Central Limit Theorem in practical terms

Learning objectives

• State the Central Limit Theorem in plain garment-floor language.
• Compute standard error and show sqrt(n) behaviour in a spreadsheet.
• Distinguish sampling-distribution normality from raw-data normality.

Core idea

Under independence and finite variance, the sampling distribution of the sample mean tends to normal as n grows, with standard error SE = s / √n.

Resampling lab

• Column A: 500 skewed repair-minute values.
• For n=2,5,10,30 use =INDEX(A:A,RANDBETWEEN(2,501)) to draw samples.
• Compute each sample mean; repeat 200 times with a data table.
• Plot histogram of sample means for each n; shape approaches normal.
• Standard error = =STDEV.S(sample_means) ≈ =STDEV.S(A2:A501)/SQRT(n).

Garment-factory example

Downtime per stop is heavily right-skewed. Averages of 30 stops per day look approximately normal on the daily control chart, which is why Xbar charts work even when individual events do not.

Method

1. Sample repeatedly; record means.
2. Compare histogram of raw data vs histogram of means.
3. Verify SE shrinks with √n, not with n.

Common mistakes

• Believing CLT makes the raw process normal.
• Assuming large n cures selection bias.