Statistics Calculator

Standard Deviation Calculator

See the spread behind the average.

Enter your data

Exact statistics.

Dataset values

Separate signed decimals, scientific e notation or integer fractions with commas, spaces, semicolons, pipes or new lines. Decimal point only; commas separate values, never thousands.

Input method

Each method keeps its own data. Frequency counts represent repeated observations, not arbitrary weights.

Conventions and display

Unit label

A literal label only; all observations must already use the same measurement unit. No unit conversion is performed.

Highlighted estimate

Changes the main result only. Both sample and population calculations remain available below.

Rounded decimal places

Exact rational answers do not change. Tiny nonzero views use scientific notation. Halfway values round away from zero.

Import a local CSV or text file

Imports into the selected method. Files must be UTF-8, at most 64 KiB, and contain only supported numeric data plus an optional frequency header. Invalid files leave current data unchanged.

Worked datasets

Show step-by-step solutionShow squared-deviation tableShow coefficients of variation

Supports up to $2{,}000$ source observations or frequency rows, and total count $n\leq10^9$ . Nonzero source magnitudes range from $10^{-100}$ through $10^{100}$ , with forty significant digits per number. Combined exact arithmetic is bounded to $1{,}500$ numerator or denominator digits. Results describe the supplied data; no missing values or outliers are removed automatically.

5 observations; 4 distinct exact values. Both sample and population estimates shown.

Sample standard deviation

\sqrt{7}

$\approx 2.645751$ points

Sample and population comparison

Population variance divides by $n$ ; sample variance divides by $n-1$ . Variances and squared-deviation totals use squared source units. Standard deviations return to the original unit. Exact radicals and their separately rounded decimal views are both shown.

Sample standard deviation

s

\sqrt{7}

\approx 2.645751

points

Population standard deviation

\sigma

\sqrt{\frac{28}{5}}

\approx 2.366432

points

Sample variance

s^2

7

\approx 7

Squared source unit: points

Population variance

\sigma^2

\frac{28}{5}

\approx 5.6

Squared source unit: points

Arithmetic mean

\bar{x}

5

\approx 5

points

Sum of squared deviations

\mathrm{SSD}

28

\approx 28

Squared source unit: points

Estimated standard error of mean

\widehat{\mathrm{SE}}(\bar{x})

\sqrt{\frac{7}{5}}

\approx 1.183216

points

Sum of observations

S

25

\approx 25

points

Minimum

x_{(1)}

2

\approx 2

points

Maximum

x_{(5)}

9

\approx 9

points

Range

R

7

\approx 7

points

Mean absolute deviation about the mean

\mathrm{AAD}

2

\approx 2

points

Sum of squared observations

\sum_j f_jv_j^2

153

\approx 153

Squared source unit: points

Observation count: $5$
Distinct exact values: $4$
Zero-frequency rows excluded: $0$

The estimated standard error has its usual sampling interpretation only with independent, comparably distributed observations. Mean absolute deviation here means the average absolute distance from the mean, not median absolute deviation.

Step-by-step standard deviation solution

1. Validate the observation model
Every active source is checked. Integer frequencies count actual repetitions; zero-frequency rows are excluded and reported. The selected highlight does not hide the other denominator convention.
$n=\sum_j f_j=5,\quad u=4$
2. Find the exact arithmetic mean
The unrounded total and observation count determine the center. Exact decimal and fraction arithmetic preserves close values and cancellation.
$\bar{x}=\frac{S}{n}=\frac{\left(25\right)}{5}=5$
3. Center, square and weight each observation
Subtract the same exact mean from every distinct value. Square that deviation and multiply by its repetition count. The audit table retains every contribution.
$\mathrm{SSD}=1\left(2-\left(5\right)\right)^2+2\left(4-\left(5\right)\right)^2+1\left(6-\left(5\right)\right)^2+1\left(9-\left(5\right)\right)^2=28$
4. Choose the variance denominator
The population treats these observations as the complete target set. The sample convention divides by one fewer observation when estimating a larger population variance.
$\sigma^2=\frac{28}{5}=\frac{28}{5},\qquad s^2=\frac{28}{4}=7$
5. Take the nonnegative square roots
The square-root forms are exact. Perfect rational squares simplify to rational values. Decimal views are rounded from the full rational square using integer comparisons, without floating-point overflow or underflow.
$\sigma=\sqrt{\frac{28}{5}},\quad s=\sqrt{7}$
6. Check the centered arithmetic independently
Signed weighted deviations must total zero. A separate exact second-moment identity must reproduce the same squared-deviation sum. No rounded intermediate is reused in either audit.
$\sum_j f_j(v_j-\bar{x})=0,\qquad \mathrm{SSD}=\sum_j f_jv_j^2-\frac{S^2}{n}=28$
7. Keep standard error and relative spread distinct
The estimated standard error uses the sample spread and presumes independent, comparably distributed observations for its usual interpretation. Optional coefficients of variation require a positive mean and meaningful ratio-scale zero. Neither output is a confidence interval or a guarantee of measurement accuracy.
$\widehat{\mathrm{SE}}(\bar{x})=\frac{s}{\sqrt{n}},\qquad\mathrm{CV}_s=\frac{s}{\bar{x}}\times100\%$

