Standard Deviation

Standard Deviation:

  • is a measure of the amount of dispersion in a set of values

  • measures the distance of values from the mean of the set of values

  • is the square root of its variance

  • is often represented by the lower case greek symbol for sigma

  • is approximated by the 68-95-99.7 Rule

Normal Distribution Example

The graph below illustrates Standard Deviation applied to a Normal Distribution:

68-95-99.7 Rule

The 68-95.99.7 Rule (aka: 68-95-100 Rule) is a shorthand for the percentage of values that lie within Standard Deviation bands around the Mean in a Normal Distribution.

  • 1 band: approximately 68%

  • 2 bands: approximately 95%

  • 3 bands: approximately 99.7% or rounded to 100%

Standard Score (Z-score)

The Standard Score (aka: z-score) is the number of standard deviations by which the value of a measured data point is above or below the mean value, for example:

  • 3 sigma = 3.0 standard score

  • 2 sigma = 2.0 standard score

  • 1 sigma = 1.0 standard score

  • 0 sigma = 0 standard score

  • -1 sigma = -1.0 standard score

  • -2 sigma = -2.0 standard score

  • -3 sigma = -3.0 standard score

Mean and Variance

Mean and Variance values change the shape of the probability distribution: