Key Concepts

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Central Limit Theorem

According to the Central Limit Theorem, the sampling distribution of the mean:

  • is normally distributed
  • has a mean equal to the population mean
  • has standard deviation (also called standard error) equal to the population standard deviation divided by the square root of the sample size

In the plots provided, the left plot shows the population distribution of salmon weights, and the right plot shows the sampling distribution of the mean salmon weights.

Sampling Distributions
Lesson 1 of 1
  1. 1
    In statistics, we often want to learn about a large population. Since collecting data for an entire population is often impossible, researchers may use a smaller sample of data to try to answer the…
  2. 2
    Now that we’ve generated some random samples from a population using an applet, let’s code this ourselves in Python. The numpy.random package has several functions that we could use to simulate ran…
  3. 3
    As we saw in the last example, each time we sample from a population, we will get a slightly different sample mean. In order to understand how much variation we can expect in those sample means, we…
  4. 4
    So far, we’ve defined the term sampling distribution and shown how we can simulate an approximated sampling distribution for a few different statistics (mean, maximum, variance, etc.). The *Centr…
  5. 5
    Now that we’ve examined the CLT from a high level, let’s get into the details. The CLT not only establishes that the sampling distribution will be normally distributed, but it also allows us to de…
  6. 6
    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 (often denoted …
  7. 7
    According to the Central Limit Theorem, the mean of the sampling distribution of the mean is equal to the population mean. This is the case for some, but not all, sampling distributions. Remember, …
  8. 8
    Once we know the sampling distribution of the mean, we can also use it to estimate the probability of observing a particular range of sample means, given some information (either known or assumed) …
  9. 9
    Let’s recap what we’ve learned in this lesson: - A sampling distribution is obtained by taking a random sample of a certain size multiple times, taking a sample statistic, and plotting the distrib…

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