Sorted squared-deviation audit

Each row keeps one distinct exact value and its full repetition count. The same unrounded mean is subtracted in every row. Squared and frequency-weighted contributions use squared source units; contribution percentages are undefined when total dispersion is zero.

Showing distinct rows 1–4 of 4. Printing includes this page; copying and CSV include every row.

Observed value	Frequency	Signed deviation	Squared deviation	Weighted square	Weighted signed deviation	Share of squared-deviation total
Value $2$ $\approx 2$ points	$1$	$-3$ $\approx -3$	$9$ $\approx 9$	$9$ $\approx 9$	$-3$ $\approx -3$	$\approx 32.142857\%$
Value $4$ $\approx 4$ points	$2$	$-1$ $\approx -1$	$1$ $\approx 1$	$2$ $\approx 2$	$-2$ $\approx -2$	$\approx 7.142857\%$
Value $6$ $\approx 6$ points	$1$	$1$ $\approx 1$	$1$ $\approx 1$	$1$ $\approx 1$	$1$ $\approx 1$	$\approx 3.571429\%$
Value $9$ $\approx 9$ points	$1$	$4$ $\approx 4$	$16$ $\approx 16$	$16$ $\approx 16$	$4$ $\approx 4$	$\approx 57.142857\%$

What Can You Create?

See how each observation contributes to spread

Compare both variance denominators

A sample and a complete population can contain the same numerical observations while serving different statistical purposes. The comparison keeps their shared squared-deviation total visible and changes only the appropriate denominator. Choosing the highlighted estimate does not delete the other result. A small dataset can describe a complete target set, and a large dataset can still be a sample, so the tool does not choose the interpretation from count alone.

Audit every weighted squared deviation

The sorted audit connects an exact observed value to its repetition count, signed deviation, squared deviation and weighted contribution. Rows with identical numerical values are combined without changing their total multiplicity. Percentages show how much of the squared total each row contributes, while a constant dataset receives an explicit no-dispersion statement. All rows remain in copying and CSV even when the browser displays only one page.

Retain small differences beside large values

Large measurement offsets can hide a small real numerical spread when rounded binary values are subtracted. Exact rational sources preserve those differences before squaring. The standard-deviation view then rounds directly from the rational square, without using an overflowing or underflowing floating-point intermediate. These arithmetic guarantees concern the entered numbers; they do not create additional measurement precision or establish that a dataset represents a wider population.

Standard deviation formulas and a complete worked example

n=\sum_jf_j,\qquad\bar{x}=\frac{\sum_jf_jv_j}{n},\qquad\mathrm{SSD}=\sum_jf_j(v_j-\bar{x})^2

\sigma^2=\frac{\mathrm{SSD}}{n},\quad\sigma=\sqrt{\frac{\mathrm{SSD}}{n}},\qquad s^2=\frac{\mathrm{SSD}}{n-1},\quad s=\sqrt{\frac{\mathrm{SSD}}{n-1}}

The population formula treats the supplied values as the complete target set, whose center is numerically $\mu=\bar{x}$ . The sample formulas require $n>1$ . For $2,4,4,6,9$ , the mean is $5$ and the squared-deviation total is $9+2+1+16=28$ . Therefore $\sigma^2=28/5$ , $\sigma=\sqrt{28/5}$ , $s^2=7$ and $s=\sqrt7$ .

