Sampling Distributions for Sample Means (College Board AP® Statistics): Revision Note

Sampling distributions for sample means

What is the sampling distribution for sample means?

• If you take all possible samples of size $n$ from a population and calculate the sample mean, $\overline{x}$, for each

  • then you would have all possible values of the sample mean

    • This collection of all possible sample means is called the sampling distribution for sample means

• This sampling distribution is often shown on a graph to see its shape

  • e.g. a relative frequency chart or histogram

What are the mean and standard deviation of the sampling distribution for sample means?

• If the population has a population mean of $\mu$ and a population standard deviation of $\sigma$

  • then the sampling distribution for sample means, $\overline{x}$, for samples of size $n$:

    • has a mean of $\mu$

    • and a standard deviation of $\frac{\sigma }{\sqrt{n}}$

  • Note how the standard deviation of sample means depends on $n$

    • A larger sample size gives a smaller standard deviation

• The standard deviation of $\frac{\sigma }{\sqrt{n}}$ assumes sampling was done with replacement

• If the sampling is carried out without replacement then the sample must satisy:

  • The randomization conditon

    • a random sampling method should be used

  • The 10% condition

    • the sample size is less than 10% of the population size

    • $n\le 10\%N$

What conditions are needed for normality?

• If in addition to the above, the population is known to be normally distributed

  • then the sampling distribution for sample means is also normally distributed

    • with mean $\mu$ and standard deviation $\frac{\sigma }{\sqrt{n}}$

• You can use these properties to calculate probabilities involving sample means, $\overline{x}$, which follow a normal distribution

  • The standardized z-statistic is $\frac{\overline{x}-\mu }{{\displaystyle \frac{\sigma }{\sqrt{n}}}}$

    • $\mu$ and $\sigma$ will be given in the question

What do I do if the population is not normally distributed?

• If the population is not normally distributed, then the sampling distribution for sample means is not guaranteed to be normally distributed

• However, despite not knowing its shape, the sampling distribution for sample means still has a

  • mean of $\mu$ and a standard deviation of $\frac{\sigma }{\sqrt{n}}$

    • i.e. you can always write these down, even though the distribution is unknown

Can I use the Central Limit theorem if the population is not normally distributed?

• If the population is not normally distributed, but the sample size is large ($n\ge 30$ )

  • then the Central Limit theorem can be applied

  • meaning the sampling distribution for the sample means is approximately normally distributed with the parameters above

    • i.e. mean $\mu$ and standard deviation $\frac{\sigma }{\sqrt{n}}$

• You can use these properties to estimate probabilities involving sample means, $\overline{x}$, as they follow an approximate normal distribution

  • Its standardized z-statistic is $\frac{\overline{x}-\mu }{{\displaystyle \frac{\sigma }{\sqrt{n}}}}$