Standard Deviation Calculator
Paste a list of numbers to calculate the sample standard deviation, population standard deviation, variance, mean, and a deviation table in one place.
Standard Deviation Calculator
Enter a list of numbers separated by commas, line breaks, or spaces to see the sample standard deviation, population standard deviation, variance, mean, and sum of squared deviations on one screen.
Use the sample option when you estimate population variability from a subset of observations, so the divisor becomes n – 1.
Commas, line breaks, spaces, and semicolons are all accepted as separators. Negative values and decimals are supported, and empty entries are ignored automatically.
Adjust the display precision to match a report, homework, or documentation format.
It is often enough to remember that the sample basis divides by n – 1 while the population basis divides by n.
Each example button updates both the number list and the basis so you can inspect the calculation flow immediately.
- A smaller standard deviation means the values cluster more tightly around the mean, while a larger one means they are more spread out.
- Use the sample basis when a subset is used to estimate the population, and the population basis when the full dataset is available.
- The standard deviation uses the same unit as the input values, so exam scores stay in points and length data stays in length units.
- The deviation table helps you spot which values contribute the most to the overall variance.
Enter the data and click Calculate to refresh the result.
For these 8 values, the mean is 5.00, the sample standard deviation is 2.14, and the population standard deviation is 2.00.
| Current basis | Sample |
|---|---|
| Number of values | 8 |
| Sum | 40.00 |
| Mean | 5.00 |
| Sum of squared deviations | 32.00 |
| Sample variance | 4.57 |
| Sample standard deviation | 2.14 |
| Population variance | 4.00 |
| Population standard deviation | 2.00 |
| Minimum / Maximum | 2.00 / 9.00 |
- The mean is 40.00 ÷ 8 = 5.00.
- Subtract the mean from each value, square the result, and add them together to get a sum of squared deviations of 32.00.
- For the sample basis, divide 32.00 by 7 (n – 1), then take the square root to get 2.14.
This dataset typically sits about 2.14 away from the mean of 5.00. The values 2 and 9 contribute heavily to the spread.
| No. | Value x | Deviation x – mean | Squared deviation (x – mean)² |
|---|---|---|---|
| 1 | 2.00 | -3.00 | 9.00 |
| 2 | 4.00 | -1.00 | 1.00 |
| 3 | 4.00 | -1.00 | 1.00 |
| 4 | 4.00 | -1.00 | 1.00 |
| 5 | 5.00 | 0.00 | 0.00 |
| 6 | 5.00 | 0.00 | 0.00 |
| 7 | 7.00 | +2.00 | 4.00 |
| 8 | 9.00 | +4.00 | 16.00 |
What is a Standard Deviation Calculator?
A standard deviation calculator helps you see at a glance how widely a list of numbers is spread around the mean. The average alone does not tell you whether the data is tightly grouped or widely scattered, but the standard deviation reveals that variation more clearly.
This tool calculates both the sample and population standard deviation from a list of numbers separated by commas or line breaks, then organizes the variance, mean, sum of squared deviations, and a value-by-value deviation table on the same screen. It is especially useful when you want to move directly from calculation to interpretation for homework checks, score analysis, measurement comparisons, or return-volatility reviews.
Where this helps
Standard deviation is useful in almost any situation where you need to understand how far data points sit from the mean, including score comparisons and measurement-quality checks. It is also convenient for quick verification before opening a spreadsheet because even a short list of values can be calculated instantly.
- Exam score analysis – When class averages look similar but you want to see whether the score spread is actually similar too
- Experiment and sensor checks – When you want a quick read on the stability and spread of repeated measurements
- Return volatility review – When you want to see how much monthly returns or price changes moved around
- Homework and statistics study – When you want to compare the sample and population formulas directly
- Quick pre-cleaning check – When you want to paste a few values from a CSV and inspect the mean and variance first
Key features
This calculator does more than return a single number. It also shows the supporting details you need to understand the calculation, so you can move from checking the standard deviation to identifying which values drive the spread the most without leaving the page.
- Sample and population side by side – The active basis and the alternate basis are shown together for easy comparison.
- Flexible separators – Paste number lists separated by commas, line breaks, spaces, or semicolons without reformatting.
- Variance, mean, and range included – Key supporting statistics are organized alongside the standard deviation.
- Value-by-value deviation table – See each deviation and squared deviation to find the values that influence the variance most.
- Quick examples and copyable results – Load common examples instantly and copy a short summary sentence for sharing.
How to use it
Start by choosing whether the data should be treated as a sample or a population, then enter the number list and calculate. Because the count, sum of squared deviations, variance, and standard deviation are shown together, it is easy to verify the formula. You can also change the decimal precision to suit a report or message.
- Choose the basis – Pick Sample for a subset and Population when you already have the full list.
- Enter the number list – Paste values separated by commas or line breaks.
- Click Calculate – Read the standard deviation for the selected basis in the top result card.
- Check the summary – Compare the mean, variance, sum of squared deviations, and the alternate basis together.
- Review the deviation table – Use it to see which values make the spread wider.
Sample vs. population standard deviation
Standard deviation is calculated by finding the mean, measuring how far each value sits from that mean, squaring those deviations, and then converting the total into an average spread. Population standard deviation assumes the full dataset is present and divides the sum of squared deviations by n. Sample standard deviation treats the data as a subset and divides by n - 1 to correct for degrees of freedom.
That divisor difference is why the sample standard deviation often comes out slightly larger than the population standard deviation for the same data. If you have every score from a class, the population basis is the natural choice. If you only sampled a few students to estimate the full class pattern, the sample basis is more appropriate. When you are unsure, start by asking whether you truly have the whole list.
Variance is the squared form of the standard deviation, so it preserves the spread but uses squared units. Standard deviation takes the square root again, which brings the result back to the same unit as the input values. That is why most people read the standard deviation first and then look at the variance and deviation table when they need more detail.
- Population basis – Divide by
nwhen the complete dataset is available. - Sample basis – Divide by
n - 1when the data is a subset used for estimation. - Variance uses squared units, while standard deviation uses the original input unit.
- The deviation table makes it easy to see which values drive the variance the most.
Frequently asked questions
When should I use the sample basis instead of the population basis?
Use the population basis when you already have the full dataset. Use the sample basis when you only have a subset and want to estimate the wider population. The sample formula uses n - 1 to reduce estimation bias.
Can I mix commas and line breaks in the same list?
Yes. This tool treats commas, line breaks, spaces, and semicolons as separators. One line can use commas while the next line uses line breaks and the list will still be parsed correctly.
Can it handle decimals and negative numbers?
Yes. Standard deviation only depends on the values themselves and their distance from the mean, so decimals and negative numbers are valid inputs. If non-numeric text appears in the list, the tool shows a warning so you can fix the incorrect entry.
Why can’t a sample standard deviation be calculated from only one value?
The sample formula divides by n - 1, so a single value makes the divisor equal to zero. In that case the population basis can still return 0, but the sample basis cannot produce a statistically meaningful estimate.
What is the difference between variance and standard deviation?
Variance is the average of squared deviations, so it is expressed in squared units. Standard deviation is the square root of that variance, which brings it back to the original unit and often makes it easier to interpret.
What does the copied result include?
The copy action creates a one-line summary with the current basis, value count, mean, standard deviation, variance, and range. It is useful for quickly sharing the result in a message or dropping it into a document.
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