📈 Standard Deviation Calculator
Enter a list of numbers to instantly calculate the mean, variance, and standard deviation — for both the full population and a data sample.
Last Updated: July 10, 2026
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Results
📊 Worked Example: Daily Sales Figures
Data set: 42, 45, 38, 50, 41, 47, 44 (n = 7)
| Value | Deviation from Mean (43.86) | Squared Deviation |
|---|---|---|
| 42 | −1.86 | 3.45 |
| 45 | 1.14 | 1.31 |
| 38 | −5.86 | 34.31 |
| 50 | 6.14 | 37.73 |
| 41 | −2.86 | 8.16 |
| 47 | 3.14 | 9.88 |
| 44 | 0.14 | 0.02 |
💡 Population vs. Sample — Which Do I Use?
Use population standard deviation (÷n) when your data set IS the entire group you care about. Use sample standard deviation (÷n−1) when your data is a sample used to estimate a larger population — dividing by n−1 (Bessel's correction) prevents underestimating the true variability.
📈 How to Calculate Standard Deviation
Step 1: Find the Mean
Mean = Sum of all values ÷ Count. For 42, 45, 38, 50, 41, 47, 44: sum = 307, n = 7, mean = 307 ÷ 7 = 43.86.
Step 2: Find Squared Deviations
Subtract the mean from each value and square the result, then sum all squared deviations. Sum of squared deviations = 94.86.
Step 3: Population Variance and Standard Deviation
Population Variance = Sum of Squared Deviations ÷ n = 94.86 ÷ 7 = 13.55. Population SD = √13.55 = 3.68.
Step 4: Sample Variance and Standard Deviation
Sample Variance = Sum of Squared Deviations ÷ (n − 1) = 94.86 ÷ 6 = 15.81. Sample SD = √15.81 = 3.98.
💡 Real-World Examples & Use Cases
See how population and sample standard deviation are used in real analysis.
Weekly sales consistency
A store tracks daily sales of $42, $45, $38, $50, $41, $47, $44 over one full week (the complete data of interest).
Result: Mean = $43.86, population SD = 3.68 — sales vary by about $3.68 from the average day.
Sampling exam scores from a large class
A teacher samples 7 exam scores from a class of 200 students to estimate overall variability.
Result: Using the same 7 values, the sample SD is 3.98 (÷6 instead of ÷7) — slightly higher, correcting for sampling bias.
Comparing two investments' volatility
Investment A has monthly returns with a standard deviation of 2%; Investment B has a standard deviation of 8%.
Result: Investment B is 4x more volatile than Investment A, even if both have the same average return.
⚠️ Common Mistakes & Pro Tips
- Using the wrong denominator: Dividing by n when you have a sample (not the full population) understates true variability — use n−1 for samples.
- Forgetting to square the deviations: Simply averaging (value − mean) always equals zero — squaring is what makes the measure meaningful.
- Confusing variance and standard deviation: Variance is in squared units; standard deviation (the square root of variance) is in the original units and easier to interpret.
- Ignoring outliers: A single extreme value can dramatically inflate standard deviation — check your data for entry errors first.
🔍 People Also Ask
What does a low standard deviation mean?
It means the data points are clustered tightly around the mean, indicating low variability.
Is standard deviation always positive?
Yes — because it is a square root of a sum of squares, standard deviation is always zero or positive, never negative.
How is standard deviation used in finance?
It measures volatility or risk — a higher standard deviation of returns means a more volatile, riskier investment.
❓ Frequently Asked Questions
Why do we square the differences from the mean?
Squaring removes negative signs so differences don't cancel out, and it weights larger deviations more heavily than smaller ones.
When should I use sample standard deviation instead of population?
Use sample standard deviation when your data is a subset of a larger population and you want to estimate that population's variability; use population standard deviation when your data represents the entire group of interest.
What does a standard deviation of zero mean?
It means every value in the data set is identical — there is no variation at all.
How is variance related to standard deviation?
Variance is the standard deviation squared; standard deviation is the square root of variance, expressed in the same units as the original data.
What is considered a "high" standard deviation?
It depends on context relative to the mean; a common rule of thumb is comparing the coefficient of variation (SD ÷ mean) — above 30% is often considered high variability.
Does standard deviation work with negative numbers?
Yes. Standard deviation treats all values by their squared distance from the mean, so negative numbers are handled the same as positive ones.
Can outliers affect standard deviation significantly?
Yes. Because differences are squared, a single extreme outlier can dramatically inflate the standard deviation compared to more typical data points.