Constructing Charts for Means

XCHART Statement

Constructing Charts for Means

The following notation is used in this section:

$\mu$ process mean (expected value of the population of measurements)

$\sigma$ process standard deviation (standard deviation of the population of measurements)

$\bar{X}_{i}$ mean of measurements in i^th subgroup

R_i range of measurements in i^th subgroup

n_i sample size of i^th subgroup

N number of subgroups

$\overline{\overline{X}}$ weighted average of subgroup means

z_p 100p^th percentile of the standard normal distribution

Plotted Points

Each point on an $\bar{X}$ chart indicates the value of a subgroup mean ( $\bar{X}_{i}$ ). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is

$\bar{X}_{10}=\frac{12 + 15 + 19 + 16 + 14}5 = 15.2$

Central Line

By default, the central line on an $\bar{X}$ chart indicates an estimate for $\mu$ , which is computed as

$\hat{\mu}=\overline{\overline{X}} = \frac{n_{1}\bar{X_{1}} + ... + n_{N}\bar{X_{N}}} {n_{1} + ... + n_{N}}$

If you specify a known value ( $\mu_{0}$ ) for $\mu$ ,the central line indicates the value of $\mu_{0}$ .

Control Limits

You can compute the limits in the following ways:

as a specified multiple (k) of the standard error of $\bar{X}_{i}$ above and below the central line. The default limits are computed with k=3 (these are referred to as $3\sigma$ limits).
as probability limits defined in terms of $\alpha$ , a specified probability that $\bar{X}_{i}$ exceeds the limits

The following table provides the formulas for the limits:

Table 42.22: Limits for $\bar{X}$ Charts

Control Limits

LCL = lower limit $= \overline{\overline{X}} - k\hat{\sigma}/ \sqrt{n_{i}}$

UCL = upper limit $= \overline{\overline{X}} + k\hat{\sigma}/ \sqrt{n_{i}}$

Probability Limits

LCL = lower limit $= \overline{\overline{X}} - z_{\alpha/2}(\hat{\sigma}/ \sqrt{n_{i}})$

UCL = upper limit $= \overline{\overline{X}} + z_{\alpha/2}(\hat{\sigma}/ \sqrt{n_{i}})$

Note that the limits vary with n_i. If standard values $\mu_{0}$ and $\sigma_{0}$ are available for $\mu$ and $\sigma$ , respectively, replace $\overline{\overline{X}}$ with $\mu_{0}$ and $\hat{\sigma}$ with $\sigma_{0}$ in Table 42.22.

You can specify parameters for the limits as follows:

Specify k with the SIGMAS= option or with the variable _SIGMAS_ in a LIMITS= data set.
Specify $\alpha$ with the ALPHA= option or with the variable _ALPHA_ in a LIMITS= data set.
Specify a constant nominal sample size $n_{i} \equiv n$ for the control limits with the LIMITN= option or with the variable _LIMITN_ in a LIMITS= data set.
Specify $\mu_{0}$ with the MU0= option or with the variable _MEAN_ in a LIMITS= data set.
Specify $\sigma_{0}$ with the SIGMA0= option or with the variable _STDDEV_ in a LIMITS= data set.

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$\mu$	process mean (expected value of the population of measurements)
$\sigma$	process standard deviation (standard deviation of the population of measurements)
$\bar{X}_{i}$	mean of measurements in i^th subgroup
R_i	range of measurements in i^th subgroup
n_i	sample size of i^th subgroup
N	number of subgroups
$\overline{\overline{X}}$	weighted average of subgroup means
z_p	100p^th percentile of the standard normal distribution