Standard Deviation Calculator
Values
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Example 1
Five students scored 85, 90, 78, 92, and 88 on a quiz. The quiz scores are a sample of the students who could take this course, so compute the sample standard deviation and sample variance of the scores.
Solution
The mean is
Each deviation from the mean: 85 − 86.6 = −1.6, 90 − 86.6 = +3.4, 78 − 86.6 = −8.6, 92 − 86.6 = +5.4, 88 − 86.6 = +1.4.
Squared deviations: 2.56, 11.56, 73.96, 29.16, and 1.96, which sum to 119.2.
Answer
The sample variance is 29.8 and the sample standard deviation is 5.46 points. Most quiz scores fall within about 5.5 points of the 86.6 mean, so this class was fairly consistent.
Source: OpenStax Introductory Statistics, ch. 2
Example 2
A machine fills cereal boxes, and quality control measures every box produced on Monday – all 8 of them, a small trial run. The fill weights in grams are 501, 498, 500, 503, 497, 502, 499, and 500. Because the entire run was weighed, treat the data as a population and compute σ.
Solution
The mean is
Deviations: +1, −2, 0, +3, −3, +2, −1, 0. Squared: 1, 4, 0, 9, 9, 4, 1, 0. The sum is 28.
Answer
The population standard deviation is 1.87 g. A fill weight more than about 4 g from the 500 g target would be unusual for this machine.
Source: NIST/SEMATECH e-Handbook of Statistical Methods
Example 3
Two investment funds both average a 9% annual return over five years. Fund A returned 7%, 9%, 11%, 8%, and 10%. Fund B returned −1%, 19%, 2%, 21%, and 4%. Compare their risk using the population standard deviation of returns.
Solution
Both means are
Fund A deviations from 9%: −2, 0, +2, −1, +1. Squared: 4, 0, 4, 1, 1 – sum 10.
Fund B deviations from 9%: −10, +10, −7, +12, −5. Squared: 100, 100, 49, 144, 25 – sum 418.
Answer
Fund A's standard deviation is 1.41% versus Fund B's 9.14%. Same average return, but Fund B's yearly outcome is about six and a half times less predictable – the textbook definition of higher risk.
Source: OpenStax Introductory Statistics, ch. 2
Example 4
Scores on a standardised test are normally distributed with a mean of 500 and a standard deviation of 100. A test taker scores 650. Using the empirical rule, estimate what share of test takers scored above 650.
Solution
The score 650 sits at
The empirical rule places about 68% of scores within ±1σ (400 to 600). Half of the remaining 32% – that is 16% – lies above 600.
Between 1σ and 2σ sits most of the difference between 68% and 95%, about 27% of all scores, split roughly 13.5% per side between 600 and 700.
Above 650, which is halfway through that band, a reasonable estimate is about 13.5% × ½ + 2.5% = 9.25%.
Answer
Roughly 7% to 9% of test takers score above 650 – the empirical rule estimate of a value 1.5 standard deviations above the mean. (Exact normal-table value: 6.68%.)
Source: PennState STAT 500, Applied Statistics
Published:
Updated:
References
- [1]NIST, NIST/SEMATECH e-Handbook of Statistical Methods. https://www.itl.nist.gov/div898/handbook
- [2]PennState Eberly College of Science, STAT 500: Applied Statistics. https://online.stat.psu.edu/stat500
- [3]OpenStax, Rice University, OpenStax Introductory Statistics. https://openstax.org/details/introductory-statistics
Glossary
- Standard deviation – The square root of the average squared distance of values from the mean, expressed in the same units as the data. It is the most widely used measure of statistical dispersion.
- Variance – The average of the squared deviations from the mean. Additive across independent sources of noise, which makes it the base quantity in probability theory, though its squared units limit direct interpretation.
- Sample standard deviation (s) – The spread estimate computed from a subset of a population, using the n − 1 divisor of Bessel's correction to stay unbiased.
- Population standard deviation (σ) – The spread of a complete data set containing every member of the group, computed with the N divisor. The lowercase Greek sigma is its conventional symbol.
- Bessel's correction – Replacing the divisor N with N − 1 when estimating population spread from a sample. It compensates for the tendency of sample deviations, measured around the sample mean, to underestimate true dispersion.
- Mean (x̄) – The arithmetic average: the sum of all values divided by their count. The reference point from which deviations are measured.
- Median – The middle value of the sorted data set, or the average of the two middle values for an even count. Unlike the mean, it is unaffected by extreme values.
- Mode – The most frequently occurring value. A data set can have one mode, several modes, or none when every value occurs only once.
- Range – The difference between the largest and smallest values. The simplest spread measure, but sensitive to outliers because it depends on only two data points.
- Coefficient of variation (CV) – The standard deviation expressed as a percentage of the mean. A unit-free measure that allows variability comparisons across different scales.
- Standard error of the mean – The standard deviation of the sample mean, equal to σ divided by the square root of n. It quantifies how precisely a sample mean estimates the population mean.
- Empirical rule – For normal data, about 68% of values lie within one, 95% within two, and 99.7% within three standard deviations of the mean. Also called the 68-95-99.7 rule.
How to Use?
- 1.
Count your values: Count how many numbers are in your data set and select that number from the Number of Values dropdown. The input table grows or shrinks to match.
- 2.