For the repeated-value example $2,4,4,4,5,5,7,9$ , the mean is $5$ and the squared total is $32$ . Its population standard deviation is exactly $2$ , while its sample standard deviation is $\sqrt{32/7}$ . The same observations produce different denominators without changing their center or individual contributions.

\widehat{\mathrm{SE}}(\bar{x})=\frac{s}{\sqrt n},\qquad\mathrm{CV}_s=\frac{s}{\bar{x}}\times100\%

Standard error presumes a suitable independent sampling model for its usual interpretation. Relative spread additionally requires a meaningful ratio-scale zero and a positive mean. No normal-distribution coverage percentages or confidence intervals are inferred automatically.

Why Users Love This Tool

Why exact arithmetic and interpretation stay separate

The numerical audit is reproducible

Invalid source tokens and missing punctuated cells never disappear or become zero.
The mean and each deviation retain exact rational values through the squared total.
Signed weighted deviations close to zero exactly, and a second-moment identity independently checks the total.
Square-root views use exact rounding boundaries, while full certificates remain available in CSV.

The statistical model remains explicit

Both sample and population results remain visible under a clear denominator convention.
A singleton's undefined sample estimates are distinguished from genuine zero spread.
Frequency counts represent repeated observations, with zero-count exclusions reported.
Standard error and optional relative spread include their own interpretation and scale assumptions.

Perfect For

A practical spread check for learning and numerical review

Students following the variance calculation

Work from a shared mean into signed differences, squared distances and the final denominator. The default dataset contains repeated values and unequal deviations, so the frequency audit explains why some rows contribute more than others. Compare exact roots with rounded numerical views without treating a decimal approximation as an algebraic certificate. Constant and singleton examples make the boundaries of the two formulas visible.

Teachers comparing conventions and assumptions

Use the same data to explain the distinction between describing a complete target set and estimating a population variance. Compare the individual-observation spread with the estimated standard error of the mean. Frequency tables make the difference between distinct rows and total observations concrete. Exact large-offset and small-scale examples also separate numerical accuracy from the separate question of whether a statistical sampling model is appropriate.

People reviewing numerical measurement lists

Paste supported values or import a local numerical file to inspect dispersion before moving into a larger analysis. Literal labels distinguish a source unit from its squared counterpart without guessing unit conversions. The audit can expose a large contribution that merits checking against the source record, but it never deletes that observation automatically. Range and average absolute distance provide additional descriptive context without claiming to replace a complete distributional analysis.

How It Works

How it works in three quick steps.

Enter the observations or their repetition counts

Paste a signed numerical list, or choose value and frequency rows. The frequency method uses whole-number multiplicities, not arbitrary reliability weights or grouped intervals. A local CSV or text file can populate either editor. Check that every observation already uses the same unit. Unsupported tokens and oversized sources are rejected explicitly instead of being removed or truncated silently.

Compare the sample and population interpretations

Choose the estimate to highlight while retaining the other convention in the comparison. The calculator reports exact variance and square-root forms, plus separately rounded numerical views. Follow the common mean into each squared deviation and inspect the frequency-weighted contributions. A singleton has zero population dispersion but no defined sample variance or sample standard deviation.

Audit the working and retain a complete record

Review the signed-deviation and second-moment checks, then inspect the sorted audit through its pages. Copy or download the result to preserve every distinct observation and contribution, even when rows or working are hidden. Share the active source when it fits the encoded-link limit. Printing retains visible mathematics and the selected audit page, with complete long certificates available in CSV.

Download & Print

Save the denominator, assumptions and complete contribution audit

Copy both exact and rounded estimates

The readable summary keeps the highlighted convention, both sample and population results, auxiliary estimates and method notes together. Ordered working and all sorted contributions remain included, even when hidden on screen. Mathematical expressions become plain text for destinations that cannot render visual LaTeX. This makes the copied record useful for reviewing the calculation rather than relying on a screenshot of one isolated standard deviation.

