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Central Limit Theorem

The central limit theorem, in layman’s terms, says that regardless of the shape of the underlying distribution, in most cases, the mean of samples taken from the distribution will approximate a normal distribution.

It is probably easiest to explain this concept using a deck of cards. If you assign a value of 1 to 13 to each card, and draw a single card, the distribution would be flat, and a random sample of 20 cards might look something like this:

Central-Limit-Theorem-1-card

But what happens when you draw 20 hands of 5 cards each and average out the results?

Central-Limit-Theorem-5-cards

The results start to cluster around the expected average value, 7, of the entire deck.

The effect gets even more pronounced as you use increasingly larger sample sizes. With 20 hands of 10 cards, the distribution is even more clearly normal. Again, this was the result of a real-life card-dealing data collection effort.

Central-Limit-Theorem-10-cards

Remember, the distribution curve for a deck of cards is completely flat, but the resulting distribution of the averages assumes a normal distribution curve.

Note that this effect would still be seen regardless of the shape of the underlying distribution, as long its shape is well-defined. That really means that the level of variation has to be consistent. If you change the underlying process, you change the output and the central limit theorem won’t apply. It would be like taking all the face cards out of a deck. You’d expect a shift in the look of the curves, so a mix of before and after samples would not be normal.

In practice, you might be looking at the cycle time of a process. You’d probably have a skewed distribution. Imagine you had a 10 minute average for a process. You’d be unlikely to have many times below 7 or 8 minutes, even if everything went perfect. But you could have some times in the mid-to-high teens, and maybe even an occasional 20 minute cycle if things got really crazy.

If you took a sample of each day’s cycle times and averaged them out, though, the resulting distribution would be normal. This is the basis of SPC. If there is a shift in the process, the shape of the average distribution shifts, allowing you to pick up on subtle changes.

 

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