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Seven Basic Quality tools documents

Definition of Quality Management -- it is a method for ensuring that all the activities necessary to design, develop and implement a product or service are effective and efficient with respect to the system and its performance. It is also a principle set by the company to endure the continuous advocacy of quality services and products, or the further improvement of it.

Welcome to QT-charts knowledge base section. Hopefully you will find some of them useful in your work.
(Read articles below to learn more.)

Jim Waite, OM 380, 10/14/2004

x bar and R  Charts

orginal text on

            Statistical process control is an effective method for improving a firm’s quality and productivity. There has been an increased interest in their effective implementation in American industry, brought about by increased competition and improvements in quality in foreign-made products.  Many tools may be utilized to gain the desired information on a firm’s quality and productivity.  Some of the more commonly used tools are control charts, which are useful in determining any changes in process performance.  These include a variety of charts such as p charts, c charts and x bar and R charts.  In this paper, I will be focusing on the latter two mentioned.

X bar Charts defined

            An x bar chart is used to monitor the average value, or mean, of a process over time.  For each subgroup, the x bar value is plotted.  The upper and lower control limits define the range of inherent variation in the subgroup means when the process is in control. 

R Chart defined

            An R Chart is a control chart that is used to monitor process variation when the variable of interest is a quantitative measure.  Now, what does all this mean?  These charts will allow us to see any deviations from desired limits within the quality process and, in effect, allow the firm to make necessary adjustments to improve quality.

The Chart Construction Process

            In order to construct x bar and R charts, we must first find our upper- and lower-control limits.  This is done by utilizing the following formulae:

            UCL = μ+   3σ      

                LCL = μ -   3σ



            While theoretically possible, since we do not know either the population process mean or standard deviation, these formulas cannot be used directly and both must be estimated from the process itself.  First, the R chart is constructed.  If the R chart validates that the process variation is in statistical control, the x bar chart is constructed.

Steps in Constructing an R chart

Select k successive subgroups where k is at least 20, in which there are n measurements in each subgroup. Typically n is between 1 and 9. 3, 4, or 5 measurements per subgroup is quite common.

Find the range of each subgroup R(i) where R(i)=biggest value - smallest value for each subgroup i.

Find the centerline for the R chart, denoted by

Find the UCL and LCL with the following formulas: UCL= D(4)RBAR and LCL=D(3)RBAR with D(3) and D(4) can be found in the following table:

Table of D(3) and D(4)

n    D(3)    D(4)       n    D(3)    D(4)

2     0      3.267      6     0      2.004

3     0      2.574      7    .076    1.924

4     0      2.282      8    .136    1.864

5     0      2.114      9    .184    1.816

Plot the subgroup data and determine if the process is in statistical control. If not, determine the reason for the assignable cause, eliminate it, and the subgroup(s) and repeat the previous 3 steps. Do NOT eliminate subgroups with points out of range for which assignable causes cannot be found.

Once the R chart is in a state of statistical control and the centerline RBAR can be considered a reliable estimate of the range, the process standard deviation can be estimated using:

d(2) can be found in the following table:

n  d(2)     n   d(2)

2  1.128    6   2.534

3  1.693    7   2.704

4  2.059    8   2.847

5  2.326    9   2.970


Steps in Constructing the XBAR Chart

Find the mean of each subgroup XBAR(1), XBAR(2), XBAR(3)... XBAR(k) and the grand mean of all subgroups using:

Find the UCL and LCL using the following equations:

A(2) can be found in the following table:

n   A(2)     n   A(2)

2   1.880    6   .483

3   1.023    7   .419

4    .729    8   .373

5    .577    9   .337

Plot the LCL, UCL, centerline, and subgroup means

Interpret the data using the following guidelines to determine if the process is in control:

a.  one point outside the 3 sigma control limits

b.  eight successive points on the same side of the centerline

c.  six successive points that increase or decrease

d.  two out of three points that are on the same side of the centerline,

    both at a distance exceeding 2 sigma’s from the centerline

e.  four out of five points that are on the same side of the centerline,

    four at a distance exceeding 1 sigma from the centerline

f. using an average run length (ARL) for determining process anomalies



The following data consists of 20 sets of three measurements of the diameter of an engine


n       meas#1  meas#2  meas#3          Range   XBAR

1       2.0000  1.9998  2.0002          0.0004  2.0000

2       1.9998  2.0003  2.0002          0.0005  2.0001

3       1.9998  2.0001  2.0005          0.0007  2.0001

4       1.9997  2.0000  2.0004          0.0007  2.0000

5       2.0003  2.0003  2.0002          0.0001  2.0003

6       2.0004  2.0003  2.0000          0.0004  2.0002

7       1.9998  1.9998  1.9998          0.0000  1.9998

8       2.0000  2.0001  2.0001          0.0001  2.0001

9       2.0005  2.0000  1.9999          0.0006  2.0001

10      1.9995  1.9998  2.0001          0.0006  1.9998

11      2.0002  1.9999  2.0001          0.0003  2.0001

12      2.0002  1.9998  2.0005          0.0007  2.0002

13      2.0000  2.0001  1.9998          0.0003  2.0000

14      2.0000  2.0002  2.0004          0.0004  2.0002

15      1.9994  2.0001  1.9996          0.0007  1.9997

16      1.9999  2.0003  1.9993          0.0010  1.9998

17      2.0002  1.9998  2.0004          0.0006  2.0001

18      2.0000  2.0001  2.0001          0.0001  2.0001

19      1.9997  1.9994  1.9998          0.0004  1.9996

20      2.0003  2.0007  1.9999          0.0008  2.0003




RBAR  = 0.0005                                         


UCL=D(4)*RBAR = 2.574 * .0005 = 0.001287               

LCL=D(3)*RBAR = 0.000 * .0005 = 0.000          



XBAR CHART LIMITS:                                                     


XDBLBAR  = 2.0000                                      


UCL  =  XDBLBAR + A(2)*RBAR  =  2.000+1.023*.0005 = 2.0005115

LCL  =  XDBLBAR - A(2)*RBAR  =  2.000-1.023*.0005 = 1.9994885






R - Chart:

XBAR - Chart:














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