Enter the numbers whose spread you want to measure, separated by commas (,). The population standard deviation σ and the sample standard deviation s are calculated together with the mean, the variances and the sum of squared deviations.
Enter 2 or more numbers separated by commas (,), for example 10, 12, 23, 23, 16. Decimals and negative numbers are OK.
Result and graph
Enter your numbers separated by commas in the field on the left and press "Calculate". The result will appear here.
Table of Contents
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What you can do on this page
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What is this calculation used for?
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How to Use
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Formulas and figures
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Symbols and terms
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Good to know before you start
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How to calculate it in Excel
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How to calculate it in Google Sheets
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How to calculate it in Python
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How to write it in LaTeX and other math languages (copy and paste)
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How to have ChatGPT do the calculation
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DataChef Features
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Related Features
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NumberChef Calculators List
What you can do on this page
- Enter numbers separated by commas (,) and you get the standard deviation on the spot
- Both the population standard deviation \(\sigma\) (divide by \(N\)) and the sample standard deviation \(s\) (divide by \(N-1\)) are shown together
- The mean \(\mu\), the sum of squared deviations \(\sum (x_i-\mu)^2\) and the variances \(\sigma^2,\ s^2\) are calculated too, so you can follow every step at a glance
- A plain-language explanation of the formulas and copy-and-paste formulas for Excel, Google Sheets and Python are all on this page
If your data is the whole group you want to study, use the population standard deviation \(\sigma\). If your data is a part (a sample) taken from a larger group and you want to estimate the spread of that whole group, use the sample standard deviation \(s\). If you are not sure, compare the two. You need at least 2 numbers.
What is this calculation used for?
The basis of z-scores and grading on a curve (school and tests)
A z-score (standard score) tells you how far a score is from the mean, measured in standard deviations: \(z = \dfrac{x - \text{mean}}{\text{standard deviation}}\). The standard deviation is right there in the denominator.
If the class mean is 70 and the standard deviation is 10, a score of 85 has \(z = (85 - 70) \div 10 = 1.5\), which is 1.5 standard deviations above the mean. Teachers who "grade on a curve" and standardized test reports use this same idea. It also explains why the same score can stand out more on a test where scores are tightly packed (small spread).
Keeping variation low in manufacturing and quality control (industry)
No matter how precisely they are made, screw lengths and part weights vary a little. Factories measure this variation with the standard deviation and manage the process so that "the mean ± 3 standard deviations" stays within the customer's specifications.
A smaller standard deviation means more consistent, better products. The standard deviation is so basic that improving quality can almost be described as the work of making it smaller.
Measuring investment risk (money and finance)
Two investments may both have an average return of 5%, but the one whose returns have a larger standard deviation has some years of big gains and some years of big losses, so it is judged to be riskier.
In finance, the standard deviation of returns is called volatility, and it is the most common measure of risk. For important money decisions, look at the spread (the standard deviation) as well as the average.
Comparing how steady climates are (weather and reading data)
Two cities may have the same average yearly temperature. If the standard deviation of their monthly temperatures is small, the city is mild all year; if it is large, summer and winter are very different.
The standard deviation turns "steady or changeable" into a number, which the average alone cannot show. It is used to compare climates and to plan crops and tourism.
Setting reference ranges for medical tests (health checkups)
The reference range shown on blood test results is sometimes set by collecting data from many healthy people and taking about "the mean ± 2 standard deviations" (in practice, each test uses a method that fits the shape of its distribution).
Knowing about the standard deviation helps you stay calm: being slightly outside the reference range does not automatically mean you are ill, because some healthy people fall outside it too.
Formulas and figures
Mean
Figure
Standard notation
(the usual math form)
\(\mu\)
\(=\)
\((x_1 + x_2 + \cdots + x_N)\)
\(\div\)
\(N\)
In words
(symbols replaced with words)
③
\(\mu\): mean
\(=\)
①
sum of the data
\(\div\)
②
\(N\): number of values
The formula in words
①
Take the
sum of the data
②
divide it by the
\(N\): number of values
③
and you get the
\(\mu\): mean
Quick example
For the data 10, 12, 23, 23, 16, 23, 21, 16 (8 values), the mean is the sum 144 divided by 8, so
\(\mu\): mean
\(=\)
sum (144)
\(\div\)
count (8 values)
\(144 \div 8 = 18\)
Key idea
The standard deviation measures how far each value is from the mean, so the first step is to find the mean.
