The central limit theorem
Why sums and averages take a bell shape, how much a sample mean tightens as the sample grows, when you can rely on it and how to work out a probability on a sum.
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Central limit theorem: what is it?
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The result saying that the sum (or the mean) of many independent variables with the same distribution is approximately normally distributed, whatever the starting distribution.
The bell shape comes from adding itself, not from the starting distribution: a flat die, a skewed exponential or a coin all give the same bell once added up.
You roll 30 dice and note the mean of the 30 results, then repeat 500 times. What shape is the histogram of the 500 means?
- A bell centred on 3.5
- A flat histogram, like that of a single die
- A bell centred on 105, the mean total of the 30 dice
- Two humps, near 1 and near 6
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A bell centred on 3.5
A die has expected value 3.5 and means of 30 dice gather around it in a bell. Flatness belongs to one roll, not to the mean; 105 would be the centre of the totals, not of the means.
The mean of n independent variables with the same distribution, expected value μ and standard deviation σ, approximately follows a … distribution with expected value … and standard deviation ….
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The mean of n independent variables with the same distribution, expected value μ and standard deviation σ, approximately follows a normal distribution with expected value μ and standard deviation σ/√n.
The mean stays centred on μ: only its spread shrinks, as 1/√n.
Sampling distribution of the mean: what is it?
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The distribution of the mean you would get by drawing n values over and over again.
A sample mean is itself random: it has an expected value and a standard deviation. This is the distribution the theorem describes, not that of a single value.