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
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
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
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
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.
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
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
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
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