Descriptive Statistics
Standard Deviation Calculator
Sample standard deviation divides the squared deviations by n minus 1 and keeps the unit of the data, unless you tick the population box.
Worksheet
Press Calculate to see the result.
Sample standard deviation divides the squared deviations by n minus 1 and keeps the unit of the data, unless you tick the population box.
What this calculator answers
Quiz scores of 72, 80, 85, 90, and 93 sit around a mean that was estimated from those same five papers. The sample standard deviation charges one degree of freedom for that estimate, so the squared gaps are divided by 4 rather than by 5. The result stays in points, the unit of the papers. Use the population box only when these five scores are the entire group you will ever judge. The page keeps the definition fixed while the ticket updates, so a hand check can match every intermediate term. The worksheet asks for Numbers, Population divisor (divide by n), and it refuses a blank that the formula cannot interpret. Treat the loaded example as a rehearsal for the live numbers you will type next, not as a claim about your own data.
The headline is the standard deviation under the divisor you selected. The matching variance is shown beside it, and the other divisor is shown so the checkbox has a visible effect. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Standard deviation uses this relationship.
s = square root of (sum of squared deviations) / (n - 1). Tick the box to divide by n instead.
- s. Sample standard deviation, using divisor n minus 1.
- sigma. Population standard deviation, using divisor n.
- SS. Sum of squared deviations from the mean.
- n. How many numbers were typed.
Square each gap so distances above and below the mean cannot cancel. Divide by n minus 1 when the mean was estimated from the same list, then take the square root to return to the original unit. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The form is prefilled with a small example so every arithmetic step can be checked by hand. Change one field after you finish the walkthrough and confirm the ticket moves the way the formula predicts.
There are 5 values: 72, 80, 85, 90, 93. Their sum is 420 and their mean is 84. The sum of squared deviations from that mean is 278.
The sample variance divides 278 by 4, giving 69.5, and its square root is 8.3367. The population variance divides the same 278 by 5, giving 55.6, and its square root is 7.4565. The population box is off, so the headline uses divisor 4 and the standard deviation is 8.3367.
The unit of both the standard deviation and the raw data is the same. Squaring stops gaps above and below the mean from cancelling before the root brings the unit back.
| Sample standard deviation | 8.3367 |
|---|---|
| Sample variance | 69.5 |
| Population standard deviation | 7.4565 |
| Mean | 84 |
| Count | 5 |
A second situation
The same five scores, with the population divisor n
There are 5 values: 72, 80, 85, 90, 93. Their sum is 420 and their mean is 84. The sum of squared deviations from that mean is 278.
The sample variance divides 278 by 4, giving 69.5, and its square root is 8.3367. The population variance divides the same 278 by 5, giving 55.6, and its square root is 7.4565. The population box is ticked, so the headline uses divisor 5 and the standard deviation is 7.4565.
The unit of both the standard deviation and the raw data is the same. Squaring stops gaps above and below the mean from cancelling before the root brings the unit back.
| Population standard deviation | 7.4565 |
|---|---|
| Population variance | 55.6 |
| Sample standard deviation | 8.3367 |
| Mean | 84 |
| Count | 5 |
With the population box ticked, the same sum of squared deviations, 278, is divided by 5 instead of by 4. The headline falls from the sample value 8.3367 points to the population value 7.4565 points. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The population box is ticked, so the divisor changes from n minus 1 to n. The headline result went from 8.3367 to 7.4565.
The nudged inputs
There are 5 values: 72, 80, 85, 90, 93. Their sum is 420 and their mean is 84. The sum of squared deviations from that mean is 278.
The sample variance divides 278 by 4, giving 69.5, and its square root is 8.3367. The population variance divides the same 278 by 5, giving 55.6, and its square root is 7.4565. The population box is ticked, so the headline uses divisor 5 and the standard deviation is 7.4565.
The unit of both the standard deviation and the raw data is the same. Squaring stops gaps above and below the mean from cancelling before the root brings the unit back.
| Population standard deviation | 7.4565 |
|---|---|
| Population variance | 55.6 |
| Sample standard deviation | 8.3367 |
| Mean | 84 |
| Count | 5 |
How to read the result
Say the unit and the divisor in the same breath as the number. A result of 8.3367 points on a sample divisor is a wider figure than 7.4565 points on the population divisor, and the data did not change. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Write the result with its unit, the inputs that produced it, and the name of the formula. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- Every score you mean to include is typed once, in the same unit.
- The mean is a sensible center for this scale.
- The population box is ticked only when the list is the whole population.
- A blank cell is missing, not a zero, unless a zero was actually observed.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Dividing by n for a sample and calling the smaller number more precise.
- Comparing a standard deviation in points with one in percentages.
- Dropping the square root and reporting the variance as a standard deviation.
- Letting one mistyped extra zero dominate the squared gaps.
Where a different calculator is the honest choice
Use the variance calculator when the squared unit is the quantity you need to pool or to add. Use the coefficient of variation when you need spread relative to the mean. Use the mean calculator when the equal share is the only question. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Variance Calculator, Coefficient of Variation Calculator, Mean Calculator.
Check yourself
Scores are 72, 80, 85, 90, and 93. What is the sample standard deviation?
The sample standard deviation is 8.3367 points, and the sample variance is 69.5.
Using the result in a sentence
Lab repeats, class scores, and process readings are the usual homes for this number. Keep the sample size next to it. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
Five scores cannot describe a whole course. One miskeyed score moves a standard deviation more than it moves a median. Pair the figure with the minimum and the maximum before you set a grading curve from it. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Standard Deviation Calculator commit to?
This page uses Standard deviation: s = square root of (sum of squared deviations) / (n - 1). Tick the box to divide by n instead.. Square each gap so distances above and below the mean cannot cancel. Divide by n minus 1 when the mean was estimated from the same list, then take the square root to return to the original unit. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Numbers, Population divisor (divide by n), and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the sample standard deviation is 8.3367. Say the unit and the divisor in the same breath as the number. A result of 8.3367 points on a sample divisor is a wider figure than 7.4565 points on the population divisor, and the data did not change. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Dividing by n for a sample and calling the smaller number more precise. Five scores cannot describe a whole course. One miskeyed score moves a standard deviation more than it moves a median. Pair the figure with the minimum and the maximum before you set a grading curve from it. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Use the variance calculator when the squared unit is the quantity you need to pool or to add. Use the coefficient of variation when you need spread relative to the mean. Use the mean calculator when the equal share is the only question. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The sample standard deviation is 8.3367 points, and the sample variance is 69.5. Name the formula, the sample size or the inputs, and the unit before you compare the figure with someone else's. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.