Moving Average (MA) and Moving Range Charts
Moving Average Charts
A moving average chart is a configuration for evaluating the process level in terms of
an unweighted averaging of the latest w observations in which the current observation
has replaced the oldest of the preceding w observations.
These charts are particularly used when it is not practical to take more than a single
observation per sample or when there is no basis for forming a rational subgroup, such
as in chemical processes. It is, however, desirable to get some advantage of averaging
to detect process level shifts and to damp out some of the effect of random errors on
individual observations.
This chart has an advantage over control charts for individual measurements in that it
reduces the effect of random errors such as measurement error. The major
disadvantage of the moving average chart is that of the lag effect, and successive
moving averages have correlated contributions to the random error. The computation
of appropriate control limits is also not easy, and warrants the use of computers.
Spurious cycling can also occur in a moving average chart due to correlated errors.
Control limits for Moving Average Chart
Let us denote:
w - length or span of the moving average.
t - time period.
t-
the current subgroup average.
 - the standard deviation of the individual observations in the population.
Mt - moving average of span w ( > 0 ) at time t. That is,
.
2
If
t are independent random variables with a common variance  /n the variance of
the moving average is given by
.
As usual,
denotes the central line and the 3 sigma limits are set at
where
= /c4 or /d2. As soon as a sample mean is available, Mt will be
calculated and plotted. When dealing with individual measurements (ie n = 1) or oneat-a-time data, the above formulae can be easily modified and used.
The choice of w is in general decided by the ability of the process to detect the
process level shift. In fact the level of shift and w are inversely related. For detecting
small shifts, a large value of w is used. When compared to the
-chart, the moving
average chart is found to detect small shifts in process levels early.
For the initial periods of time (t < w), the control limits will use t in place of w. The
control limits for the moving average chart will describe in steps till the time and
moving average span become equal.
Example: Consider the Table 1 giving 40 random numbers from N(38,1) which will
be used to explain the construction of a moving average chart:
Table 1 Data for Moving Average Chart
38.4 37.1 38.6 38.5 37.4 37.3 39.0 37.7 39.5 37.4
38.3 37.7 37.4 37.1 36.5 36.3 38.0 39.2 37.0 38.2
36.1 37.6 38.3 39.2 38.7 38.2 36.2 38.8 39.5 39.2
39.8 40.8 38.1 37.8 36.7 38.3 39.0 38.3 36.9 38.8
(Read from left to right)
Consider the data in Table 1 as individual measurements, ie subgroup size n =1. First
we find
= 38.073. Let w, the moving average span, be 3. We then compute:
M1 = 38.4,
M2 = (38.4+37.1)/2 = 37.75,
M3 = (38.4+37.1+38.6)/3 = 38.0333,
M4 = (37.1+38.6+38.5)/3 = 38.0667,
…….
M40 = (38.3+36.9+38.8)/3 = 38.
The true process sigma is estimated using the sample standard deviation S as
=
/c4 = 1.0639 / 0.9936 = 1.0708.
Hence the control limits are:
LCL =
= 38.073 - 3(
) = 36.218 and
UCL =
= 38.073 + 3(
) = 39.928.
M1 and M2 are not based on three observations and hence the control limits in their
case will be different from the control limits of other moving averages. That is the
control limits for M1 are:
LCL =
= 38.073 - 3(
UCL =
= 38.073 + 3(
) = 34.861 and
) = 41.285.
Similarly the control limits for M2 are:
LCL =
= 38.073 - 3(
UCL =
= 38.073 + 3(
) = 37.316 and
) = 38.830.
The Mi values are then plotted in the form of a moving average chart as given in
Figure 1
Figure 1 Moving Average Chart Output
MINITAB will draw a moving average chart with a subgroup size ( 1). Figure 2
provides two moving average charts for the above random data with subgroup sizes
one and two with a moving average span of 3. The historical sigma value is assumed
as one for drawing the charts. Note that the subgroup size one gives a false alarm.
Figure 2 Moving Average Charts
Moving Range Chart
Moving range chart is a configuration for evaluating the variability within a process
using a range in which the current observation has replaced the oldest of the preceding
w observations.
This chart is also used when there is difficulty in forming rational subgroups and as an
accompanying chart for individual measurements (I-chart). The moving range chart
suffers from the disadvantage that successive moving ranges have correlated
contributions to random error, and spurious cycling can occur.
Following the notation used for the moving average chart, a moving range of span w
at time t is given by
The population range is estimated as
, which is the average of all MRt. The
population standard deviation of the original X values is estimated as
/d2 (d2 for
subgroup size w) and hence, as in the case of the R-chart, the control limits for the
moving range chart are set at
LCL = D3
The central line continues to be
and UCL = D4
.
CL =
The moving range MRt is calculated and plotted on the chart. Strictly speaking, the
successive ranges are not independent and hence some authors do not recommend the
use of moving range charts. The usually preferred span for the moving range is two
for which d2 = 1.128, D3 = 0 and D4 = 3.267.
From the data presented for the moving average chart, the moving range chart drawn
using MINITAB is given in Figure 3
Figure 3 Moving Range Chart
MINITAB also provides for the historical sigma to be used as an input value. The
spurious cycling occurring in the moving average charts is clear in the above example.
Estimation of Sigma Using Moving Range
Cryer and Ryan (1990) have shown that the method of estimating the population
sigma as
/d2 is very inefficient when compared to the estimation using the
sample standard deviation as S/c4. They found that the moving range method of
estimation has over 60% more variance than the estimation based on S. Their
recommendations are:
1. For future process monitoring, estimation should be based on S.
2. It is desirable to use two charts concurrently: one based on moving range and
the other based on S. If the estimates of sigma agree well, the resulting charts
will be meaningful. If they do not agree, it may be due to one of the following
reasons:
 Process is out of control - a change in mean level or dispersion.
 Process is in control - but exhibits correlation over time.