Mean and Standard Deviation

A random variable is defined by a distribution that has one or more variables that describe location, shape and scaling. (The term distribution is used in six sigma to denote a probability density function). Practically, a distribution can be described by:

• mean 
• variance or standard deviation 
• skewness 
• kurtosis 

Once mean, standard deviation, skewness, and kurtosis are calculated or assumed, the relevant location, shape, and scaling variables can be computed.

Mean
The expected (or average) value for the distribution of a random variable x_bar is the mean and it can be calculated from sample data as follows:

where x_i are the value for n data points. Mean is also called the first moment of a distribution about zero (this is the same things as a centroid – the distribution is rotated around zero).

Means (red lines) for different distributions.

Variance and Standard Deviation
The measurement of spread of a random variable is called variance σ^2 and the square root of variance is called standard deviation σ. This is equivalent to taking the second moment of a distribution around it’s mean.

Distributions with increasing standard deviation (a) to (c). Red lines are means