Measures of Variation variation A set of data exhibits variation if all of the data are not the same value.

Range range The range is a measure of variation that is computed by finding the difference between the maximum and minimum values in the data set. R = Maximum Value - Minimum Value

Interquartile Range interquartile range The interquartile range is a measure of variation that is determined by computing the difference between the first and third quartiles. Interquartile Range = Third Quartile - First Quartile

Variance & Standard Deviation variance The population variance is the average of the squared distances of the data values from the mean. standard deviation The standard deviation is the positive square root of the variance.

Population Variance where: = population mean N = population size  2 = population variance (sigma squared)

Sample Variance where: = sample mean n = sample size s 2 = sample variance

Sample Standard Deviation where: = sample mean n = sample size s = sample standard deviation

The Empirical Rule If the data distribution is bell-shaped, then the interval: contains approximately 68% of the values in the population or the sample contains approximately 95% of the values in the population or the sample contains approximately 99.7% of the values in the population or the sample

The Empirical Rule (Figure 3-11) X 68% 95%

Tchebysheff’s Theorem Regardless of how the data are distributed, at least (1 - 1/k 2 ) of the values will fall within k = 1 standard deviations of the mean. For example: ä At least (1 - 1/1 2 ) = 0% of the values will fall within k=1 standard deviation of the mean ä At least (1 - 1/2 2 ) = 3/4 = 75% of the values will fall within k=1 standard deviation of the mean ä At least (1 - 1/3 2 ) = 8/9 = 89% of the values will fall within k=1 standard deviation of the mean

6 Sigma Quality 4Specification for a quality characteristic is six standard deviation away from the mean of the process distribution. 4Translates into process output that does not meet specifications two out of one billion times.

Sigma Quality Levels Sigma (  ) Quality Level 1  2  3  4  5  6  , Defects per Million Opportunities for Defects 317,400

Sigma Quality Level Concepts Sigma (  )Equated to Quality LevelRelative Area 1  Floor space of a typical factory 2  Floor space of a typical supermarket 3  Floor space of a small hardware store 4  Floor space of a typical living room 5  Area under a typical desk telephone 6  Top surface of a typical diamond 7  Point of a sewing needle

Standardized Data Values standardized data value A standardized data value refers to the number of standard deviations a value is from the mean. The standardized data values are sometimes referred to as z-scores.

Standardized Data Values STANDARDIZED SAMPLE DATA where: x = original data value = sample mean s = sample standard deviation z = standard score