Download a complete numerical report

CSV retains the literal active source, source-unit label, precision and interpretation choices alongside each exact result. A rectangular contribution section includes frequencies, signed deviations, squares, weighted totals and percentage contributions. Formula-like text cells are protected against spreadsheet execution. Unavailable estimates retain their reason instead of an invented numerical zero, and hidden rows or optional relative-spread calculations remain distinguishable through the saved display and assumption flags.

Print a focused mathematical worksheet

Use Print / Save PDF and choose a browser PDF destination. Editing controls and supporting article sections are removed, while visible exact working and the selected audit page remain. Short contribution tables stay together and repeated table headings support longer pages. Very long exact expressions point to their complete CSV forms instead of running beyond the page width. For larger audits, use CSV to retain every row rather than assuming one printed page contains the full dataset.

FAQ

Frequently Asked Questions

Should I use sample or population standard deviation?

Use the population convention when the supplied observations constitute the complete target set you want to describe. Use the sample convention when the observations are treated as a sample for estimating a larger population's variance. The same squared-deviation total is divided by

n

n-1

, respectively. This calculator shows both results rather than inferring the statistical meaning from the number of values or from a measurement label.

What is the difference between variance and standard deviation?

Variance averages squared distances from a center under the chosen denominator convention, so it uses squared source units. Standard deviation is its nonnegative square root and returns to the original source unit. For the default dataset, the sample variance is

s^2=7

, while sample standard deviation is

s=\sqrt{7}

. The exact expression and the independently rounded view are displayed separately, allowing a rational or radical result to remain inspectable.

Why does sample variance divide by one fewer observation?

The sample mean is estimated from the same observations, which constrains the signed deviations to sum to zero. Dividing their squared total by

n-1

gives the familiar unbiased sample-variance estimator under the usual independent, identically distributed sampling assumptions and finite variance. Taking a square root does not automatically make sample standard deviation unbiased. This tool implements the standard sample formula and does not silently apply distribution-specific corrections or infer that the sampling assumptions hold.

What happens with a singleton or a constant dataset?

A singleton has zero population variance and population standard deviation, because its only observation equals its dataset mean. The sample denominator is zero, so sample variance, sample standard deviation and the estimated standard error are unavailable. A constant dataset with at least two observations has zero variance and standard deviation under both conventions. These cases remain different from an empty dataset, invalid input or a zero-frequency table with no observed values.

How do frequency rows change the calculations?

A nonnegative integer frequency represents repeated observations of the row's exact value. Each squared deviation is multiplied by that count before entering the total, and the denominator uses the sum of counts rather than the number of distinct rows. Zero-frequency rows are excluded explicitly. The program never allocates an expanded list for large counts. Probability weights, reliability weights and unknown observations within grouped class intervals require different models and are not guessed from this input format.

How are very large offsets and tiny spreads kept accurate?

Parsing, sorting, mean calculation, centered differences and variance arithmetic retain exact integer fractions. Closely spaced large values therefore remain distinct, and tiny squared magnitudes are not lost to binary floating-point subtraction or underflow. Square-root decimal views use integer comparisons against rounding boundaries. The tool also verifies an exact second-moment identity. Input and combined-arithmetic bounds remain explicit; an unsupported calculation reports an error instead of returning a partly computed or silently rounded result.

What does the estimated standard error mean?

The displayed estimate is

\widehat{\mathrm{SE}}(\bar{x})=s/\sqrt{n}

. It describes estimated variation of a sample mean under an appropriate independent sampling model, rather than dispersion of the individual observations themselves. Repeating a frequency count does not establish independence in the real collection process. The estimate is unavailable for a singleton. It is not a confidence interval, a guarantee of measurement accuracy or a substitute for checking how the observations were obtained.

When are coefficients of variation useful?

The optional coefficients express standard deviation relative to the positive mean and are displayed as percentages. Their interpretation requires ratio-scale measurements with a meaningful zero. A label alone cannot establish that assumption, and interval scales such as ordinary Celsius temperatures do not support the same interpretation. A zero or negative mean is reported as unavailable under this tool's stated convention. Near a zero mean, even small source changes can make the percentages highly sensitive.

What input, precision and import limits apply?

