Measures of spread (range, variance, standard deviation)

Range

Range is the difference between the largest and smallest values in a data set.

• The Min is the smallest value in a data set

• The Max is the largest value in a data set

• The Range is the difference between the Max and the Min

  • Ex: Here are ten ages in ascending order: 32, 33, 34, 34, 35, 37, 37, 39, 41, 44

  • Min = 32 , Max = 44

  • Range = Max − Min = 44 − 32 = 12

Variance

Variance tells you how far off the data values are from the mean overall.

1. Calculate the mean of all the data points, Xbar

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   Calculate the difference between each data point and the average (X _i —Xbar)

3. Square those figures for all data points

   • This ensures that you'll always be dealing with a positive number— otherwise , all of the values would cancel each other out and sum to zero

4. Add the squared values together (a value called the sum of squares in statistics)

5. Divide that total by n-1 (the number of data values minus 1)

Note that the equation above follows statistical conventions (p. 105) for describing sample statistics. Variance for a population uses a sigma as shown here.

Though more people are familiar with standard deviation ( see below), variance has one big advantage: it is additive while standard deviations are not. That means, for example, that the total variance for a process can be determined by adding together the variances for all the process steps.

• So to calculate a standard deviation for an entire process, first calculate the variances for each process step, add those variances together, then take the square root. Do not add together the standard deviations of each step.

A drawback to using variance is that it is not in the same units of measure as the data points. Ex: for cycle times, the variance would be in units of "minutes squared," which doesn't make logical sense.

Standard deviation

Think of standard deviation as the "average distance from each data point to the mean." Calculate the standard deviation for a sample or population by doing the same steps as for the variance, then simply taking the square root. Here's how the equation would look for the 10 ages listed on the previous page:

Just as with variance, the standard deviation of a population is denoted with sigma instead of "s", as shown here: