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Standard deviation calculator

Calculate standard deviation without mixing formulas

Enter a numerical data set and choose whether it is a sample or a full population. The denominator and notation stay visible in every step.

Separate values with commas, spaces, or new lines.
Data role
Try an example

Result

Using the sample formula, a typical observation is about 2.138 units from the sample mean of 5.

s=327≈2.138s=\sqrt{\frac{32}{7}}\approx2.138
Conditions
  • All entries must be finite real numbers measured on the same scale.
  • Sample standard deviation requires n at least 2 and uses n minus 1 in the denominator.
  • Population standard deviation uses N only when the entered values are the full population of interest.
  • The calculator keeps full available numeric precision through the variance step and rounds only the displayed result.

Steps

  1. Find the sample mean The eight values total 40.xˉ=408=5\bar{x}=\frac{40}{8}=5
  2. Square each deviation Subtract 5 from every observation, then square the result.∑(xi−5)2=9+1+1+1+0+0+4+16=32\begin{aligned}\sum(x_i-5)^2&=9+1+1+1\\&\quad+0+0+4+16=32\end{aligned}
  3. Calculate the sample variance Because these data are a sample, divide by n minus 1, which is 7.s2=328−1=327s^2=\frac{32}{8-1}=\frac{32}{7}
  4. Take the square root The square root restores the original measurement units.s=327≈2.138s=\sqrt{\frac{32}{7}}\approx2.138
Independent check

Squaring the exact result returns 32/7, and multiplying that variance by n minus 1 returns the independently summed squared deviations, 32.

Scope

What this sample-standard-deviation covers

For 2, 4, 4, 4, 5, 5, 7, 9, the mean is 5 and the squared deviations total 32. The population standard deviation is 2. If the same values are a sample, the sample standard deviation is the square root of 32/7, about 2.138.

Sample standard deviation

Use the sample mean and the n minus 1 denominator, with s for the result.

Examples: 1, 3, 5, 10, 12, 2, 4, 4, 4, 5, 5, 7, 9

Population standard deviation

Use the population mean and N denominator, with sigma for the result.

Examples: -3, 0, 3, 5, 5, 5, 2, 4, 4, 4, 5, 5, 7, 9

Variance and squared deviations

Show the deviation table and variance before taking the final square root.

Examples: \sum(x_i-\bar{x})^2=32, s^2=32/7

Calculated and reference forms

Use the live decimal result for calculation, then compare it with the exact radical shown in the worked reference when available.

Examples: \sqrt{32/7}, 2.138 rounded to three decimal places

How to use it

Enter enough information for one clear task

  1. 1
    Enter every observation

    Use commas or spaces and keep repeated observations because each occurrence contributes to the result.

  2. 2
    Choose sample or population

    Base the choice on the role of the data, not on which answer looks better. A sample comes from a larger population; a population contains the whole group of interest.

  3. 3
    Inspect the denominator

    Confirm that the variance step uses n minus 1 for a sample or N for a population.

  4. 4
    Interpret the units

    Standard deviation uses the same units as the observations. Variance uses squared units.

Worked inputs

Examples to try

Use these examples to recognize the method, compare equivalent forms, and check your own work.

Classic population example

Divide the squared-deviation total 32 by N = 8.

2, 4, 4, 4, 5, 5, 7, 92,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9

Expected result

σ=32/8=2\sigma=\sqrt{32/8}=2

The same values treated as a sample

Divide the same total 32 by n minus 1, which is 7.

2, 4, 4, 4, 5, 5, 7, 92,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9

Expected result

s=32/7≈2.138s=\sqrt{32/7}\approx2.138

Three-value sample

The mean is 3 and the squared deviations total 8; divide by 2.

1, 3, 51,\ 3,\ 5

Expected result

s=8/2=2s=\sqrt{8/2}=2

Three-value population

The mean is 3 and the squared deviations total 8; divide by 3.

1, 3, 51,\ 3,\ 5

Expected result

σ=8/3≈1.633\sigma=\sqrt{8/3}\approx1.633

Constant data

Every deviation from the mean is zero.