When you work with the spread of a sample, the mean is called the sample mean and is often written \(\bar{x}\) (x-bar), but it is calculated in exactly the same way.
Sum of squared deviations
Figure
Standard notation
(the usual math form)
\(S\)
\(=\)
\(\displaystyle\sum_{i=1}^{N}\)
\((x_i - \mu)\)
\(2\)
In words
(symbols replaced with words)
④
\(S\): sum of squared deviations
\(=\)
③
\(\displaystyle\sum\): add them all up
①
difference between each value \(x_i\) and the mean \(\mu\) (the deviation)
②
squared
The formula in words
①
Take the
difference between each value and the mean (the deviation)
②
square it
(so that every value becomes positive),
③
then
add them all up \(\displaystyle\sum\)
④
and you get the
\(S\): sum of squared deviations
Quick example
For the data 1, 5 (mean 3), the deviations are \(1-3=-2\) and \(5-3=2\). Square each one and add them up
\(S\): sum of squared deviations
\(=\)
deviation (−2)
squared
\(+\)
deviation (2)
squared
\((-2)^2 + (2)^2 = 4 + 4 = 8\)
Key idea
If you simply add up the deviations (the differences between each value and the mean), you always get 0, because the positive and negative ones cancel out. So you square them first to make them all positive, and then add them up. This is the sum of squared deviations, the foundation for turning spread into a number.
The population variance, the sample variance, the population standard deviation and the sample standard deviation are all built from this \(S\). The only difference between a population and a sample is whether you divide by \(N\) or by \(N-1\).
Variance (population and sample)
Figure
Standard notation
(the usual math form)
\(\sigma^2\)
\(=\)
\(S\)
\(\div\)
\(N\)
\(s^2\)
\(=\)
\(S\)
\(\div\)
\((N-1)\)
In words
(symbols replaced with words)
③
\(\sigma^2\): population variance
\(=\)
①
\(S\): sum of squared deviations
\(\div\)
②
\(N\): count
⑥
\(s^2\): sample variance
\(=\)
④
\(S\): sum of squared deviations
\(\div\)
⑤
\((N-1)\): one less than the count
The formula in words
①
Divide the
\(S\): sum of squared deviations
②
by the
\(N\): count
③
and you get the
\(\sigma^2\): population variance
.
④
Divide the
same sum of squared deviations \(S\)
⑤
by
\((N-1)\): one less than the count
⑥
and you get the
\(s^2\): sample variance
Quick example
For the data 2, 8 (mean 5, sum of squared deviations 18), the population variance is 18 divided by the count 2, and the sample variance is 18 divided by (2 − 1)
\(\sigma^2\): population variance
\(=\)
18
\(\div\)
2
\(\sigma^2 = 18 \div 2 = 9\)
\(s^2 = 18 \div (2 - 1) = 18 \div 1 = 18\)
Key idea
The variance shows how large the spread is, but because the data is squared, its unit is squared too (for test points, "points²"), which makes it hard to picture. Taking the square root in the next step brings the unit back, and that is the standard deviation.
The sample variance divides by \(N-1\) instead of \(N\) because, when you estimate the spread of a population from a sample, dividing by \(N\) makes the spread come out a little too small. Dividing by one less corrects this (it is called Bessel's correction, and \(N-1\) is called the degrees of freedom).
Some statistics textbooks call the version divided by \(N\) the sample variance and call the version divided by \(N-1\) the unbiased sample variance. On this page, as in Excel and Python functions, the version divided by \(N-1\) is called the sample variance \(s^2\).
Standard deviation (population and sample)
Figure
Standard notation
(the usual math form)
\(\sigma\)
\(=\)
\(\sqrt{\sigma^2}\)
\(s\)
\(=\)
\(\sqrt{s^2}\)
In words
(symbols replaced with words)
②
\(\sigma\): population standard deviation
\(=\)
①
square root of the population variance \(\sigma^2\)
④
\(s\): sample standard deviation
\(=\)
③
square root of the sample variance \(s^2\)
The formula in words
①
Take the
square root of the population variance \(\sigma^2\) (the number that gives the variance when squared)
②
and you get the
\(\sigma\): population standard deviation
.
