Standard Deviation Calculator
Calculate Mean, Variance, SD & visualize the Bell Curve.
Use Sample if your data is a part of a larger group.
Accepts commas, spaces, or new lines.
Data Count (N): 0
Sorted: -Sample Standard Deviation (s)
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Central Tendency
Mean (Average)
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Median (Middle)
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Mode (Frequent)
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Dispersion & Range
Variance
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Range (Max - Min)
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Sum (Σ)
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Sum of Squares (SS)
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Normal Distribution (Bell Curve)
Step-by-Step Calculation
| # | Value ($x$) | Difference ($x - \bar{x}$) | Squared Diff $(x - \bar{x})^2$ |
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What is Standard Deviation?
[Image of normal distribution bell curve]Standard Deviation (SD) is a measure of how spread out numbers are in a dataset. It tells you how far each number is from the mean (average).
- Low SD: Data points are clustered close to the mean (consistent).
- High SD: Data points are spread out over a wide range (volatile).
Sample vs. Population SD
Choosing the right mode is crucial for accuracy:
- Population SD ($\sigma$): Use this when you have data for the *entire* group you are studying (e.g., the heights of *every single student* in a class). The formula divides by N.
- Sample SD ($s$): Use this when you only have a *subset* of the group (e.g., a survey of 10 people representing a city). The formula divides by N - 1 (Bessel's Correction).
The Formulas
1. Calculate Mean ($\bar{x}$)
$\bar{x} = \frac{\sum x}{N}$
2. Calculate Variance
Sample Variance ($s^2$): $\frac{\sum (x - \bar{x})^2}{N - 1}$
Population Variance ($\sigma^2$): $\frac{\sum (x - \bar{x})^2}{N}$
3. Calculate Standard Deviation
It is simply the square root of the Variance.
SD = $\sqrt{Variance}$