6, 6, 66,\ 6,\ 6

Expected result

σ=0\sigma=0

Two-value sample

The mean is 11, the squared deviations total 2, and n minus 1 equals 1.

10, 1210,\ 12

Expected result

s=2≈1.414s=\sqrt{2}\approx1.414

Complete example

Find the sample standard deviation of eight values

The list is explicitly a sample, so the calculation uses the sample mean, the symbol s, and the n minus 1 denominator.

2, 4, 4, 4, 5, 5, 7, 92,\ 4,\ 4,\ 4,\ 5,\ 5,\ 7,\ 9
  1. 1
    Compute the center

    Divide the total 40 by eight observations.

    xˉ=40/8=5\bar{x}=40/8=5
  2. 2
    Compute the spread total

    The deviations are -3, -1, -1, -1, 0, 0, 2, and 4.

    9+1+1+1+0+0+4+16=329+1+1+1+0+0+4+16=32
  3. 3
    Use the sample denominator

    There are seven degrees of freedom after estimating the mean from eight sample values.

    s2=32/7s^2=32/7
  4. 4
    Return to original units

    Take the square root and round only for display.

    s=32/7≈2.138s=\sqrt{32/7}\approx2.138
s=327≈2.138s=\sqrt{\frac{32}{7}}\approx2.138

Verification: The deviation sum is -3 - 1 - 1 - 1 + 0 + 0 + 2 + 4 = 0. Also, 7 times the exact variance 32/7 equals the squared-deviation total 32.

Avoidable errors

Common mistakes and how to fix them

Choosing the denominator by habit

Problem: Always dividing by the number of observations.

Why it matters: That is the population calculation and does not produce the usual sample variance estimate.

Better approach: Identify the data role first, then use N for a population or n minus 1 for a sample.

Stopping at variance

Problem: s^2=32/7\text{ reported as the standard deviation}

Why it matters: Variance is the mean squared deviation, while standard deviation is its square root.

Better approach: Take the positive square root and keep the variance label separate.

Rounding the mean too early

Problem: Rounding the center before finding deviations.

Why it matters: Every later squared deviation inherits that rounding error.

Better approach: Keep exact values through the variance step and round the final display only.

Ignoring measurement units

Problem: Reporting variance and standard deviation with the same units.

Why it matters: Variance squares the original units; standard deviation does not.

Better approach: Attach squared units to variance and original units to standard deviation.

Trust the result for the right reasons

Checks, assumptions, and limits

How results are checked

  • The arithmetic mean is recomputed from the full sum and count.
  • The deviations are checked to sum to zero before squaring.
  • The squared-deviation total is calculated independently from the displayed variance.
  • Squaring the reported standard deviation and restoring the selected denominator reproduces the same spread total.

When to stop and revise the input

  • A one-value sample has no sample standard deviation because n minus 1 is zero.
  • Standard deviation is sensitive to extreme observations because deviations are squared.
  • A small standard deviation describes closeness to the mean, not whether the data are accurate or unbiased.
  • The calculation alone does not determine whether the data collection method supports inference to a population.
  • Inputs whose total, span, or squared deviations exceed the browser's finite numeric range are rejected instead of displaying an infinite result.

Common questions

Calculate standard deviation without mixing formulas FAQ

What is the difference between sample and population standard deviation?

Population standard deviation describes the full population and divides by N. Sample standard deviation estimates population spread from a sample and divides the squared-deviation total by n minus 1.

Why is sample standard deviation usually larger for the same data?

Its variance uses the smaller denominator n minus 1 instead of n. The squared-deviation total is unchanged, so the sample variance and its square root are larger unless every deviation is zero.

Can standard deviation be negative?

No. Squared deviations are nonnegative, and standard deviation is the nonnegative square root of their average or adjusted average.

Does zero standard deviation mean the data are zero?

No. It means all observations are equal. The shared value can be zero, positive, or negative.

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Created by Mathos AI. Methods, conditions, and checks are shown so you can review the mathematical reasoning.