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| calculators.standard-deviation.xHeader | calculators.standard-deviation.xMinusMean | calculators.standard-deviation.squaredDeviation |
|---|---|---|
| 10 | -8.00 | 64.00 |
| 12 | -6.00 | 36.00 |
| 23 | 5.00 | 25.00 |
| 23 | 5.00 | 25.00 |
| 16 | -2.00 | 4.00 |
| 23 | 5.00 | 25.00 |
| 21 | 3.00 | 9.00 |
| 16 | -2.00 | 4.00 |
| calculators.standard-deviation.sumLabel: | 0 | 192.0000 |
Sample vs Population
calculators.standard-deviation.interpretingStdDev
출처 및 방법론
σ = √[Σ(x - μ)² / N]Population standard deviation
How to Use the Standard Deviation Calculator
The standard deviation calculator computes both population and sample standard deviation, showing every step of the calculation. Visualize your data on a bell curve and understand what standard deviation really means.
What is Standard Deviation?
Standard deviation (σ or s) measures how spread out numbers are from the mean. A low standard deviation means data points are close to the mean; a high standard deviation means they're spread out.
The Formula
Population Standard Deviation (σ):
σ = √[Σ(xᵢ - μ)² / N]
Sample Standard Deviation (s):
s = √[Σ(xᵢ - x̄)² / (n-1)]
Use sample when working with a subset; use population when you have all data.
Step-by-Step Calculation
- Calculate the mean (average)
- Subtract the mean from each data point (deviation)
- Square each deviation
- Find the average of squared deviations (variance)
- Take the square root of the variance
The 68-95-99.7 Rule
For normally distributed data:
- 68% of data falls within 1 standard deviation of the mean
- 95% falls within 2 standard deviations
- 99.7% falls within 3 standard deviations
Variance vs Standard Deviation
Variance is the standard deviation squared. Standard deviation is preferred because it's in the same units as the original data.
Related Calculators
For mean, median, and mode, use our mean median mode calculator. For hypothesis testing, try our p-value calculator.
자주 묻는 질문
Population standard deviation (σ) divides by N (total count). Sample standard deviation (s) divides by n-1 (degrees of freedom). Use population when you have all data points; use sample when working with a subset. The n-1 correction makes sample SD an unbiased estimator.
A high standard deviation means data points are spread far from the mean. For test scores: SD of 5 means most scores are within 5 points of average (consistent); SD of 20 means scores vary widely. Context matters - compare to similar datasets.
Variance = Standard Deviation². Standard Deviation = √Variance. Variance measures spread in squared units, making it hard to interpret directly. Standard deviation is preferred because its in the same units as the original data.
For normal distributions: about 68% of data falls within 1 SD of the mean, 95% within 2 SDs, and 99.7% within 3 SDs. This is also called the empirical rule. Values beyond 3 SDs are rare outliers (less than 0.3% of data).
No, standard deviation is always zero or positive. Its calculated by squaring deviations (making them positive), averaging, then taking the square root (also positive). A standard deviation of zero means all values are identical to the mean.
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