| = | $\sigma$ | = | $n$ | = |
| $N$ | is | $N$ | = |
| $f_A$ | = | $f_B$ | = | $f_C$ | = | $f_D$ | = | $f_E$ | = | $f_F$ | = |
A sampling distribution is the probability distribution of a sample statistic. So, for example, the sampling distribution of the sample mean ($\bar{x}$) is the probability distribution of $\bar{x}$. There are three things we need to know to fully describe a probability distribution of $\bar{x}$: the expected value, the standard deviation and the form of the distribution. First, the expected value of the sample mean is equal to the population mean ($\mu$). The idea here is that if we took all possible samples from the population and computed their sample means, they would average to equal the population mean.
| Expected Value |
| $E(\bar{x}) = {\color{Black}\mu}$ |
Calculation of the standard deviation depends on whether we're sampling from a finite population or an infinite population. Note that the formulas below have two standard deviations. One of them, $\sigma_\bar{x}$, is the standard deviation of the sample mean while the other one, $\sigma$, is the standard deviation of the population. Two avoid confusing the two, $\sigma_\bar{x}$ is referred to as the standard error of the mean. To switch from an infinite population to a finite population, click on $\boxed{\text{Infinite}}$ and select $\boxed{\text{Finite}}$.
| Finite Population | Infinite Population | |
| Standard Deviation | $\sigma_\bar{x} = \sqrt{\dfrac{N-n}{N-1}} \left(\dfrac{\sigma}{\sqrt{n}} \right) $ | $\sigma_\bar{x} = \dfrac{\sigma}{\sqrt{n}} $ |
The standard error of the mean measures the amount of error in using $\bar{x}$ to estimate $\mu$. The sample size, n, being in the denominator of the formulas indicates that increasing the sample size decreases this error. Note that the only difference between the two formulas above is the term $\sqrt{\frac{N-n}{N-1}}$. This is referred to as the finite population correction factor. Plugging in a large value for N and a relatively small value for n into the finite population correction factor, we get a value close to one. This allows us to use the following rule of thumb.
| Rule of Thumb |
| Use $\sigma_\bar{x} = \dfrac{\sigma}{\sqrt{n}} $ whenever |
| 1. The population is infinite, or |
| 2. The population is finite and n/N $\leq$ .05 |
The form of the sampling distribution of the sample mean depends on the form of the population. If the population has a normal distribution, the sampling distribution of $\bar{x}$ is a normal distribution. If the population does not have a normal distribution, we must use the Central Limit Theorem. It says that as the sample size becomes large, the sampling distribution of $\bar{x}$ approximates a normal distribution. Generally, a sample size of at least thirty is sufficient for the sampling distribution of $\bar{x}$ to be approximately normal.
One of the benefits of knowing the sampling distribution of the sample mean is that we can calculate the probability that $\bar{x}$ will be within a certain range of the population mean. Since the sample mean follows a normal distribution, calculating probabilities for $\bar{x}$ simply involves converting to the standard normal distribution. The formula for converting from normal to standard normal involves subtracting by the mean and dividing by the standard deviation: $z=\frac{x-\mu}{\sigma}$. In the case of the sampling distribution of sample mean, the mean is the population mean, $\mu$, and the standard deviation is the standard error of the mean, $\sigma_{\bar{x}}$.
Point estimation is the process of estimating a population parameter. The term "point" is used because it's a single-value estimate, as opposed to a range of values like in a confidence interval. The point estimatar of a population parameter is simply the sample statistic corresponding to the population parameter. For example, the point estimator of the population mean is the sample mean. The point estimator of the population variance is the sample variance and the point estimator of the population standard deviation is the sample standard deviation.
| Population Mean | Population Variance | |
| Point Estimator | $\bar{x} = \dfrac{\sum x}{n}$ | $ s^2 = \dfrac{\sum (x - \bar{x})^2}{n-1}$ |
Sampling is the process of selecting the elements of a sample from a population. The sampling procedure depends on whether you're sampling from a finite or infinite population. When sampling from a finite population, a simple random sample is preferred. In a simple random sample, each element from the population has the same probability of being selected. When sampling from an infinite population, a random sample is preferred. In a random sample, the elements are chosen independently and come from the same population.
| Finite Population | Infinite Population | |
| Method | Simple Random Sample | Random Sample |
If the sample size is large enough, the sampling distribution of the sample mean approximates a normal distribution. The normal distribution is one example of a continuous probability distribution. Probabilities for continuous distributions can be calculated using the Continuous Distribution Calculator. Sampling distributions form the theoretical foundations for more advanced statistical inferennce, such as confidence intervals. Confidence intervals can be calculated using the Confidence Interval Calculator.