③
Take the
square root of the sample variance \(s^2\)
④
and you get the
\(s\): sample standard deviation
Quick example
For the same data 2, 8, take the square roots of the population variance 9 and the sample variance 18
\(\sigma\): population standard deviation
\(=\)
\(\sqrt{9}\)
\(\sigma = \sqrt{9} = 3\)
\(s = \sqrt{18} \approx 4.243\)
Key idea
The standard deviation is the square root of the variance, so its unit is the same as the data (for test points, the standard deviation is in points too). That makes it easy to picture "how far values typically are from the mean".
The larger the standard deviation, the more spread out the data; the smaller it is, the more the data gathers near the mean. If all the values are the same, there is no spread, so the standard deviation is 0.
To find the standard deviation: (1) find the mean, (2) square each difference from the mean (each deviation) and add them all up (the sum of squared deviations), (3) divide by the count N (population) or N − 1 (sample) to get the variance, and (4) take the square root. The only difference between a population and a sample is whether you divide by N or by N − 1.
Symbols and terms
Symbols
| \(x_1, x_2, \ldots, x_N\) | x sub 1, x sub 2, …, x sub N | The individual values whose spread you want to measure. (Example - the test scores of 8 students) |
| \(N\) | N | The number of values. (Example - for 8 students, \(N = 8\)) |
| \(\mu\) | mu | The mean. This Greek letter stands for the mean of a population. The mean of a sample is written \(\bar{x}\) (x-bar), but it is calculated the same way. |
| \(x_i - \mu\) | x sub i minus mu | The deviation. It shows how far each value is from the mean. |
| \(S\) | S | The sum of squared deviations, \(\sum (x_i-\mu)^2\), found by squaring each deviation and adding them all up. The variance and the standard deviation are built from it. |
| \(\sigma^2\) | sigma squared | The population variance, the sum of squared deviations divided by the count \(N\). |
| \(\sigma\) | sigma | The population standard deviation, the square root of the population variance. It is the best-known symbol for spread. |
| \(s^2\) | s squared | The sample variance (unbiased sample variance), the sum of squared deviations divided by \(N-1\). |
| \(s\) | s | The sample standard deviation, the square root of the sample variance. |
| \(\sum\) | sigma (summation sign) | A symbol that means "add them all up". \(\sum_{i=1}^{N} x_i\) is the sum from \(x_1\) to \(x_N\). |
Terms
| standard deviation | A number that shows how spread out the data is around the mean. It is the square root of the variance, and because its unit is the same as the data, it works well as a guide to "the typical distance from the mean". It is the most common measure of spread. |
| variance | The average of the squared deviations (the differences between each value and the mean). It shows how large the spread is, but its unit is squared, so it is usually turned into the standard deviation by taking the square root. |
| deviation | The difference between a value and the mean (\(x_i - \mu\)). Positive means above the mean and negative means below it. Adding them all up as they are always gives 0, so they are squared before use. |
| sum of squared deviations | The deviations squared and all added up. The variance and the standard deviation are all built from this value. |
| population | The whole group you want to study. If your data is this whole group, use the population standard deviation \(\sigma\). |
| sample | A part of the data taken from a population (the whole group). When you estimate the spread of the whole group from a sample, use the sample standard deviation \(s\). |
| unbiased sample variance | The variance used to estimate the variance of a population from a sample. Divide the sum of squared deviations by \(N-1\). Dividing by \(N\) would make the spread of the population come out too small, so dividing by one less corrects it. |
| Bessel's correction | The correction of dividing by \(N-1\) instead of \(N\) in the sample variance. \(N-1\) is also called the degrees of freedom. |
Good to know before you start
Here is what helps you use the calculation on this page with real understanding, not just by pressing the button.
| The mean (Grade 6) |
|
| Negative numbers and parentheses (Grades 6–7) |
|
| Exponents (Grade 6) |
|
| Square roots (Grade 8) |
|
| Distribution and spread of data (Grade 6 to high school) |
|
How to calculate it in Excel
Copy the whole table below and paste it into cell A1 in Excel. It works as is.
