The following a brief discussion of statistics found in the Explore routines. Confidence intervals are also available for the Mean, Standard Deviation, and Variance.

Included on this page are:

Measures of Location
Measures of Position
Measures of Variability
Measures of Shape
Distribution Testing Statistics
Natural Tolerance Statistics
Capability and Performance Statistics

The mean is also known as the average, calculated as follows.

50% of the data fall at or above the median and 50% of the data fall at or below the median. The median is found in the (n+1)/2 position in the data, when the data is sorted from low to high.

When data are grouped an interpolated median may be used. This is also useful for low resolution data as implemented in MVPstats, where several values fall at the median. The following formula is used.

Where

- n = sample size,
- Fcum = sum of the frequencies up to, but not including, the median value,
- w = resolution, found by finding the minimum non-zero differences between sorted values, and
- fm = frequency at the median,
- Lm = Lower limit of the the median interval, which is the median minus w/2.

Lowest value in the data set.

X_{low}

Highest value in the data set.

X_{high}

Q1 is the first quartile. It is the value at or below which 25% of the data fall. Q1 is using the (n+1)/4 position in a data set sorted from low to high and interpolating.

Q3 is the third quartile. It is the value at or above which 25% of the data fall. Q3 is found in the 3(n+1)/4 position in a data set sorted from low to high and interpolating.

The Pth percentile is the value that P% of the values fall at or below, and (100-P%) fall above it.

Percentiles are found by sorting the data from low to high, and interpolating when necessary. The Pth percentile may be found in the P(n+1)/100th position, where P is expressed as a percent. If the percentile position is not an integer the percentile falls between two values. In this case, the fraction generated in the position calculation yields the fraction of the difference between the two values. If P is less than 100/(n+1) or greater than 100 n /(n+1), P is undefined.

The Highest minus the lowest value.

R = Xhigh - Xlow

The standard deviation is a measures of variability. It is the square-root of the average squared distance each value falls from the mean.

The variance is simply the standard deviation squared.

The coefficient of variation is calculated as follows.

The is another estimate of the standard deviation, using the average Moving Range (the average of the moving absolute difference of sequential values).

where

- is the average moving range, and
- is a constant equal to 1.128 (using moving ranges between two points).

This is another method to estimate the standard deviation, similar to Std(MR), except the median of the moving ranges is used.

where

- is the median moving range, and
- is a constant equal to 0.954 (using moving ranges between two points).

This is the range of the middle 50% of the data.

IR = Q3 - Q1

This is another method to estimate standard using the Interquartile Range divided by a constant, assuming a normal distribution.

PSD = (Q3 - Q1) / 1.3489795

This is simply 1/2 the Interquartile range.

SIR = IR / 2

This measure is related to the standard deviation, except it measures absolute deviations around the median. The MD(Med) is calculated as follows.

This measure is related to the standard deviation, except it measures absolute deviations around the mean or average. The MD(Avg) is calculated as follows.

Where

A-D - Anderson-Darling Test for Normality Test Statistic

L-M - Lin-Mudholkar Test for Normality Test Statistic

W(E) - Shapiro-Wilk Test for Exponentiality Test Statistic

Poisson Test Statistic - See Poisson Distribution Test.

These are the values corresponding to given standard deviations from the mean, as follows.

A*s* = Mean + A*s*

where “A” is a positive or negative number

These may be used for natural tolerances, assuming normality.

NT = As

This is the count of observations below 3 standard deviations, or above 3 standard deviations.

These are the Entered Specifications.

These are the count or percent of observations below, above, or total outside of specifications.

These are estimated proportions outside of specifications, assuming normality.

These are capability measures, estimated using sample statistics.

These are performance measures, which are calculated the same as capability measures using sample statistics.

See Ppm, Ppk, Pp, %Off-Target