Sample Mean

Definition of Sample Means

The mean of a set of numbers is just the average, that is, the sum of all the elements in the set divided by the number of elements in the set.

It is easy to see that if two sets are different, they will most likely also have different means.

Calculation of Sample Means

The sample mean is denoted by \(\overline{x}\), and is calculated by adding up all the values obtained from the sample and dividing by the total sample size \(n\). The process is the same as averaging a data set. Therefore, the formula is \[\overline{x}=\frac{x_1+\ldots+x_n}{n},\]

where \(\overline{x}\) is the sample mean, \(x_i\) is each element in the sample and \(n\) is the sample size.

Standard Deviation and Variance of the Sample Mean

Since the variance is the square of the standard deviation, to calculate either value, two cases must be considered:

1. You know the population standard deviation.

2. You do not know the population standard deviation.

The following section shows how to calculate this value for each case.

The Mean and Standard Deviation Formula for Sample Means

The mean of the sample mean, denoted by \(\mu_\overline{x}\), is given by the population mean, that is if \(\mu\) is the population mean, \[\mu_\overline{x}=\mu.\]

To calculate the standard deviation of the sample mean (also called the standard error of the mean (SEM)), denoted by \(\sigma_\overline{x}\), the two previous cases must be considered. Let's explore them in turn.

Calculating the Sample Mean Standard Deviation using the Population Standard Deviation

If the sample of size \(n\) is drawn from a population whose standard deviation \(\sigma\) is known, then the standard deviation of the sample mean will be given by \[\sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}.\]

Calculating the Sample Mean Standard Deviation without using the Population Standard Deviation

Sometimes, when you want to estimate the mean of a population, you do not have any information other than just the data from the sample you took. Fortunately, if the sample is large enough (greater than \(30\)), the standard deviation of the sample mean can be approximated using the sample standard deviation. Thus, for a sample of size \(n\), the standard deviation of the sample mean is \[\sigma_\overline{x}\approx\frac{s}{\sqrt{n}},\] where \(s\) is the sample standard deviation (see the article Standard Deviation for more information) calculated by:

\[s=\sqrt{\frac{(x_1-\overline{x})^2+\ldots+(x_n-\overline{x})^2}{n-1}},\]

where \(x_i\) is each element in the sample and \(\overline{x}\) is the sample mean.

❗❗ The sample standard deviation measures the dispersion of data within the sample, while the sample mean standard deviation measures the dispersion between the means from different samples.

Sampling Distribution of the Mean

Recall the sampling distribution definition.

If \(\overline{x}\) is the sample mean of a sample of size \(n\) from a population with mean \(\mu\) and standard deviation \(\sigma\). Then, the sampling distribution of \(\overline{x}\) has mean and standard deviation given by \[\mu_\overline{x}=\mu\,\text{ and }\,\sigma_\overline{x}=\frac{\sigma}{\sqrt{n}}.\]

Furthermore, if the distribution of the population is normal or the sample size is large enough (according to the Central Limit Theorem, \(n\geq 30\) is enough), then the sampling distribution of \(\overline{x}\) is also normal.

When the distribution is normal, you can calculate probabilities using the standard normal distribution table, for this you need to convert the sample mean \(\overline{x}\) into a \(z\)-score using the following formula

\[z=\frac{\overline{x}-\mu_\overline{x}}{\sigma_\overline{x}}=\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}.\]

Let's see an example of a graph of a sampling distribution of the mean.

Examples of Sample Means

Let's see an example of how to calculate probabilities.