÷ 9 = 671.56 ms² s = √671.56 = 25.9 ms SE = s ÷ √n = 25.9 ÷ √10 = 8.2 ms

Interpretation. Trial-to-trial variability of about 26 ms is typical for simple reaction time. The slowest trial, 355 ms, is 1.70 standard deviations above the mean, well inside the 2.5 or 3 SD cut-offs many labs use to drop lapses of attention, so no trial is excluded. The standard error answers a different question: the participant's true mean is known to within roughly ±8 ms. Keep the two apart when writing up; the standard error calculator explains the difference.

8. Survey: Likert-scale answers

Twenty employees answered "I have the tools I need to do my job" on a 1–5 scale. The answers are summarised as a frequency table.

Which formula. Sample, s. The 20 respondents are a sample of the company's staff, so the divisor is n − 1 = 19. With a frequency table, each squared deviation is multiplied by how many people gave that answer.

n = Σf = 20 and Σf·x = 67, so x̄ = 67 ÷ 20 = 3.35.

s² = 30.55 ÷ 19 = 1.6079 s = √1.6079 = 1.27 points

Interpretation. An average of 3.35 reads as "mildly positive", but a standard deviation of 1.27 on a 5-point scale is large: opinion is divided, with 5 of 20 disagreeing and 10 agreeing. A question with the same mean and an SD of 0.5 would describe a workforce that broadly agrees. When the SD is this large, report the percentage in each category alongside the mean. The frequency table standard deviation calculator accepts the table directly.

9. Grouped data: commute times

A survey of 50 commuters recorded journey times in 10-minute classes: 6 people took 0–10 minutes, 14 took 10–20, 18 took 20–30, 9 took 30–40 and 3 took 40–50.

Which formula. Sample, s, on class midpoints. The 50 commuters are a sample of the town's commuters, so the divisor is 49. The exact times are lost, so each person is placed at the midpoint of their class.

n = 50 and Σf·m = 1,140, so x̄ = 1,140 ÷ 50 = 22.8 minutes.

s² = 5,658 ÷ 49 = 115.47 min² s = √115.47 = 10.75 minutes

Interpretation. The typical commute is 22.8 minutes give or take about 11. The distribution leans right (a long tail of 30- to 50-minute journeys), so mean ± SD describes it less well than it would bell-shaped data, and the median class, 20–30 minutes, is a useful companion figure. Because midpoints stand in for the real times, treat the result as an estimate. The grouped data standard deviation calculator takes class limits and frequencies.

10. Sample or population: the same six numbers two ways

A charity's board has six members, aged 45, 52, 38, 61, 49, 55. The ages add to 300, so the mean is 50.

Case A: describing this board. The six people are the whole group of interest, so use the population formula:

σ² = 320 ÷ 6 = 53.33 σ = √53.33 = 7.30 years

Case B: estimating trustees in general. A researcher uses the same six ages to estimate the spread of ages among charity trustees nationally. Now they are a sample:

s² = 320 ÷ 5 = 64 s = √64 = 8.00 years

Interpretation. Same data, different question, different answer: the sample figure is 9.5% larger. Dividing by n − 1 corrects for the fact that a sample's own mean sits closer to its values than the unknown population mean does, which would otherwise make the spread look too small. With six values the gap matters; with 600 it would be negligible. The full argument is in sample vs population standard deviation, and the population standard deviation calculator shows both results side by side.

What the examples have in common

Three habits run through all ten. First, the formula is chosen by the question, not by the data: the same six ages gave 7.30 or 8.00 depending on what they were meant to describe. Second, the standard deviation is rarely the last number. It fed a Cpk, an RSD, an annualised volatility, a coefficient of variation or a standard error, because each field has a version that answers its own question. Third, the interpretation always compares the SD with something: the mean, a tolerance, a scale, another player, a threshold. A standard deviation on its own is hard to read; how to interpret standard deviation covers those comparisons in more depth.

To practise the calculation rather than read it, the standard deviation practice problems give you fresh numbers and mark your answer, and the printable worksheets have answer keys for classroom use.