Measures of VariationMeasures of Variation

A set of data exhibits variationvariation if all of the data are not the same value.

RangeRange

The rangerange 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 ValueR = Maximum Value - Minimum Value

Interquartile RangeInterquartile Range

The interquartile rangeinterquartile 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 Variance & Standard DeviationDeviation

The population variancevariance is the average of the

squared distances of the data values from the

mean.The standard deviationstandard deviation is the positive square root of

the variance.

Population VariancePopulation Variance

where: = population mean

N = population size

2 = population variance (sigma squared)

N

xN

ii

1

2

2

)(

Sample VarianceSample Variance

where: = sample mean

n = sample size

s2 = sample variance

1

)(s 1

2

2

n

xxn

ii

x

Sample Standard Sample Standard DeviationDeviation

where: = sample mean

n = sample size

s = sample standard deviation

1

)(ss 1

2

2

n

xxn

ii

x

The Empirical RuleThe Empirical Rule

If the data distribution is bell-If the data distribution is bell-shaped, then the interval: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

1

2

3

The Empirical RuleThe Empirical Rule(Figure 3-11)(Figure 3-11)

Xx

1x

68%

2x

95%

Tchebysheff’s TheoremTchebysheff’s Theorem

Regardless of how the data are distributed, at least (1 - 1/k2) of the values will fall within k = 1 standard deviations of the mean. For example: At least (1 - 1/12) = 0% of the values will fall within k=1 standard deviation of the mean At least (1 - 1/22) = 3/4 = 75% of the values will fall within k=1 standard deviation of the mean At least (1 - 1/32) = 8/9 = 89% of the values will fall within k=1 standard deviation of the mean

6 Sigma Quality6 Sigma Quality

Specification for a quality Specification for a quality characteristic is six standard characteristic is six standard deviation away from the mean of deviation away from the mean of the process distribution.the process distribution.

Translates into process output Translates into process output that does not meet specifications that does not meet specifications two out of one billion times.two out of one billion times.

Sigma Quality LevelsSigma Quality Levels

Sigma ()Quality Level

123456 0.002

45,4002700

630.57

Defects per MillionOpportunities for Defects

317,400

Sigma Quality Level Sigma Quality Level ConceptsConcepts

Sigma () Equated to Quality Level Relative Area

1 Floor space of a typical factory2 Floor space of a typical supermarket3 Floor space of a small hardware store4 Floor space of a typical living room5 Area under a typical desk telephone6 Top surface of a typical diamond7 Point of a sewing needle

Standardized Data ValuesStandardized Data Values

A standardized data valuestandardized 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 ValuesStandardized Data Values

STANDARDIZED SAMPLE DATASTANDARDIZED SAMPLE DATA

where: x = original data value

= sample mean

s = sample standard deviation z = standard score

s

xx z

x