Standard Deviation

Standard deviation is a measure of statistical dispersion that quantifies the variability or spread of a set of data around its mean. It is calculated by the square root of the variance.

Formula:Standard deviation (σ) = √[Variance (Var) = Σ(x - μ)² / n - 1]

where:* σ is the standard deviation* Var is the variance* x is each element in the data set* μ is the mean of the data set* n is the number of elements in the data set

Interpretation:

• Standard deviation measures the degree of variation in a data set.
• A high standard deviation indicates a wide range of values, while a low standard deviation indicates a narrow range of values.
• The standard deviation is an important measure of data dispersion, alongside the range, variance, and coefficient of variation.

Significance:

• Standard deviation is used to:
  • Describe the variability of a data set.
  • Compare the variability of different data sets.
  • Calculate confidence intervals and hypothesis tests.
  • Evaluate the performance of statistical models.

Example:

“`Data: [10, 12, 14, 16, 18]

Mean (μ) = (10 + 12 + 14 + 16 + 18) / 5 = 16

Variance (Var) = [(10 – 16)² + (12 – 16)² + (14 – 16)² + (16 – 16)² + (18 – 16)] / (5 – 1) = 8

Standard Deviation (σ) = √8 = 2.828“`

Therefore, the standard deviation of the data set is 2.828.