Standard Deviation

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Conventional Approach


Purpose

The standard deviation is used to provide a measurement yardstick that describes the variation in a set of data. It is the most important measure of variation.


Anatomy

Standard Deviation Formula Explained

Standard Deviation Formula Explained


Reference: Juran's Quality Control Handbook

Terminology

A. (sigma hat): Represents the estimator of the population standard deviation. Also, "s" is used to denote sample standard deviation.

B. Represents the sample size minus 1 (n-1). It represents the number of degrees of freedom associated with the statistic.

C. (x bar) is used to denote sample mean.

D. X1 Represents the ith value of X.

E. The summation symbol, from i = 1 to n.


Major Considerations

Standard deviation is the square root of the variance. The variance is difficult to interpret because the unit of measure is squared. When we take the square root of the variance, we obtain a measure of dispersion (standard deviation) which is in the same unit of measure as the sample data.

Application Cookbook

To calculate the standard deviation:

1. Using Excel to calculate the standard deviation of a data set, use the statistical function STDEV.

2. Another tool is the use of Minitab, using STATS>BASIC STATISTICS>DESCRIPTIVE STATISTICS

Short-Term Standard Deviation


Purpose

Short-term standard deviation estimates the standard deviation within a data set subgroup, and is a measure of how widely values are dispersed from the average value (the mean) due only to the effect of common-cause variation. It is used to calculate Short-term Process Capability (i.e.: Cp and Cpk).

Anatomy


Short Term Standard Deviation

Short-Term Standard Deviation


Reference: Juran's Quality Control handbook

Terminology

A. Calculated standard deviation Short-Term.

B. Standard deviation (hat means: Estimate of).

C. ith data point in the jth row (subgroup).

D. Arithmetic average of jth row.

E. Sum of squares for the jth row.

F. Sum of squares for all rows.

G. Number of rows (g).

H. Number of data in a row (n).

I. Degrees of freedom: (i.e. g(n-1)).


Major Considerations

  • Major concept in quality management and in statistics.
  • Metric used to evaluate short-term process variability (smallest possible for a given process)
  • Used by some customers to rate its suppliers and competitors
  • Standard Deviation Short-Term is also called Pooled Standard Deviation

Application Cookbook

1. Compute the variance for each row.

2. Average row variances.

3. Square root the average variance.

4. Short-term standard deviation can be computed with Excel or Minitab.

Long-Term Standard Deviation


Purpose

The standard deviation long-term estimates the total standard deviation of the process, and is a measure of how widely values are dispersed from the overall average value (the grand mean) due to white noise and black noise.

It's used to calculate process long-term capability (i.e.: Pp and Ppk) and PPM defective

Anatomy


Long Term Standard Deviation

Long-Term Standard Deviation


Reference: Mikel J. Harry, The vision of six sigma

Terminology

A. Calculated Standard Deviation Long-Term.

B. Standard Deviation (hat means: Estimate of).

C. ith data point in the jth row (subgroup).

D. Overall arithmetic average.

E. Sum of squares for the jth row.

F. Sum of squares for all rows.

G. Number of rows (g).

H. Number of data in a row (n).

I. Degree of freedom: (i.e. (gn-1)).


Major Considerations

  • Major concept in quality management and in statistics.
  • Metric used to evaluate long-term process variability.
  • Used by customers to rate suppliers and competitors.

Application Cookbook

1. Compute the overall mean of all data.

2. Calculate the differences between each individual data and the overall mean.

3. Square each difference.

4. Sum all squares.

5. Compute the degrees of freedom, which is the product of number of rows (n) and the number of columns (g), minus 1 (i.e. ng-1).

6. Divide the sum of squares by the degrees of freedom.

7. Square root the ratio.

Alternative: Compute directly the STD DEV of all data at once.
The standard deviation long-term may be computed with Excel or Minitab

From Standard Deviation to Six Sigma Tools.

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