Enter your data: Type each value into its own row. Decimals and negative numbers are supported. Enter all values in the same unit – grams, minutes, points – so the results are interpretable.
- 3.
Choose sample or population: Pick Sample if your values are a subset of a larger group; pick Population if you measured every member. Unsure? Sample is the safer default, and switching recalculates instantly with the matching variance shown.
- 4.
Click Calculate and read the headline result: The featured standard deviation appears at the top of the results panel, with the variance, mean, median, mode, range, count, sum, CV, and standard error beneath it.
- 5.
Audit the computation in the deviation table: Each row shows a value, its signed distance from the mean, and its squared distance. The squared-deviation column is the exact numerator of your variance – use it to verify hand calculations or spot outliers.
- 6.
Check the distribution chart: The chart overlays your data points on the normal curve built from your mean and standard deviation. If the points track the curve, the empirical rule percentages apply; if not, treat them as rough guides.
Frequently Asked Questions
Enter each value in the data table, set the number of values to match, choose Sample or Population, and click Calculate. The calculator computes the mean, subtracts it from every value, squares and sums the deviations, divides by n − 1 for a sample or N for a population, and takes the square root. Every step appears in the deviation table so you can follow the arithmetic for each value you entered.
The sample standard deviation divides the sum of squared deviations by n − 1, while the population standard deviation divides by N. The smaller divisor in the sample formula compensates for the fact that samples systematically underestimate population spread, because sample points cluster around their own mean. For the same data, the sample result is always equal to or larger than the population result. Use the population formula only when your data set includes every member of the group you are describing.
Dividing by n − 1 is Bessel's correction. A sample's deviations are measured from the sample's own mean, which sits closer to the sample points than the true population mean does. Dividing by the smaller number inflates the variance estimate just enough to cancel this downward bias, making the sample variance an unbiased estimator of the population variance. With very small samples the correction is large: for two values it inflates the result by 41%.
There is no universal good value because the answer depends on the scale and purpose of your data. A σ of 5 g is excellent on a pharmaceutical tablet press and disastrous on a 500 g cereal line. Judge spread relative to the mean using the coefficient of variation, which the calculator reports: under 10% is very consistent, 10% to 25% is moderate, and above 50% signals high variability. The right benchmark is set by what your application tolerates.
A population standard deviation needs at least one value and is zero when all values are identical. A sample standard deviation needs at least two values, because the n − 1 divisor would be zero otherwise. For meaningful estimates, aim for at least five values; the standard error of the mean, shown in the results, shrinks with the square root of your sample size, so more data makes every estimate more precise.
It tells you how far your values typically sit from the mean, in the same units as the data. Combined with the mean, it sketches the whole distribution for approximately normal data: about 68% of values fall within one standard deviation of the mean, 95% within two, and 99.7% within three. A small σ means consistent, predictable data; a large σ means wide variation. It says nothing about direction or shape – for that, check the median and the distribution chart.
It can never be negative because it is the square root of a sum of squares. It is exactly zero only when every value in the data set is identical, meaning no spread at all. If the calculator shows zero, verify that you entered distinct values rather than one value duplicated by mistake. Values that differ even slightly produce a positive σ.
The coefficient of variation expresses the standard deviation as a percentage of the mean, removing units so different processes can be compared. A CV of 2% on a fill line means the typical deviation is 2% of the target – tight control. A CV of 40% on household incomes means wide dispersion. Because it divides by the mean, the CV is undefined or misleading when the mean is zero or near zero, such as temperature data in Celsius; prefer raw σ there.
For normally distributed data, about 68% of values lie within one standard deviation of the mean, about 95% within two, and 99.7% within three. It lets you turn any mean-plus-σ pair into immediate probability statements, like estimating that a 650 SAT score sits above roughly 93% of test takers when the mean is 500 and σ is 100. The rule only holds for bell-shaped data; the distribution chart shows whether your data qualifies.
First verify they are real – most outliers are entry errors, and the deviation table makes them obvious as rows with outsized deviations. If a value is genuine, decide based on the question you are asking: keep it when describing actual variability, flag it when it distorts the story the mean tells. Standard deviation is sensitive to outliers because deviations are squared; for a resistant alternative, compare medians or report the interquartile range alongside.
Variance is the average squared deviation from the mean; standard deviation is its square root. Variance is the mathematically convenient form – it adds across independent sources of noise, which is why probability theory and error propagation work in variance units. Standard deviation is the communicable form, because taking the square root returns it to the original units of the data. This calculator reports both from the same computation.
No. The standard deviation describes the spread of the population or process you are sampling, and that spread does not change with sample size. What shrinks is the standard error of the mean – the uncertainty of your estimate of the mean – which declines as one over the square root of n. Collecting four times as much data halves the standard error while leaving σ essentially unchanged.
Yes. The calculator accepts any real numbers, positive, negative, whole, or decimal. Mixed signs are handled correctly because deviations are squared before summing. Enter values in any order – the calculator sorts internally for the median and uses every value for the mean, deviations, and variance regardless of entry sequence.
Use the median for skewed data or data with extreme outliers. Household incomes, house prices, and hospital stay lengths all have long right tails that drag the mean above what a typical member experiences. The mean and standard deviation remain valid summaries, but they describe a distribution whose centre few members actually occupy. The calculator reports the median alongside the mean so you can see immediately when they diverge – a gap between them is the classic signature of skew.
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