Standard Deviation Calculator
Calculate std dev, variance, mean, and more
Enter or paste your numbers below, separated by commas, spaces, or new lines.
When you have a dataset (maybe it's test scores, sales numbers, survey responses, or measurement readings), and you need to know how spread out those numbers really are, you need to calculate the standard deviation.
The standard deviation calculator does that for you.
Paste your numbers in, choose whether your data is a full population or a sample, and you'll get the standard deviation, variance, mean, and a bunch of other useful stats.
But a number by itself isn't very useful. The rest of this page covers how the calculation works, what your result means, and the common mistakes to watch out for.
How Standard Deviation Is Calculated
The formula looks intimidating, but the logic behind it is straightforward. You're measuring how far each value sits from the average.
Sample standard deviation:
s = √[ Σ(xi - x̄)² / (n - 1) ]
Population standard deviation:
σ = √[ Σ(xi - μ)² / N ]
The only difference is what you divide by. Samples use (n - 1). Populations use N.
How to Interpret the Results
This is the part most calculators skip. You've got a number. Now what?
Low vs. High Standard Deviation
Standard deviation tells you how spread out your data is. A low value means the numbers cluster tightly around the mean. A high value means they're scattered.
But "low" and "high" are relative. A standard deviation of 5 could be tiny if your mean is 10,000, or enormous if your mean is 8.
The best way to judge is the Coefficient of Variation (CV):
CV = (Standard Deviation / Mean) × 100%
| CV Range | What It Means |
|---|---|
| Under 15% | Low variability. Data is consistent. |
| 15% to 30% | Moderate spread. Fairly normal. |
| 30% to 50% | High. Worth investigating why. |
| Over 50% | Very high. Data points differ significantly. |
For the quiz score example: CV = (9.33 / 82) × 100% = 11.4%. That's low variability. The class performed fairly consistently.
The 68-95-99.7 Rule
If your data follows a roughly normal (bell-shaped) distribution, standard deviation gives you a built-in probability guide:
- About 68% of values fall within 1 SD of the mean
- About 95% fall within 2 SDs
- About 99.7% fall within 3 SDs
For the quiz example (mean 82, SD 9.33): roughly 68% of students scored between 73 and 91. Any score below 54 or above 110 would be a serious outlier (beyond 3 SDs).
When Standard Deviation Equals Zero
A standard deviation of zero means every value in your dataset is identical. No variation at all.
Real-World Benchmarks
How you read your result depends on what you're measuring:
- Test scores: SD of 10 to 15 points on a 100-point exam is typical. Much higher suggests the test was polarizing.
- Investment returns: Annual SD below 15% is moderate risk. Above 25% is high volatility.
- Manufacturing measurements: SD should be small relative to your tolerance range. Six Sigma methodology targets defect rates below 3.4 per million.
- Survey ratings (1-5 scale): SD around 1.0 is normal. Below 0.5 means strong consensus. Above 1.5 means sharp disagreement.
Population vs. Sample: Which Should You Pick?
This trips up a lot of people. Here's the simple version:
Use Population (σ) when your dataset includes every single member of the group you care about. For example, all 30 students in a class, or every transaction from last Tuesday.
Use Sample (s) when your data is a subset of a larger group. For example, 200 customers surveyed out of 50,000 total, or 15 products tested from a batch of 10,000.
When in doubt, use Sample. It's the safer choice. The sample formula divides by (n - 1) instead of n, which slightly increases the result. This corrects for the fact that a sample tends to underestimate the true spread of a population. With large datasets (above 30 or so values), the difference between the two is small. It matters most with small samples.
Common Mistakes to Avoid
1. Using Population SD for Sample Data
This is the most frequent calculation error. If your data is a sample (which it usually is), dividing by n instead of (n - 1) systematically underestimates the true standard deviation. For a dataset of 5 values, the difference can be significant. The calculator above shows both values so you can compare.
2. Confusing Standard Deviation with Standard Error
Standard deviation measures how spread out your individual data points are. Standard error (SE = SD / √n) measures how precisely your sample mean estimates the population mean. SE is always smaller than SD, so reporting SE instead of SD makes your data look more consistent than it actually is. This actually happens a lot in published research.
3. Comparing Standard Deviations Across Different Scales
A standard deviation of 10 means something very different when your mean is 50 versus 5,000. If you need to compare variability between datasets with different units or scales, use the coefficient of variation (CV = SD / Mean × 100%) instead.
4. Applying SD to Heavily Skewed Data
Standard deviation works best with roughly symmetric data. For skewed distributions (like income, housing prices, or website traffic), outliers inflate the SD and make it misleading. The interquartile range (IQR) or median absolute deviation works better for that kind of data.
5. Rounding Intermediate Steps Too Early
If you're doing the calculation by hand, don't round the mean or squared deviations before the final step. Rounding errors compound through each step and can throw off your final answer by a noticeable amount. Keep full precision until the end.
Also see: Calculating Standard Deviation in Excel
Some Questions You May Have
Can standard deviation be negative?
No. Standard deviation is always zero or positive. It's the square root of squared values, so a negative result isn't possible. If your hand calculation gives you a negative number, there's an arithmetic error somewhere.
What is the difference between standard deviation and variance?
Variance is the average of squared deviations from the mean. Standard deviation is the square root of variance. The practical difference is units. If your data is in dollars, variance is in "dollars squared" (not intuitive), while standard deviation is in dollars. That's why SD is reported more often.
Why does the sample formula use (n - 1) instead of n?
This is called Bessel's correction. A sample mean is always slightly closer to the sample data than the true population mean. Dividing by n would consistently underestimate the real variability. The (n - 1) adjustment fixes that. It matters most when you're working with small samples.
What counts as a "good" standard deviation?
There's no universal benchmark. It depends on your data and context. The coefficient of variation (CV = SD / Mean × 100%) is the most practical way to judge. A CV under 15% generally indicates tight consistency. Over 50% suggests high variability. But what counts as acceptable really depends on your specific situation.
How many data points do I need?
Technically, you need at least 2 values. But standard deviation becomes more reliable as your dataset grows. With fewer than 10 data points, take the result with a grain of salt. For meaningful statistical analysis, 30 or more values is a common rule of thumb.
Can I use this calculator for data I copied from Excel?
Yes. Copy your column of numbers from Excel and paste directly into the text box. The calculator handles tab-separated and newline-separated values automatically. You don't need to convert them to commas first.
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