5 8 11 14 or one value per line.What this calculator does
This mean standard deviation and variance calculator helps you quickly summarize a dataset. Enter a list of numbers, click Calculate, and the tool returns:
- Count (how many values)
- Sum
- Mean (average)
- Population variance and population standard deviation
- Sample variance and sample standard deviation
- Minimum, maximum, and range
Key formulas (simple reference)
Mean
The mean is the average:
mean = (x1 + x2 + ... + xn) / n
Population variance and standard deviation
Use these when your values represent the entire population:
population variance (σ²) = Σ(x - μ)² / n
population standard deviation (σ) = √σ²
Sample variance and standard deviation
Use these when your values are a sample from a larger population:
sample variance (s²) = Σ(x - x̄)² / (n - 1)
sample standard deviation (s) = √s²
When to use sample vs population values
A common source of confusion is choosing the right formula. A practical rule:
- Population: You measured every value in the group you care about. Example: test scores of all 30 students in one class.
- Sample: You measured only part of a larger group. Example: survey responses from 200 customers out of all customers.
Example walkthrough
Suppose your data is: 10, 12, 12, 14, 16.
- Count = 5
- Sum = 64
- Mean = 64 / 5 = 12.8
- Deviations from mean: -2.8, -0.8, -0.8, 1.2, 3.2
- Squared deviations sum = 20.8
- Population variance = 20.8 / 5 = 4.16
- Population standard deviation = √4.16 ≈ 2.0396
- Sample variance = 20.8 / 4 = 5.2
- Sample standard deviation = √5.2 ≈ 2.2804
Why standard deviation and variance matter
The mean tells you the center of your data. Variance and standard deviation tell you how spread out values are around that center. Two datasets can have the same mean but very different variability.
In finance, quality control, education, health research, and machine learning, understanding spread is often as important as understanding average performance.
Data entry tips
- Use decimal numbers if needed (for example,
2.5,7.75). - Negative values are allowed (for example,
-3). - Do not include words or units in the input box.
- If your list has only one value, sample variance/standard deviation are undefined.
Quick FAQ
Is variance always non-negative?
Yes. Because it is based on squared differences, variance can never be negative.
Why is standard deviation easier to interpret than variance?
Standard deviation is in the same units as your original data, while variance is in squared units.
Can I use this for large datasets?
Yes. You can paste long numeric lists directly into the input area.