Constructing Charts for Medians and Ranges

MRCHART Statement

Constructing Charts for Medians and Ranges

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 the number of subgroups

x_ij j^th measurement in the i^th subgroup, j = 1,2,3, ... , n_i

x_i(j) j^th largest measurement in the i^th subgroup. Then
$x_{i(1)} \leq x_{i(2)} \leq ... \leq x_{i(n_{i})}$

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

M_i median of the measurements in the i^th subgroup:
$M_i = \{ x_{i((n_i + 1)/2)} & {if n_{i} is odd} \ (x_{i(n_i/2)} + x_{i((n_i/2)+1)})/2 & {if n_{i} is even} .$

$\bar{M}$ average of the subgroup medians:
$\bar{M} = (n_1M_1 + ... + n_NM_N)/(n_1+ ... +n_N)$

$\tilde{M}$ median of the subgroup medians. Denote the j^th largest median by M_(j) so that $M_{(1)} \leq M_{(2)}\leq ... \leq M_{(N)}$ .
$\tilde{M} = \{ M_{((N+1)/2)} & {if N is odd} \ (M_{(N/2)} + M_{(N/2)+1})/2 & {if N is even} .$

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

Q_p(n) 100p^th percentile (0<p<1) of the distribution of the median of n independent observations from a normal population with unit standard deviation

d₂(n) expected value of the range of n independent normally distributed variables with unit standard deviation

d₃(n) standard error of the range of n independent observations from a normal population with unit standard deviation

z_p 100p^th percentile of the standard normal distribution

D_p(n) 100p^th percentile of the distibution of the range of n independent observations from a normal population with unit standard deviation

Plotted Points

Each point on a median chart indicates the value of a subgroup median (M_i). For example, if the tenth subgroup contains the values 12, 15, 19, 16, and 14, the value plotted for this subgroup is M₁₀ = 15. Each point on a range chart indicates the value of a subgroup range (R_i). For example, the value plotted for the tenth subgroup is R₁₀=19-12=7.

Central Lines

On a median chart, the value of the central line indicates an estimate for $\mu$ , which is computed as

$\bar{M}$ by default
$\overline{\overline{X}}$ when you specify MEDCENTRAL=AVGMEAN
$\tilde{M}$ when you specify MEDCENTRAL=MEDMED
$\mu_{0}$ when you specify $\mu_{0}$ with the MU0= option

On the range chart, by default, the central line for the i^th subgroup indicates an estimate for the expected value of R_i, which is computed as $d_{2}(n_{i})\hat{\sigma}$ ,where $\hat{\sigma}$ is an estimate of $\sigma$ . If you specify a known value ( $\sigma_{0}$ ) for $\sigma$ , the central line indicates the value of $d_{2}(n_{i})\sigma_{0}$ .The central line on the range chart varies with n_i.

Control Limits

You can compute the limits

as a specified multiple (k) of the standard errors of M_i and R_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 M_i or R_i exceeds its limits

The following table provides the formulas for the limits:

Table 36.22: Limits for Median and Range Charts

Control Limits

Median Chart	LCL = lower limit $= \bar{M} - k\hat{\sigma}e_{M}(n_i)$
	UCL = upper limit $= \bar{M} + k\hat{\sigma}e_{M}(n_i)$
Range Chart	LCL = lower control limit $= {max}(d_{2}(n_{i})\hat{\sigma} - kd_{3}(n_{i})\hat{\sigma},0)$
	UCL = upper control limit = $d_{2}(n_{i})\hat{\sigma} + kd_{3}(n_{i})\hat{\sigma}$
Probability Limits

Median Chart	LCL = lower limit $= \bar{M} - Q_{\alpha/2}(n_i)\hat{\sigma}$
	UCL = upper limit $= \bar{M} + Q_{1-\alpha/2}(n_i)\hat{\sigma}$
Range Chart	LCL = lower limit $= D_{\alpha/2}\hat{\sigma}$
	UCL = upper limit $= D_{1-\alpha/2}\hat{\sigma}$

In Table 36.22, replace $\bar{M}$ with $\overline{\overline{X}}$ if you specify MEDCENTRAL=AVGMEAN, and replace $\bar{M}$ with $\tilde{M}$ if you specify MEDCENTRAL=MEDMED. Replace $\bar{M}$ with $\mu_{0}$ if you specify $\mu_{0}$ with the MU0= option, and replace $\hat{\sigma}$ with $\sigma_{0}$ if you specify $\sigma_{0}$ with the SIGMA0= option.

The formulas assume that the data are normally distributed. Note that the limits for both charts vary with n_i and that the probability limits for R_i are asymmetric around the central line.

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 the LIMITS= data set.
Specify $\sigma_{0}$ with the SIGMA0= option or with the variable _STDDEV_ in the LIMITS= data set.

Chapter Contents
Previous
Next
Top

$\mu$	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	the number of subgroups
x_ij	j^th measurement in the i^th subgroup, j = 1,2,3, ... , n_i
x_i(j)	j^th largest measurement in the i^th subgroup. Then $x_{i(1)} \leq x_{i(2)} \leq ... \leq x_{i(n_{i})}$
$\overline{\overline{X}}$	weighted average of subgroup means
M_i	median of the measurements in the i^th subgroup: $M_i = \{ x_{i((n_i + 1)/2)} & {if n_{i} is odd} \ (x_{i(n_i/2)} + x_{i((n_i/2)+1)})/2 & {if n_{i} is even} .$
$\bar{M}$	average of the subgroup medians: $\bar{M} = (n_1M_1 + ... + n_NM_N)/(n_1+ ... +n_N)$
$\tilde{M}$	median of the subgroup medians. Denote the j^th largest median by M_(j) so that $M_{(1)} \leq M_{(2)}\leq ... \leq M_{(N)}$ . $\tilde{M} = \{ M_{((N+1)/2)} & {if N is odd} \ (M_{(N/2)} + M_{(N/2)+1})/2 & {if N is even} .$
e_M(n)	standard error of the median of n independent, normally distributed variables with unit standard deviation (the value of e_M(n) can be calculated with the STDMED function in a DATA step)
Q_p(n)	100p^th percentile (0<p<1) of the distribution of the median of n independent observations from a normal population with unit standard deviation
d₂(n)	expected value of the range of n independent normally distributed variables with unit standard deviation
d₃(n)	standard error of the range of n independent observations from a normal population with unit standard deviation
z_p	100p^th percentile of the standard normal distribution
D_p(n)	100p^th percentile of the distibution of the range of n independent observations from a normal population with unit standard deviation