Statistics – Building Intuition About The Central Limit Theorem Using Excel

March 26, 2010 at 4:48 pm Leave a comment

Here’s how I introduced the Central Limit Theorem in class on Thursday. We took an exam on binomial, Poisson, and normal probabilities, so I began with a review question – Find the probability that a randomly selected adult has an IQ of 97 or below (mean: 100, std dev: 15). The students worked through the problem, and found that P(z < -0.2) = 0.4207.

I then asked the class whether it would be more likely or less likely that a group of 100 people would have a mean IQ of 97 or lower. My students weren’t really sure how to think about this. I reminded them of the law of large numbers, and a student commented that as the sample size increased the mean should be drawn towards the population mean (100), so she said it would be less likely that the mean would be below 100.

Before introducing the formulas from the Central Limit Theorem, I opened up Excel for some simulation work. I generated a column with 100 random values in it, sampled from a normal distribution with a mean of 100 and standard deviation of 15. This can be done using the data analysis toolpak. I calculated the mean and standard deviation for this sample – the mean was around 99 and the standard deviation was around 14.5. We talked about these values making sense, the mean and standard deviation should be close to those of the population. We sorted the values and saw that 40 of them were 97 or lower, which was close to the 42% we expected.

I then used the data analysis toolpak to generate 100 samples in the same fashion. We looked at the mean of each sample and noticed that they seemed to be centered around 100, but did not vary as much as the individual values did. We sorted these means and noticed that only 2 of the sample means were 97 or lower, much lower than the number of individual values in that category. We calculated the mean of the sample means (99.97) and standard deviation of the sample means (around 1.5), showing that the mean was roughly equal to the population mean and that the standard deviation was significantly lower than the population standard deviation. I finished by creating a histogram of these sample means, and we noticed that the shape was roughly bell shaped.

At this point I introduced the Central Limit Theorem and the formulas. My students understood the Central Limit Theorem before the formulas were ever presented, and that should help to understand confidence intervals conceptually.

-George

I am a math instructor at College of the Sequoias in Visalia, CA. If there’s a particular statistics topic you’d like me to address, or if you have a question or a comment, please let me know. You can also reach me through the contact page on my website – http://georgewoodbury.com.

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