Standard Deviation Calculator

Last updated: April 2026

Enter your dataset

Separate numbers with commas, spaces, new lines, or tabs. Use sample for most real-world datasets where your data is a subset of a larger group.

Data values

8 numbers entered

Standard deviation (s)

2.13809

Variance (s²)

4.571429

Supporting stats

Count (n): 8
Mean: 5
Sum of squares: 32
Variance: 4.571429
Std deviation: 2.13809

Deviation table

x	Mean	x - mean	(x - mean)^2
2	5	-3	9
4	5	-1	1
4	5	-1	1
4	5	-1	1
5	5	0	0
5	5	0	0
7	5	2	4
9	5	4	16
Sum		0 OK	32

Normal distribution visual

The 68-95-99.7 Rule

In a normal distribution, 68% of data falls within +/-1 standard deviation, 95% within +/-2, and 99.7% within +/-3.

+/-1s: 2.86191 to 7.13809

+/-2s: 0.72382 to 9.27618

Step-by-step working

Find the mean: (2 + 4 + 4 + 4 + 5 + 5 + 7 + 9) / 8 = 40 / 8 = 5.
(2 - 5)^2 = (-3)^2 = 9
(4 - 5)^2 = (-1)^2 = 1
(4 - 5)^2 = (-1)^2 = 1
(4 - 5)^2 = (-1)^2 = 1
(5 - 5)^2 = (0)^2 = 0
(5 - 5)^2 = (0)^2 = 0
(7 - 5)^2 = (2)^2 = 4
(9 - 5)^2 = (4)^2 = 16
Sum the squared deviations: 9 + 1 + 1 + 1 + 0 + 0 + 4 + 16 = 32.
Sample variance with Bessel's correction: 32 / (8 - 1) = 32 / 7 = 4.571429.
s = sqrt(4.571429) = 2.13809.

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Population vs sample standard deviation

Population standard deviation divides by N. Sample standard deviation divides by n - 1 because sample data usually underestimates the spread of the full population.

Why divide by n - 1?

Bessel's correction adjusts for the fact that a sample mean is estimated from the same data, giving a less biased estimate of population variance.

What does standard deviation tell you?

A small standard deviation means values cluster near the mean. A large standard deviation means values are more spread out.

Standard deviation vs variance

Variance is the average squared deviation. Standard deviation is the square root of variance, so it is expressed in the same units as the original data.

About this calculator

The Standard Deviation Calculator measures how spread out a set of numbers is around the mean. It calculates mean, variance, population standard deviation, sample standard deviation, and related summary values where supported. Use this expanded guide when you need more than a quick result. It explains the assumptions behind the Standard Deviation Calculator, the records to gather, and the decisions the estimate can support. It is especially useful for students, analysts, teachers, researchers, and business users measuring how spread out a dataset is around its mean. The strongest use of the page is scenario comparison: change one input at a time, compare the output, and keep a note of which assumption changed.

Standard deviation method

Standard deviation is the square root of variance. Population standard deviation divides by the number of values, while sample standard deviation divides by one less than the number of values. The calculator result depends on the quality of the inputs and on the rule set or formula selected in the calculator above. For practical use, treat the output as a structured estimate: start with the core inputs, review the main outputs, then test the decision points that matter most to your situation. Key decisions include whether to use sample or population formula, whether outliers are driving spread, how to compare variation between datasets.

mean = sum of values / n
population variance = sum((x - mean)^2) / n
sample variance = sum((x - mean)^2) / (n - 1)
better estimate = accurate inputs + correct rule set + realistic assumptions
scenario difference = revised result - original result

How to use the standard deviation calculator

Enter the data values separated by commas, spaces, or lines.
Choose sample or population if the calculator asks.
Review the mean and variance.
Review standard deviation and count.
Use the same units as the original data when interpreting spread.
Gather the main inputs first: data values, sample or population choice, decimal precision.
Check supporting records such as raw data and data source before relying on a final number.
Enter one realistic scenario first, using conservative assumptions where the future is uncertain.
Review the main outputs: mean, variance, sample standard deviation.
Run at least one alternative scenario so you can see which input changes the answer most.
Compare the result with standard statistical formulas or the relevant contract, bill, statement, or professional document.
Keep the calculation date and assumptions with your notes so you can revisit the estimate when rates, rules, or circumstances change.

Worked example

Small dataset

Input: Values 2, 4, 4, 4, 5, 5, 7, 9

Calculation: Mean is 5; population variance is 4; square root of 4 is 2

Result: Population standard deviation is 2.

Quality-control scenario

Input: A factory measures fill weights for a sample of bottles.

Calculation: Sample standard deviation measures spread around the sample mean.

Result: A higher value suggests less consistent fills.

Outlier scenario

Input: A dataset has one value far from the rest.

Calculation: Squared deviations make the outlier influential.

Result: The calculator result rises, and the user should inspect the data.

