Constructing Box Charts

The following notation is used in this section:

process mean (expected value of the population of measurements)

process standard deviation (standard deviation of the population of measurements)

mean of measurements in th subgroup

sample size of th subgroup

the number of subgroups

th measurement in the th subgroup,

th largest measurement in the th subgroup:

     

weighted average of subgroup means

median of the measurements in the th subgroup:

     

average of the subgroup medians:

     

median of the subgroup medians. Denote the th largest median by so that .

     

standard error of the median of independent, normally distributed variables with unit standard deviation (the value of can be calculated with the STDMED function in a DATA step)

th percentile of the distribution of the median of independent observations from a normal population with unit standard deviation

th percentile of the standard normal distribution

th percentile of the distribution of the range of independent observations from a normal population with unit standard deviation

Elements of Box-and-Whisker Plots

A box-and-whisker plot is displayed for the measurements in each subgroup on the box chart. Figure 15.14 illustrates the elements of each plot.

Figure 15.14 Box-and-Whisker Plot
Box-and-Whisker Plot

The skeletal style of the box-and-whisker plot shown in Figure 15.14 is the default. You can specify alternative styles with the BOXSTYLE= option; see Example 15.2 or the entry for BOXSTYLE= in Dictionary of Options: SHEWHART Procedure.

Control Limits and Central Line

You can compute the limits in the following ways:

  • as a specified multiple () of the standard error of (or ) above and below the central line. The default limits are computed with (these are referred to as limits).

  • as probability limits defined in terms of , a specified probability that (or ) exceeds the limits

The CONTROLSTAT= option specifies whether control limits are computed for subgroup means (the default) or subgroup medians. The following tables provide the formulas for the limits:

Table 15.5 Control Limits and Central Line for Box Charts

CONTROLSTAT=MEAN

CONTROLSTAT=MEDIAN

LCLX lower limit =

LCLM lower limit =

Central Line =

Central Line =

UCLX upper limit =

UCLM upper limit =

Table 15.6 Probability Limits and Central Line for Box Charts

CONTROLSTAT=MEAN

CONTROLSTAT=MEDIAN

LCLX lower limit =

LCLM lower limit =

Central Line =

Central Line =

UCLX upper limit =

UCLM upper limit =

In the preceding tables, replace with if you specify MEDCENTRAL=AVGMEAN in addition to CONTROLSTAT=MEDIAN. Likewise, replace with if you specify MEDCENTRAL=MEDMED in addition to CONTROLSTAT=MEDIAN. If standard values and are available for and , replace with and with in Table 15.5 and Table 15.6.

Note that the limits vary with . The formulas for median limits assume that the data are normally distributed.

You can specify parameters for the limits as follows:

  • Specify with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.

  • Specify with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.

  • Specify a constant nominal sample size for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.

  • Specify with the MU0= option or with the variable _MEAN_ in a LIMITS= data set.

  • Specify with the SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.

Note: You can suppress the display of the control limits with the NOLIMITS option. This is useful for creating standard side-by-side box-and-whisker plots.