THE FREQUENCY OF THE SAMPLE

No one will deny that sampling is important. However, for that importance to be of significance, the sample must be random and representative of the whole. To make sure that this is actually happening, statisticians have developed many techniques to test the appropriateness of the sample. In Volume III we discussed some of these techniques. Here, however, we will address some simple and straightforward approaches.

To begin with, the questions that we should pose are as follows :

• Is the process stable?

• What is the confidence level desired? (90%, 95%, or 99%)

• What is the size of difference interested in A?

• What is the amount of variation (unexplained) being lived with ( s )?

If we have an idea of the answers to these questions, then we may proceed to use one of the following formulas.

Condition one: When a difference of two averages of size A is considered economically significant, use

Condition two: When a standard deviation must be known within A for economic reasons, use

where t =	1.64	‰ˆ	1.6 ƒ for 90% confidence
	1.96	‰ˆ	2 ƒ for 95% confidence
	2.58	‰ˆ	3 ƒ for 99% confidence

Example: How many samples?

We have a process that the setting time in minutes is the following:

150	145
160	140
150	160
140	155
155	150

Average = 1505/10 = 150.5

How many samples should be taken to detect a difference of 10 with 95% confidence level?

How would n change if we wanted to increase our confidence level from 95 to 99%?

How many samples should be taken to detect a difference of 5 with 95% confidence level?

Now that we have established the sample, we have to determine the frequency. A rule of thumb is to determine the frequency by comparing the intrinsic process variability with the difference between the process average and the critical specification limit to which the process is subject. If there were both an upper and lower specification limit, the critical one would be that closest to the process average. To calculate the optimum frequency, we define a factor k:

k = (Critical Specification - Process Average)/ S _p

where

S _p

=

process variability.

The steps for calculating k are:

1. Take present frequency data (40 to 60 samples and covering a sufficiently long operation cycle to represent typical ups and downs ) and calculate moving ranges, MR .

2. Calculate average MR = & pound ; MR/n .

3. Check for odd readings and delete any MRs exceeding 3.27 — .

4. If there are deletions, continue steps 2 and 3 until there are no deletions.

5. Calculate S _PCL = /1.128 on final .

6. Calculate S _P = . Be sure S _T is based on at least 10 estimates. If S _T is not available, set up program to measure it.

7. Calculate k .

8. If k is between 2.7 and 3.3, present frequency of sampling is best.

9. If k is greater than 3.3, present frequency is too high.

   1. Calculate by getting differences between every other sample from existing data.

   2. Delete odd readings.

   3. When no more deletions are necessary, calculate S _PCL (step 5).

   4. Calculate S _P (step 6).

   5. Calculate k .

   6. If k is between 2.7 and 3.3, best sampling frequency is 1/2 present frequency.

   7. If k is greater than 3.3, calculate MRs from every third sample; calculate MR after there are no more deletions; calculate S _PCL (step 5), S _P (step 6), and k .

   8. If k is between 2.7 and 3.3, best sampling frequency is 1/3 present frequency.

   9. If k is greater than 3.3, repeat process of determining by skipping a larger and larger number of samples from the existing data until k = 2.7 to 3.3.

10. If original k is less than 2.7, present sampling frequency is too low.

    1. Double present sampling frequency for 20 time periods covering sufficient ups and downs in operation.

    2. Calculate k as before, i.e., .

    3. If k is 2.7 “3.3, doubling present frequency is proper.

    4. If k is greater than 3.3, correct sampling frequency is 1.5 times present frequency.

    5. If k in 10b is less than 2.7, increase present sampling frequency 4 times.

    6. Calculate k .

       1. If k = 2.7 “3.3, best sampling frequency is 4 times present sampling frequency.

       2. If k is less than 2.7, keep increasing sampling frequency until k is 2.7 “3.3.

       3. If k is greater than 3.3, then best sampling frequency is halfway between 4 times and 2 times present frequency = 3 times present frequency.