Sample versus population

Use population standard deviation when the data includes every value in the group being studied. Use sample standard deviation when the data is a sample used to estimate a wider population.

What to check before relying on the result

A useful Standard Deviation Calculator result starts with the same evidence you would use if you were checking the answer manually. The calculator can organise the arithmetic, but it cannot know whether a payslip is final, a bill is estimated, a quote excludes fees, or a personal circumstance has changed since the last statement.

Before making a decision, compare the calculator result with the source document that controls the real outcome. For this topic, that usually means checking standard statistical formulas. If there is a difference between the calculator and an official statement, contract, assessment, or professional advice, treat the official document as the stronger source.

raw data: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.
data source: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.
cleaning notes: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.
calculation method: Use this as supporting evidence for the calculation. If it is out of date, estimated, or based on a different period, the calculator output may look precise while still being wrong for the decision.

Inputs that usually change the answer

The most important input is not always the largest number on the form. Sometimes a date, threshold, percentage, eligibility flag, or timing assumption changes the result more than the headline amount. This is why scenario testing is more useful than a single calculation.

Input	Why it matters	What to double-check
data values	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.
sample or population choice	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.
decimal precision	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.
grouping or raw-data format	It feeds directly into the estimate or changes which rule is applied.	Check the period, units, eligibility, and whether the figure is final or estimated.

How to interpret the output

The output should be read as a decision aid, not just a number. For Standard Deviation Calculator, the useful question is often what the result means for timing, affordability, eligibility, comparison, or next steps.

mean: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
variance: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
sample standard deviation: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
population standard deviation: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.
count: Use this output alongside the other results rather than in isolation. A monthly amount, percentage, date, or payback figure can look acceptable until fees, timing, evidence, or eligibility conditions are added.

Scenarios worth comparing

A single estimate is a snapshot. A better approach is to save a base case, then adjust one assumption at a time. This shows whether the result is stable or whether a small change in timing, rate, usage, income, or cost creates a very different answer.

Scenario	Change one assumption	What the comparison shows
Base case	Use the best current evidence.	Shows the result you would expect if nothing important changes.
Conservative case	Use lower income, higher cost, slower growth, or less favourable timing.	Shows whether the decision still works with less optimistic assumptions.
Improved case	Use the realistic upside, such as lower cost, better rate, higher usage, or stronger evidence.	Shows the potential benefit without treating it as guaranteed.

Common mistakes and edge cases

Most errors come from using the right formula with the wrong assumption. Dates can be counted differently, rates can change, official thresholds can move, and real bills or contracts often include conditions that a simple calculator cannot infer automatically.

Sample and population formulas differ.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.
Outliers can dominate the result.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.
Standard deviation is most intuitive for roughly symmetric data.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.
Units match the original data.: Check this point before using the estimate for a payment, claim, purchase, application, employment decision, or health-related decision.

Next steps after calculating

Once you have a result, write down the key assumptions and compare them with standard statistical formulas. If the number affects a deadline, tax return, benefit claim, employment issue, medical question, finance agreement, or major purchase, use the calculator as preparation for a more formal check.

For lower-stakes use, the next step may simply be comparing two or three scenarios. For higher-stakes use, the next step should be checking the official guidance, speaking to the relevant organisation, or getting qualified advice before acting.

Important edge cases

Sample and population formulas differ.
Outliers can dominate the result.
Standard deviation is most intuitive for roughly symmetric data.
Units match the original data.

Limitations

This calculator provides statistical calculations only. This is educational statistics information, not professional statistical advice. The calculator is designed to support understanding and planning, but it cannot verify documents, predict future rule changes, or account for every exception. Use it as an estimate and check the official source before acting where the result matters.

Outliers can strongly affect standard deviation.
It does not prove causation or data quality.
Some skewed data is better described with percentiles or interquartile range.
Check standard statistical formulas for current rules, rates, definitions, and eligibility where relevant.
Do not rely on a single scenario where income, costs, dates, rates, usage, or health circumstances may change.
Keep records of the inputs used so that the estimate can be reviewed later.

Frequently asked questions

What does a higher standard deviation mean?

It usually means the values are more spread out from the mean.

Should I use sample or population?

Use sample when estimating from part of a larger group, and population when you have the whole group.

Is standard deviation in the same unit as the data?

Yes. Variance is squared units, but standard deviation returns to the original unit.

Why square deviations?

Squaring avoids positive and negative differences cancelling out and gives more weight to larger deviations.

What is variance?

Variance is the average squared deviation; standard deviation is its square root.

Can standard deviation be negative?

No. It is zero or positive because it measures spread.

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Mean, Median, Mode and Range Calculator
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What does this mean?

This calculator is designed to help you understand the likely number before you make a decision or start an application.

Your result should be checked against official UK guidance, especially if your circumstances include dependants, exemptions, prior leave, or a complex immigration history.

Treat the figure as a planning tool rather than legal advice. Where the answer affects an application deadline or major payment, speak to an authorised adviser.

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