Standard Error

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The second part of the Central Limit Theorem is:

The sampling distribution of the mean is normally distributed, with standard deviation equal to the population standard deviation (σ) divided by the square root of the sample size (often denoted as n):

$$ \frac{σ}{\sqrt{n}} $$

The standard deviation of a sampling distribution is also known as the standard error of the mean. In any instances, we cannot know the population standard deviation, so we estimate the standard error using the sample standard deviation:

$$ \frac{standard \ deviation\ of\ our\ sample}{\sqrt{sample\ size}} $$

Two important things to note about this formula is that:

• As sample size increases, the standard error will decrease
• As the population standard deviation increases, so will the standard error

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1. In the workspace, you can see a population distribution and a sampling distribution. The sample size is set to 10. Increase the sample size to 50 and note the change in the shape of the sampling distribution. A smaller standard error means that the distribution will be taller and skinnier.

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2. Now increase the standard deviation of the population to 30. This means that the population distribution will have more variation (and will therefore appear wider and flatter). The sample distribution will also become wider and flatter because the standard error will increase (due to the large numerator).