Definition of Standard Deviation
Standard deviation is a statistical measure that quantifies the amount of variation or dispersion in a set of values. It describes how spread out the values are from the mean (average) of the dataset. A low standard deviation indicates that values cluster closely around the mean, while a high standard deviation shows greater spread.
Key Principles and Calculation
Standard deviation is calculated as the square root of the variance. Variance is the average of the squared differences from the mean. For a population, the formula is σ = √[Σ(x - μ)² / N], where μ is the population mean and N is the number of observations. For a sample, it uses n-1 in the denominator to provide an unbiased estimate. This process ensures the measure accounts for both positive and negative deviations equally.
Practical Example
Consider test scores: 80, 85, 90, 95, 100. The mean is 90. Differences from the mean are -10, -5, 0, 5, 10. Squared differences: 100, 25, 0, 25, 100. Average squared difference (variance) is 50. Standard deviation is √50 ≈ 7.07, showing moderate spread. If scores were 85, 90, 90, 90, 95, the standard deviation would be ≈ 3.54, indicating tighter clustering.
Importance and Applications
Standard deviation is crucial for assessing data reliability and making inferences. In finance, it measures investment risk (volatility). In quality control, it evaluates process consistency. In research, it helps determine if results are statistically significant, such as in hypothesis testing where it informs confidence intervals.