Control Charts Introduction: the x-bar R Chart
An x-bar R chart is meant to plot variable data, that is, numerical quantities for which the idea of average makes sense. Other charts, such a the p and c charts, monitor attribute data. Control limits are values established 3 standard deviations above and 3 standard deviations below the average of all values. Traditionally, the average range has been used to estimate 3 standard deviations, because in the era BC (before calculators) the standard deviation was too hard to calculate. Also the range is a more understandable quantity than the standard deviation.
An x-bar R chart consists of
  • The original data, from which the averages and ranges are calculated.
  • Two charts— a chart of sample means ( 's) and a chart of ranges.
  • Supporting information, including the title block or header information and notes on the process.

The chart below chart records glue amounts applied to veneer layers in making plywood. Since the glue is expensive it is important to apply just the amount of glue needed for bonding. The blue labels show the parts common to x-bar R charts. The blank chart is from the American Society for Quality.

Observations on the example control chart

The header information is not completely filled in for this chart, but we see the process is the "glue line", the data are from booth 2 in the glue line, and the specification limits are 27 grams plus or minus 2 grams.

Time and date information has been recorded for all but one sample. Five samples were taken on January 2 during the day shift, three on the swing shift, and one on graveyard shift.

Each sample consists of three measurements. The average and range calculations go in the boxes below each sample. The average of the Jan.3 10:30 AM sample is 27 and the range of the 1:30 PM sample is 2.

The sample means ( 's) and ranges (r's) are then plotted as points in the chart area. They are usually connected with a lines because the data have been taken in sequence over time. The vertical scale for each chart should be chose to allow all data points to fit on the chart. Usual scales go by 1's, 2's, 5's, 10's and so forth-- numbers which are easy to read and plot.

The dashed line in the middle of the upper chart shows the mean of all the sample means () , also called x double bar or the grand average. The grand average is the same as the average of all individual measurements.
Another dashed line shows the average range, , in the lower chart.

The control limits, corresponding to the grand average plus or minus 3 standard deviations, are drawn as solid lines. Formulas, factors and computations for the control limits are written in forms on the back of this chart. You can see some of them showing through is this computer scanned version.

Notes about the process, such as changes in material or adjustments need to be marked on the chart. this chart lacks good supporting notes-- there is one note on the first day, and the shift information. This chart is missing another important item-- the initials or name of the person taking each sample.

Below in yellow is a spreadsheet-type representation of the first 20 samples
Glue booth #2 All shifts
obs.# 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
date/shift 1/2 Day       Sw   R   1/3 Day     Sw     R      
time 10:00 11:15 1:30 3:00 3:45 6:45   11:15 3:30 8:30 10:30 1:30 3:30 5:55 7:40 10:25 12:45 2:30 5:15 4:45
east 31 28 28 26 30 29 28 24 25 26 26 29 26 30 29 28 27 26 28 26
center 33 28 28 29 35 36 36 26 29 29 28 28 27 34 33 31 29 29 29 29
west 29 27 27 27 31 30 28 24 26 27 27 27 26 29 29 27 27 27 27 28
average 31.0 27.7 27.7 27.3 32.0 31.7 30.7 24.7 26.7 27.3 27.0 28.0 26.3 31.0 30.3 28.7 27.7 27.3 28.0 27.7
range 4 1 1 3 5 7 8 2 4 3 2 2 1 5 4 4 2 3 2 3
remark start
up
                                     
Other methods for charting the same information: at the left is an x-bar R chart of different data done using Excel.
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