# Notation for Sample Variance and Sample Standard Deviation

Notation for Sample Variance and Sample Standard Deviation

Sample Variance

• The sample variance, denoted by ‘s²’, is a measure of dispersion in a data set.
• It quantifies the spread of data points from the mean.
• To calculate the sample variance, firstly, the mean (average) of the sample must be found. This is denoted by ‘x̄’.
• Subsequently, subtract each data value in the sample from the mean and square the result, forming a squared deviation.
• The average of these squared deviations provides the sample variance.

Sample Standard Deviation

• The sample standard deviation, denoted by ‘s’, is the square root of the sample variance, and also measures the spread of data.
• Standard deviation has the advantage of being in the same units as the data, unlike variance which is in squared units, making it easier to interpret in relation to the data.
• The smallest possible value of the standard deviation is zero, which occurs only when all data points are the same value (i.e., there is no spread).
• Larger values indicate greater dispersion in the sample data.

Notation Summary

• ‘n’ is used to represent the number of items in the sample.
• ‘Σ’ means the sum of the following terms.
• ‘xi’ denotes each individual value in the sample.
• ‘x̄’ is the sample mean.
• ‘s²’ is the sample variance.
• ’s’ signifies the standard deviation.
• So the formula for the sample variance is ‘s² = Σ (xi - x̄)² / (n-1)’ and for standard deviation it is ‘s = √s²’.

Interpreting the Variance and Standard Deviation

• A large sample variance, and hence a large standard deviation, indicates that individual sample values deviate significantly from the mean on average, suggesting a wide range of data.
• A small variance and standard deviation indicate that the data points in the sample are close to the mean, showing a tightly grouped set of data.
• Variance and standard deviation are most meaningful when studying data drawn from a population that follows a normal distribution.

Properties

• The sample variance and standard deviation are always positive or zero.
• They are vulnerable to the influence of outliers. An extreme data value can increase the average squared deviation, which consequently would increase both the variance and standard deviation.
• They provide more information about a dataset than just the range would; they consider how every data point behaves in relation to the mean, not just the extremes.