Variation is present in everything we do. Indeed, no two things are exactly alike. One of the most widely accepted and quoted statistics for quantifying this variability is the standard deviation. However, there are many ways of calculating standard deviation and, therefore, different ways of applying process potential and performance indices. Estimates of standard deviation can be in the form of either short-term or long-term variation. The difference between the two is subtle, but important.

The short-term estimate of variability is often referred to as the “process standard deviation” or (see Table 1). The long-term estimate of variation is referred to as the “sample standard deviation” or s (see Equation 1). Although these two standard deviation statistics each describe variation, their calculations, interpretation and use are vastly different.

Short-Term Standard Deviation ( )

If a control chart’s range, moving range or s chart is in-control, it indicates that the variation from subgroup-to-subgroup is consistent and unchanging. The range, moving range and s statistics are all measures of within subgroup variation; a measure of short-term variability.

With variables data control charts, the standard deviation for individual data is estimated from different charts by using the formulas found in Table 1. Table 1’s formulas are viable techniques for calculation of the short-term standard deviation ( ) and for use with process capability indices Cp, Cpk, Cr, and Cpm (see Table 2). InfinityQSÔ can calculate using any of the formulas found in Table 1.

Control Chart

Standard Deviation

Formula

Moving Range (MR)

Short-term

Range

Short-term

S

Short-term

Range_Within

Short-term

S_Within

Short-term

3D (Total Variation)

Pooled short-term

Table 1 Table of short-term standard deviation formulas

Long-Term Standard Deviation, s

The long-term standard deviation s, (often called “sample standard deviation”), is calculated in part by summing the differences between the individual data points and their data set’s overall average (Equation 1). The inclusion of the overall average, , in the formula is what makes s considered to be a long-term estimate of variability, and it is the critical difference between short-term and long-term standard deviation.

Equation 1 Long-term estimate of variation, s

Long-term standard deviation, s, is used in calculating process performance indices like Pp, Ppk, Ppm, and Pr.

What are the Differences Between s and ?

Take a look at the control chart in Figure 1. The moving range chart is in-control, indicating that short-term variability is unchanging. However, the IX chart shows a distinct trend downward. This downward trend is an example of the average changing over a long period of time. That is, the long-term variability is significantly changing.

Figure 1 IX-MR chart with changes in long-term variability.

If both IX and MR charts remain consistent through time, the short-term and long-term estimates of variation will be approximately the same. If, however, the average changes significantly over time, will not include this variability since is calculated using only a short-term estimate of variability, (either or ). The process standard deviation, ignores and any changes in its value. Therein lies the primary difference between and s.

There also is an issue with the interpretation of the capability ratios. Sometimes Cp and Cpk values can be misleading. Note that the Cp and Cpk values on the right of the chart are both larger than 1.33. This would typically indicate that although the trend exists, all products were produced within specification. However, this is not the case. The Pp ratio (calculated using the long-term estimate of variability, s) is equal to 0.76 while Ppk=0.52. Each of these indices is substantially less than the corresponding Cp and Cpk values, indicating the production of out-of-specification product. Since long-term variability is changing, the Pp and Ppk ratios more accurately represent the performance of the process.

If one is unaware of the underlying variability calculations in their capability ratios, erroneous data could be reported and costly decisions might be made.

Process Capability Indices for Bi-lateral Tolerances

Index

Variability Estimate Used

Formula

Index

Variability Estimate Used

Formula

Cp

Pp

s

Cpk

Ppk

s

Cr

Pr

s

Cpm

Ppm

s

Table 2 Process capability indices using both short-term and long-term standard deviation estimates.

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