Calculating Sample Sizes For Lean Six Sigma

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Many situations call for sampling from a larger population and drawing inferences from the sample. But how big of a sample size do you need? There's two different requirements for sample sizes:

1. Suppose you manufacture bottle caps of a certain diameter?
   How many samples do you need to get an accurate estimate of the mean (average) diameter at a specified level of precision (+/- .01 inch)?
2. Suppose you have a lake filled with trout and you want to know the percentage of fish over 12 inches in length. How many samples do you need to get an accurate estimate of the proportion (i.e., percentage) within a population.

The QI Macros for Excel contains a sample size calculator. Just select Anova and Analysis Tools then Sample Size. A sample size input window or template will open.

QI Macros 2006 and earlier versions:

QI Macros 2007:

Example 1 (Variable data):
As it turns out, to calculate the sample size for cap diameters, you need to know three things: Confidence level, confidence interval, and standard deviation.

1. Confidence "level": Do you want to be 95% or 99% confident that your results will be within the confidence interval? There are constants (K) each of these confidence levels: % K
   80% - 1.28
   90% - 1.64
   95% - 1.96
   99% - 2.58
2. Confidence "interval" (i.e., precision) variation from sample mean (think of this as being like the upper and lower specification limit).
3. Standard Deviation (actual/estimated - the default is 1/6=0.167). You can enter the actual stdev or estimate the standard deviation as (High-Low)/6. (In this case let's say sigma=.03).

   The formula for sample size = (K^2 * sigma^2)/precision^2
   = ((1.96)^2 * (.03)^2)/(.01)^2 = 35 samples to be 95% confident that mean lies within +/- .01 of actual value.

Example 2 (attribute data):

1. Confidence "level": 95%
2. Confidence "interval" (+/-5%)
3. In this example, to calculate the number of samples required to get a percentage, you have to assume the worst case 50%/50% (.5) for sigma.

   Sample size = ((1.96^2)*(0.5)^2)/(0.05)^2 = 384

As you can see, it's not that hard to calculate the minimum sample size required to determine the mean from a sample of attribute data or the percent "good/bad" from a larger population.

There are more exotic calculations when you get into gathering samples to compare two means, but these should be enough to get you started.

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