Use signed finite decimals, scientific e notation or integer fractions, with a decimal point and supported separators. Commas separate observations rather than thousands. Source editors and local files are bounded, and the tool supports up to two thousand source values or frequency rows with total count one billion. Selected precision changes the numerical view, never the exact variance. Tiny nonzero values use scientific notation when they would otherwise display as zero, and halfway rounding moves away from zero.

What is retained in copies, CSV and printed PDFs?

Copy and CSV retain both statistical conventions, exact certificates, rounded views, assumptions, ordered working and every sorted deviation contribution, including rows hidden by pagination or display switches. CSV additionally retains the active literal source and protects formula-like text cells. Browser printing includes the currently visible result, working and audit page. Long exact expressions receive explicit CSV notices to prevent clipping. A shareable fragment preserves validated active data and display choices within its separate encoded-size limit.

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About This Tool

Spread should explain the data and the chosen denominator

A standard deviation describes dispersion through distances from an arithmetic center. Squaring those distances prevents negative and positive deviations from canceling, while also giving distant observations greater influence on the total. Taking the nonnegative square root restores the source unit. A dataset with the same mean as another can still have a very different spread, and one extreme observation can make a substantial contribution. Neither a small standard deviation nor a large sample count establishes measurement accuracy, representativeness or a particular population distribution.

This calculator keeps a numerical audit separate from the interpretation of a dataset. The exact supplied observations determine the mean and squared-deviation total. A population convention describes them as the complete target set; a sample convention uses one fewer degree of freedom in its variance denominator. Both are shown because their statistical purposes cannot be inferred from an opaque unit label. The sample formula requires at least two observations, whereas a singleton has a valid zero population spread. Integer frequencies preserve repetitions without inventing observations inside a grouped class or assuming arbitrary weights count as independent samples.

Exact rational arithmetic retains finite decimal and fraction sources through sorting, centering and squaring. Positive root certificates and certified numerical rounding then keep large, tiny and closely spaced inputs mathematically coherent within the stated bounds. The signed-deviation closure and second-moment identity provide complementary checks on the same source record. Estimated standard error, mean absolute distance and optional coefficients of variation are labelled separately because they answer different questions or require additional assumptions. Together with the complete contribution export, these distinctions help a reviewer connect a numerical spread back to the observations and model that produced it.

Editorial Transparency

Who maintains this page

Ownership and review

Written and maintained by the Toolarithm editorial team. No review date is shown without a maintained editorial record. No independent professional review is claimed.

Methodology

Equations are checked by substituting worked examples into the stated relationship and comparing the result with the interactive implementation.

Read the editorial methodology

Dates and sources

Review dates change only after a substantive method or content check.

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Standard Deviation Calculator

Enter your data

Conventions and display

Sample standard deviation

Sample and population comparison

Sample standard deviation

Population standard deviation

Sample variance

Population variance

Arithmetic mean

Sum of squared deviations

Estimated standard error of mean

Sum of observations

Minimum

Maximum

Range

Mean absolute deviation about the mean

Sum of squared observations

Step-by-step standard deviation solution

1. Validate the observation model

2. Find the exact arithmetic mean

3. Center, square and weight each observation

4. Choose the variance denominator

5. Take the nonnegative square roots

6. Check the centered arithmetic independently

7. Keep standard error and relative spread distinct

Sorted squared-deviation audit

See how each observation contributes to spread

Compare both variance denominators

Audit every weighted squared deviation

Retain small differences beside large values

Standard deviation formulas and a complete worked example

Why exact arithmetic and interpretation stay separate

The numerical audit is reproducible

The statistical model remains explicit

A practical spread check for learning and numerical review

Students following the variance calculation

Teachers comparing conventions and assumptions

People reviewing numerical measurement lists

How it works in three quick steps.

Enter the observations or their repetition counts

Compare the sample and population interpretations

Audit the working and retain a complete record

Save the denominator, assumptions and complete contribution audit

Copy both exact and rounded estimates

Download a complete numerical report

Print a focused mathematical worksheet

Frequently Asked Questions

Related calculators

Mean Median Mode Calculator

Percentile Calculator

Z Score Calculator

Spread should explain the data and the chosen denominator

Who maintains this page

Ownership and review

Methodology

Dates and sources

Explore more calculators