Table to find the mean
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Count N | =COUNT(B1:B8) |
| Sum Σx | =SUM(B1:B8) |
| Mean μ | =AVERAGE(B1:B8) |
Table to find the sum of squared deviations S
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Sum of squared deviations S | =DEVSQ(B1:B8) |
Table to find the population standard deviation σ and variance σ²
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Population variance σ² | =VARP(B1:B8) |
| Population standard deviation σ | =STDEVP(B1:B8) |
Table to find the sample standard deviation s and variance s²
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Sample variance s² | =VAR(B1:B8) |
| Sample standard deviation s | =STDEV(B1:B8) |
After pasting, B1 to B8 are the cells for your numbers, and the cells in bold green are calculated automatically.
The first table shows 18 for the mean μ. The population standard deviation σ in the third table is about 4.899, and the sample standard deviation s in the fourth table is about 5.237.
To use a different number of values, add (or remove) rows of numbers, then change "B1:B8" in the formulas to your actual data range (for 10 values, B1:B10).
STDEVP and VARP, with a "P", are for a population (divide by N). STDEV and VAR, without the "P", are for a sample (divide by N − 1). DEVSQ gives the sum of squared deviations in one step.
The first table shows 18 for the mean μ. The population standard deviation σ in the third table is about 4.899, and the sample standard deviation s in the fourth table is about 5.237.
To use a different number of values, add (or remove) rows of numbers, then change "B1:B8" in the formulas to your actual data range (for 10 values, B1:B10).
STDEVP and VARP, with a "P", are for a population (divide by N). STDEV and VAR, without the "P", are for a sample (divide by N − 1). DEVSQ gives the sum of squared deviations in one step.
How to calculate it in Google Sheets
Copy the whole table below and paste it into cell A1 in Google Sheets. It works as is.
Table to find the mean
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Count N | =COUNT(B1:B8) |
| Sum Σx | =SUM(B1:B8) |
| Mean μ | =AVERAGE(B1:B8) |
Table to find the sum of squared deviations S
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Sum of squared deviations S | =DEVSQ(B1:B8) |
Table to find the population standard deviation σ and variance σ²
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Population variance σ² | =VARP(B1:B8) |
| Population standard deviation σ | =STDEVP(B1:B8) |
Table to find the sample standard deviation s and variance s²
| Value 1 | 10 |
| Value 2 | 12 |
| Value 3 | 23 |
| Value 4 | 23 |
| Value 5 | 16 |
| Value 6 | 23 |
| Value 7 | 21 |
| Value 8 | 16 |
| Sample variance s² | =VAR(B1:B8) |
| Sample standard deviation s | =STDEV(B1:B8) |
The same functions as in Excel (AVERAGE, DEVSQ, STDEVP, VARP, STDEV, VAR) work as is. Copy the whole table, paste it into cell A1, and replace B1 to B8 with your own numbers.
In Google Sheets too, use STDEVP and VARP for a population and STDEV and VAR for a sample.
In Google Sheets too, use STDEVP and VARP for a population and STDEV and VAR for a sample.
How to calculate it in Python
import statistics
numbers = [10, 12, 23, 23, 16, 23, 21, 16] # the numbers whose spread you want to measure
mean = statistics.mean(numbers) # mean μ
pstdev = statistics.pstdev(numbers) # population standard deviation σ (divide by N)
pvariance = statistics.pvariance(numbers) # population variance σ²
sstdev = statistics.stdev(numbers) # sample standard deviation s (divide by N-1)
svariance = statistics.variance(numbers) # sample variance s²
print(f"Count: {len(numbers)} / Sum: {sum(numbers)} / Mean: {mean}")
print(f"Population variance σ²: {pvariance} / Population SD σ: {pstdev}")
print(f"Sample variance s²: {svariance} / Sample SD s: {sstdev}")
Runs with the standard library only (the statistics module). Replace the numbers list at the top with your own numbers and run it. pstdev and pvariance, starting with "p", are for a population (divide by N). stdev and variance, without the "p", are for a sample (divide by N − 1).
