Principles of Food Sanitation

whereas the Rchart is used for controlling

process variation. The following examples

show how both of these control charts are

used in a manufacturing environment.

A food manufacturer may monitor the pH

value of finished products to satisfy safety

concerns. Five samples may be pulled every

hour during an 8-hour shift and analyzed for

pH as noted in Table 8–3.

First calculate the average (X) and range

(R) for each inspection sample. For example,

sample calculations for sample 1 are:

X ......

5

=46 44 41 48 45++++= 448

Ris the highest value minus the smallest

value of the five samples.

After all of the sample Xs and Rs are cal-

culated, take the average of the Xs and Rs to

obtainXandR.

X=number of sample lotsSum of allXs ==35 58 8. 4 4475.

R=number of sample lotsSum of all Rs ==^478. 0 5875.

From the calculated, the center line for the

XandRchart can be defined to be:

XChart center line = 4.4475

RChart center line = 0.5875

In order to calculate the upper control

limits (UCL) and lower control limits (LCL),

the standard deviation for each sample lot

must be determined. Rather than perform

the lengthy calculation needed for this value,

another method can be used to determine

these values. The control limits for the previ-

ous charts were represented by:

UCL=+X 3 d

LCL=+X 3 d

By substituting a factor (A 2 ) from a statis-

tical table into the above equation for UCL

and LCL, the needed values for the control

point can be obtained. In this example, the

value for (A 2 ) for a sample size of 5 is 0.58.

The new equation becomes:

UCL=+XRA 2

LCL=-XRA 2

substituting,

UCL = 4.475 +0.58(0.5875) = 4.7883

LCL = 4.4475 −0.58(0.5875) = 4.1067

The control limits for the R chart are

determined similarly, using factors D 4 and

D 3 from the statistical reference table.

DD 43 == 211 ., 0

UCL==D 4 R 2 11 0 5876.(. ) .=1 2396

LCL==D 3 R 0 0 5875(. )= 0

Once these calculations are complete, the

values can be plotted on an X-Ychart to

obtain the XandRcharts for pH measure-

ments. Figures 8–7 and 8–8 illustrate complete

control charts from the sample data. Both

graphs show a process currently under control,

Quality Assurance for Sanitation 135

Table 8–3XandRValues for pH Measurements

Sample pH Measurement X R

1 4.6 4.4 4.1 4.8 4.5 4.48 0.7

2 4.1 4.2 4.3 4.6 4.6 4.36 0.5

3 4.6 4.6 4.3 4.2 4.5 4.44 0.4

4 4.7 4.8 4.5 4.5 4.3 4.56 0.5

5 4.1 4.1 4.0 4.6 4.9 4.32 0.8

6 4.2 4.2 4.6 4.6 4.9 4.50 0.7

7 4.6 4.5 4.6 4.7 4.7 4.62 0.2

8 4.0 3.9 4.8 4.4 4.4 4.30 0.9

Average: 4.4475 0.5875