Standard Deviation Calculator
Calculate the sample and population standard deviation, variance, mean, sum, and margin of error for a set of numbers. Enter your data points in the table below or paste from clipboard.
Results
| Statistic | Value |
|---|---|
| Count (n) | 0 |
| Sum | 0 |
| Mean (x̄) | 0 |
| Variance (σ²) | 0 |
| Standard Deviation (σ) | 0 |
| Margin of Error (95%) | 0 |
About Standard Deviation
Standard deviation is a measure of how spread out numbers are from their average value. It's one of the most important concepts in statistics and data analysis, used to quantify the amount of variation or dispersion in a set of values.
How Standard Deviation is Calculated
The standard deviation is calculated using the following steps:
- Calculate the mean (average) of the data set
- For each data point, subtract the mean and square the result
- Calculate the average of these squared differences
- Take the square root of this average
The formula differs slightly depending on whether you're working with a sample or the entire population:
Population Standard Deviation (σ):
σ = √(Σ(xᵢ - μ)² / N)
Sample Standard Deviation (s):
s = √(Σ(xᵢ - x̄)² / (n - 1))
Where:
- σ = population standard deviation
- s = sample standard deviation
- N = number of values in the population
- n = number of values in the sample
- xᵢ = each value in the data set
- μ = population mean
- x̄ = sample mean
- Σ = sum of all values
Why Standard Deviation Matters
Standard deviation is widely used in statistics, finance, quality control, and many other fields because it:
- Measures the spread of data points around the mean
- Helps identify outliers in the data
- Is used to calculate confidence intervals and margins of error
- Helps compare different data sets
- Is a key component in many statistical tests
Interpreting Standard Deviation
Understanding what standard deviation tells you about your data:
- Low standard deviation: Data points are close to the mean (less variability)
- High standard deviation: Data points are spread out over a wider range (more variability)
- For normal distributions:
- 68% of values fall within ±1σ of the mean
- 95% of values fall within ±2σ of the mean
- 99.7% of values fall within ±3σ of the mean
Practical Applications
Standard deviation is used in many real-world applications:
- Finance: Measuring investment risk and volatility
- Quality Control: Monitoring manufacturing processes
- Research: Analyzing experimental results
- Weather Forecasting: Predicting temperature variations
- Sports: Evaluating player consistency