Statistics Definitions and Reference

Descriptive statistics

Average: The mean value of the samples in the sample set
Count: The number of samples in the sample set
Maximum and Minimum: The largest and smallest values in the sample set
Standard Deviation: The sample standard deviation (equivalent to STDEV.S in Excel)

Distribution

Normal Distribution p-value: The probability value in a test for Normal distribution. A p-value smaller than 0.05 indicates that probability distribution of the sample set is most likely not Normal.

Quartiles

Quartile 1: Twenty-five percent of the samples in the sample set have values smaller than or equal to Quartile 1 (Q1).

Median: Fifty percent of the samples in the sample set have values smaller than or equal to the median.

Quartile 3: Seventy-five percent of the samples in the sample set have values smaller than or equal to Quartile 3 (Q3). Fifty percent of the samples have values between Q1 and Q3.

Capability and performance

The process performance metrics show how well an operating process consistently produces output that meets its specification limits.

Comparison tests

Comparison tests are displayed if a baseline Statistics Summary has been selected for comparison

Equal Mean p-value: The p-value is the probability that the means (the averages) of two Normal probability distributions are the same. A p-value smaller than 0.05 indicates that the means are most likely not the same.

Equal Variance p-value: The p-value is the probability that the variances of two Normal probability distributions are the same. A p-value smaller than 0.05 indicates that the variances are most likely not the same.

Whiskers and outliers

Whiskers: The lower whisker W_L on the Box Plot is located at the larger of x_min (the minimum sample value), or at the zero quartile Q₀.

The upper whisker W_U is located at the smaller of x_max (the maximum sample value) or at the fourth quartile Q₄. Figure 2 shows how these quantities are calculated.

Outliers: If any sample values lie below Q₀ or above Q₄ they are displayed on the box plot as outliers.

Process performance indexes

Figure 3 shows how the performance indexes are calculated. U and L represent the upper and lower specification limits, μ is the sample average from the Statistics Summary and σ is the sample standard deviation from the Statistics Summary.

P_p: The overall capability of a process comparing the full six-sigma spread of the data to the range between the upper and lower specification limits.
P_pl and P_pu: The performance relative to the lower and upper specification limits individually.
P_pk: The actual performance when the mean is not perfectly centered.

A low value for any of the four indexes indicates too many samples in the data set are lying beyond one or both of the upper and lower specification limits. Rules of thumb for P_pk are:

• P_pk >= 1.33 is acceptable,

• P_pk >=1.67 is good,

• P_pk >= 2.0 is world-class.

Process capability index C_pk

Figure 4 shows how the process capability index is calculated. C_pk uses estimates of within-subgroup (short term) variation rather than overall process deviation.

The signal is segmented into groups of n samples. The process average is calculated from the subgroup means, and the process variation is estimated from subgroup ranges for group sizes up to 10, or from subgroup standard deviations for groups of ten or more.

Capability indices should be interpreted only when the process is stable and in statistical control. Estimates of process variation may also become biased and artificially small when autocorrelation is present in the signal.

Normality test: The Normality test uses the Shapiro-Wilk method with Royston approximation. The test is more reliable when the sample set has fewer than 5000 samples.

Comparison test for equal means: The test is the Student t-test. The test is most reliable when the two distributions being compared both have a Normal distribution.

Comparison test for equal valiance: The test is the F-test. The test is most reliable when the two distributions being compared both have a Normal distribution.