How to write it in LaTeX and other math languages (copy and paste)
Mean
μ = Σxᵢ ÷ N
\mu = \frac{x_1 + x_2 + \cdots + x_N}{N}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>μ</mi>
<mo>=</mo>
<mfrac>
<mrow>
<msub><mi>x</mi><mn>1</mn></msub>
<mo>+</mo>
<msub><mi>x</mi><mn>2</mn></msub>
<mo>+</mo>
<mo>⋯</mo>
<mo>+</mo>
<msub><mi>x</mi><mi>N</mi></msub>
</mrow>
<mi>N</mi>
</mfrac>
</mrow>
</math>
mu = (x_1 + x_2 + cdots + x_N) / N
Mean[data]
mu := add(x[i], i = 1 .. N)/N;
mu = mean(x);
μ = (x_1 + x_2 + ⋯ + x_N)/N
Sum of squared deviations
S = Σ(xᵢ − μ)²
S = \sum_{i=1}^{N} (x_i - \mu)^2
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>S</mi>
<mo>=</mo>
<munderover>
<mo>∑</mo>
<mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow>
<mi>N</mi>
</munderover>
<msup>
<mrow><mo>(</mo><msub><mi>x</mi><mi>i</mi></msub><mo>−</mo><mi>μ</mi><mo>)</mo></mrow>
<mn>2</mn>
</msup>
</mrow>
</math>
S = sum_(i=1)^N (x_i - mu)^2
Total[(data - Mean[data])^2]
S := add((x[i] - mu)^2, i = 1 .. N);
S = sum((x - mean(x)).^2);
S = ∑_(i=1)^N (x_i − μ)^2
Variance (population and sample)
σ² = S ÷ N, s² = S ÷ (N − 1)
\sigma^2 = \frac{S}{N}, \qquad s^2 = \frac{S}{N-1}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<msup><mi>σ</mi><mn>2</mn></msup>
<mo>=</mo>
<mfrac><mi>S</mi><mi>N</mi></mfrac>
<mo>,</mo>
<mspace width="1em"/>
<msup><mi>s</mi><mn>2</mn></msup>
<mo>=</mo>
<mfrac><mi>S</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></mfrac>
</mrow>
</math>
sigma^2 = S/N, s^2 = S/(N-1)
popVar = S/n; sampleVar = Variance[data] (* N is a built-in function in Mathematica, so the count is named n *)
sigma2 := S/N; s2 := S/(N - 1);
popVar = S/N; sampleVar = var(x);
σ^2 = S/N, s^2 = S/(N−1)
Standard deviation (population and sample)
σ = √(σ²), s = √(s²)
\sigma = \sqrt{\sigma^2} = \sqrt{\dfrac{S}{N}}, \qquad s = \sqrt{s^2} = \sqrt{\dfrac{S}{N-1}}
<math xmlns="http://www.w3.org/1998/Math/MathML" display="block">
<mrow>
<mi>σ</mi>
<mo>=</mo>
<msqrt><mfrac><mi>S</mi><mi>N</mi></mfrac></msqrt>
<mo>,</mo>
<mspace width="1em"/>
<mi>s</mi>
<mo>=</mo>
<msqrt><mfrac><mi>S</mi><mrow><mi>N</mi><mo>−</mo><mn>1</mn></mrow></mfrac></msqrt>
</mrow>
</math>
sigma = sqrt(S/N), s = sqrt(S/(N-1))
popSD = Sqrt[S/n]; sampleSD = StandardDeviation[data] (* N is a built-in function in Mathematica, so the count is named n *)
sigma := sqrt(S/N); s := sqrt(S/(N - 1));
popSD = sqrt(S/numel(x)); sampleSD = std(x);
σ = √(S/N), s = √(S/(N−1))
How to have ChatGPT do the calculation
You are a statistics calculation assistant. Do the following calculation by actually running Python code, and base your answer only on the numbers from the execution result (do not answer by mental math or guessing). For the following data, find each of the values below. Data: 10, 12, 23, 23, 16, 23, 21, 16 1. The count, the sum and the mean 2. The sum of squared deviations (each value's difference from the mean, squared and all added up) 3. The population variance σ² (divide by N) and the population standard deviation σ 4. The sample variance s² (divide by N − 1) and the sample standard deviation s Show the formulas you used and the numbers from the execution result.
How to Use
-
1Enter your numbersType the numbers you want to calculate with into the input fields
-
2CalculatePress the "Calculate" button
-
3Check the resultThe result appears on the spot. The same page also explains the idea behind the calculation and the formula
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