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SOP 20

Standard Operating Procedure

for

Standard Deviation and Range Charts 

1 Introduction 

Purpose 

This procedure describes a process to be followed to monitor the statistical control 

of a measurement process using standard deviation charts or range charts for any 

calibration method where replicate measurements are performed and where it is not 

practical or feasible to maintain laboratory check standards. Standard deviation 

charts are preferred to the use of range charts for monitoring and evaluating process 

standard deviations when replicate measurements are made.

Prerequisites 

The procedure must match the calibration procedure that is being monitored. 

2 Summary 

Because of the size and cost of some laboratory standards, it is not always practical to have 

check standards remain in the laboratory for the purposes of measurement control. 

However, it is practical to maintain standard deviation (sp) charts or range (R) charts for 

each nominal size standard that is submitted to evaluate the standard deviation of replicate 

runs. Directions for preparing and using an R control chart that monitors the precision of 

the test procedure are given. It is assumed that standards of the same nominal capacity and 

design will have similar characteristics with respect to the repeatability of tests. Since it is 

not practical to run a sufficient number of tests on each unknown standard to determine the 

repeatability, the absolute difference between replicate tests are graphed on the same sp or 

R chart to reflect the repeatability of measurement of the unknown standards tested in the 

laboratory. Note: Provers of similar readability may be grouped together on the same chart 

as long as results from different sized provers pass an F-test (e.g., provers greater than and 

equal to 500 gal generally have 20 in3 graduations and equivalent repeatability).

3 Equipment 

All equipment is designated in the respective calibration SOP. 

4 Procedure 

Define: Determine appropriate measurement assurance

The SOP used for calibration should contains instructions to define the type of 

measurement assurance and check standards to be measured and analyzed to ensure 

the process and standards are in control prior to reporting measurement values from 

calibrations.
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Measure: Data Collection 

Conduct the required number of replicates per test on the unknown test item 

per the designated SOP. A minimum of 12 sets of data must be available 

before a reasonably adequate data base is established. Note: 25 to 30 

replicate sets of data are needed to determine valid uncertainties.

Tabulate the measured errors as determined by each of the two trials using 

a form such as the one contained in the Appendix. It is recommended that 

the data be maintained in a spreadsheet or other electronic program in lieu 

of a paper form. (If the unknown item is adjusted after the first run, the first 

trial reading after the adjustment is used in the evaluation).

Analyze: Calculate the standard deviation (recommended)

A standard deviation of the replicate runs, sp, may be calculated for each set 

of runs according to the appropriate SOP. An accepted standard deviation 

over time is calculated as the pooled standard deviation to represent the 

accepted standard deviation of the measurement process. This pooled 

standard deviation may be used to perform F-tests as part of the SOP to 

immediately determine pass/fail status of the runs.

Analyze: Calculate the range (optional)

Where the range is used, it represents is the absolute difference of two (or 

more) replicate runs. Be sure that only absolute values are used in the 

determination of the range and average range!

1 2 R = x − x Eqn. (1)

Calculate the average range of the trials, R , for the n tests as follows: 

n

R

R ∑= Eqn. (2)

The estimated standard deviation may be calculated using the average range 

as follows (obtain values for * d2 from NISTIR 6969, Table 9.1 and see 

NISTIR 6969, Section 8.3 for additional notes). For one set of two runs, the * d2 value is 1.414; for one set of three runs the * d2 value is 1.91:

*

2

p

R

s

d = Eqn. (3)
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Analyze: Construct Appropriate Charts

Construct Standard Deviation Charts and Range Charts using the same 

approach. You may use 2 and 3 as the respective multipliers for the Upper 

Warning Limit and Upper Control Limits on standard deviation charts for 

replicates of two. See Table 1 for limits and replicate options based on a 

Warning Limit of 95 % and an Action Limit of 99.73 %. Note that there will 

be no negative numbers when calculating standard deviations or ranges. The 

formula for calculating alternative limits for standard deviations is given 

by:

UCL s F = pooled J K J α , 1, ( 1) − − Eqn. (4)

Where the alpha is chosen based on the desired limits (e.g., 0.05 for 95 %); J-1 

represents the degrees of freedom from the replicates used in the applicable 

SOP; and K(J-1) represents the degrees of freedom corresponding to the pooled 

standard deviation. 

Table 1. Warning and Control Limits for Standard Deviation Charts.

Replicates Upper Warning Limit Upper Action (Control) Limit

2 1.960 3.00

3 1.731 2.432

4 1.614 2.172

5 1.540 2.016

10 1.371 1.675

Construct an R control chart having the following limits based on the 

number of replicate measurements used in the process where the value in 

the table is multiplied by the average Range, R or Standard Deviation: 

Table 2. Warning and Control Limit Multipliers for Range Charts.

Replicates Upper Warning Limit Upper Action (Control) Limit

2 2.512 3.267

3 2.050 2.575

4 1.855 2.282

5 1.743 2.115

10 1.777

The lower control limit is zero. Note: Using subgroup standard deviations and 

pooled standard deviations is preferable. 

Interpret the Data: Using Control Charts 

Replicate trials are run on each standard submitted to the laboratory for 

calibration according to the specific SOP. The values for standard deviation 

or range are plotted on the appropriate control chart, preferably in 
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chronological order. The limits of the charts are such that 95 % of the values 

should fall within the warning limits and rarely should a value fall outside 

the control limits, provided that the system is in a state of statistical control. 

If the values plotted on the standard deviation or range chart fall outside of 

the control limits, a decrease in precision is indicated. Problems with the 

standards or process will need to be investigated.

Even while the system is in an apparent state of control, incipient troubles 

may be indicated when the control data show short- or long-term trends, 

shifts, or runs.

Controlling the Process

No calibration data should be accepted when the system is out of control.

If a plotted value for standard deviation or range is outside of the warning 

limit but inside the control limit, a second set of replicate calibrations should 

be made. If the new value for the standard deviation or range is within the 

warning limit, the process may be considered in control. If it lies outside of 

the warning limit, lack of control is indicated. Investigation should be made, 

with suitable corrective action taken with demonstrated attainment of 

control before calibration measurements are considered acceptable.

Demonstration of "in control" indicates that the calibration process is 

consistent with the past experience of the laboratory. There is no reason to 

believe that excessive changes in precision have occurred.

The accuracy is inferred from a consideration of control of the sources of 

bias. 

To the extent appropriate, the indicated measurement precision of the 

calibration of the measured type of standards may be extended to the 

calibration of other standards of similar type, capacity and design. 
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Appendix

Standard Deviation and Range Data

Standard Capacity: Laboratory: 

Test

Number Date Run 1 Run 2 Average 

of Runs Standard deviationA

RangeB

|Run 1 - Run 2|

(Max – Min)

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

A Use of the standard deviation and pooled standard deviations are preferred to the use of Range as an estimate 

of the standard deviation.

B This is the range, R, of the two trials and is calculated as the larger value minus the smaller value.
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