What is Standard Deviation?
Standard deviation is a measure of how spread out numbers are in a dataset. It quantifies the amount of variation or dispersion from the average (mean). A low standard deviation indicates that the data points tend to be close to the mean, while a high standard deviation indicates that the data points are spread out over a wider range.
Key Concepts:
- Sample Standard Deviation (s): Used when working with a sample of the population.
- Population Standard Deviation (σ): Used when working with the entire population.
- Variance: The square of the standard deviation, representing the average squared deviation from the mean.
- Mean Absolute Deviation: The average of the absolute deviations from the mean.
- Coefficient of Variation: The ratio of the standard deviation to the mean, expressed as a percentage.
Formulas Used
Sample Standard Deviation (s):
\[ s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2} \]
Population Standard Deviation (σ):
\[ \sigma = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2} \]
Sample Variance (s²):
\[ s^2 = \frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2 \]
Population Variance (σ²):
\[ \sigma^2 = \frac{1}{n} \sum_{i=1}^{n} (x_i - \mu)^2 \]
Where:
- \( x_i \) = each value in the dataset
- \( \bar{x} \) = sample mean
- \( \mu \) = population mean
- \( n \) = number of data points
- \( \sum \) = summation (add them all up)