Estimating a Population Mean

From a Sample to a Population Mean

A sample can tell us about a population we have not measured in full. For example, a clinic might measure the waiting times of a sample of patients to estimate the average waiting time of all patients seen during a particular month. The average in the sample is calculable from observed data; the population average is the quantity the clinic wants to learn about. Keeping those two quantities distinct is the starting point for estimating a mean.

A useful estimate is not the whole story. Two samples can have the same average but differ in how spread out their measurements are. The number of observations and how they were selected also matter when judging how much confidence to place in an estimate. This tutorial identifies those ingredients; it does not construct a confidence interval.

Definition: A population mean, denoted by $\mu$, is the average value of a quantitative variable for the target population. For a finite population of $N$ values $x_1,\ldots,x_N$, $\mu=(x_1+\cdots+x_N)/N$. A sample mean, denoted by $\bar{x}$, is the average of the $n$ observed sample values: $\bar{x}=(x_1+\cdots+x_n)/n$. The population mean is a parameter; the sample mean is a statistic calculated from data.

The sample mean is used as an estimate of the population mean. Before data are collected, the sample mean can vary from one possible sample to another, so it is an estimator of $\mu$. Once a particular sample is observed and its mean calculated, that numerical result is an estimate. In a report, the estimate is often written as $\bar{x}$, alongside its value and units.

The distinction is important even when the numerical values happen to be close. The sample mean describes the observations actually collected. The population mean describes the entire target population, including values that were not observed. Unless the full population was measured, the population mean is generally unknown.

What Makes the Sample Mean a Reasonable Estimate?

If a sample is selected in a way that represents the target population, its average can provide information about the population average. In particular, under simple random sampling, each population member has an equal chance of selection. The sample mean is then centered on the population mean over repeated samples: it does not systematically tend to be above or below $\mu$. This does not guarantee that any one sample mean will be close to $\mu$.

The centering result can be justified directly. Suppose a sample of size $n$ is drawn so that each sampled value $X_i$ has expected value $\mu$. The mean of the sample is the average of those $n$ values. By linearity of expectation,

E(\bar{X}) = E\left(\frac{X_1+\cdots+X_n}{n}\right) = \frac{E(X_1)+\cdots+E(X_n)}{n} = \frac{n\mu}{n} = \mu.

Here $\bar{X}$ represents the sample mean before a particular sample is observed. The argument shows why the sample mean is a natural estimator when sampling is representative. It does not say that $\bar{x}=\mu$ in every sample, nor does it say how far a particular estimate might be from $\mu$. That remaining question is about uncertainty.

Worked Example: Estimating the Mean Height of Seedlings

In a hypothetical greenhouse, a simple random sample of five seedlings is selected from a clearly defined batch. Their heights are 12, 15, 14, 11, and 18 cm. Find the sample mean and identify what it estimates.

The sample size is $n=5$, and the sample mean is the sum of the observed heights divided by five:

\bar{x} = \frac{12+15+14+11+18}{5} = \frac{70}{5} = 14\text{ cm}.

The estimate is 14 cm. It estimates $\mu$, the mean height of all seedlings in the specified batch, not the mean height of seedlings in every greenhouse. The random selection supports using the sample to learn about that batch, although this one sample does not reveal whether its estimate is exactly right.

Information Needed to Assess Uncertainty

A reported value of $\bar{x}$ gives the location of the sample average, but it does not by itself show how precisely that average estimates $\mu$. At a minimum, a reader needs the sample size $n$ and a description of the spread of the individual measurements. For a sample, spread is commonly summarized by the sample standard deviation $s$. These summaries have different jobs: $\bar{x}$ describes the sample's center; $s$ describes how dispersed its observations are; and $n$ tells how many observations contributed to the estimate.

Definition: For sample values $x_1,\ldots,x_n$, the sample standard deviation $s$ summarizes their spread around $\bar{x}$. It is calculated as the square root of the average squared deviation using $n-1$ in the denominator: $$ s=\sqrt{\frac{\sum_{i=1}^{n}(x_i-\bar{x})^2}{n-1}}. $$ The units of $s$ are the same as the units of the original measurements.

A large $s$ means the observed measurements vary substantially around their sample mean; a small $s$ means they are more tightly clustered. This is the spread among individual observations, not a direct measurement of the error in $\bar{x}$. To assess uncertainty in the estimate, one also needs to know $n$ and how the observations were obtained. For comparable representative samples, more observations generally make the sample mean less variable across samples. But sample size alone cannot repair biased selection or a poorly defined target population.

The sampling design matters because uncertainty calculations describe variation under a sampling process. A random sample from the target population gives a defensible basis for that reasoning. A sample of volunteers, or patients selected only from one clinic when the target is much broader, may not represent the population of interest. Increasing the size of a systematically unrepresentative sample does not automatically make its mean a good estimate of the target mean.

The variable and its units must also be clear. A mean of 14 is hard to interpret if it is not stated whether the measurements are in centimetres, minutes, or another unit. And the population must be specified: “the mean waiting time” could refer to all visits at one clinic this month, all visits across a health system, or a different group entirely. An estimate cannot be interpreted correctly until the quantity it is intended to estimate is named.

Worked Example: Calculating the Sample Spread

A hypothetical random sample of four packages from a production lot has masses of 48, 50, 52, and 50 g. Calculate $\bar{x}$ and $s$, then explain what each summary says.

First calculate the sample mean:

\bar{x} = \frac{48+50+52+50}{4} = \frac{200}{4} = 50\text{ g}.

The deviations from the mean are $-2, 0, 2,$ and $0$ g. Their squared deviations sum to $4+0+4+0=8$ g$^2$. Therefore,

s = \sqrt{\frac{8}{4-1}} = \sqrt{\frac{8}{3}} \approx 1.633\text{ g}.

The sample mean is 50 g, and the sample standard deviation is about 1.633 g. The first describes the centre of these four observed masses; the second describes their spread around that centre. With $n=4$, the estimate is based on only four packages. These summaries are relevant to uncertainty, but they are not enough on their own to decide whether a conclusion is reliable: the lot must be the intended target, and the sampling method must provide a suitable basis for inference.

Same Mean, Different Information

The following comparison shows why a mean alone cannot describe how much individual observations vary. Consider two hypothetical samples, each of three measurements in the same units.

Sample	Observed values	$n$	$\bar{x}$	$s$
A	8, 10, 12	3	10	2
B	4, 10, 16	3	10	6

For sample A, the deviations from 10 are $-2,0,2$, so $s=\sqrt{(4+0+4)/2}=2$. For sample B, the deviations are $-6,0,6$, so $s=\sqrt{(36+0+36)/2}=6$. Both means are 10, but sample B's observations are more spread out. Thus a report of “mean = 10” alone leaves out relevant information about the data.

Worked Example: Identifying What a Report Still Needs

A hypothetical report says that a sample of 25 patients had a mean appointment wait of 32 minutes. It gives no other details. What information would help a reader assess the estimate of the mean wait for all patients visiting the clinic during the same month?

The reported estimate is $\bar{x}=32$ minutes and the stated sample size is $n=25$. The report should also give the sample standard deviation $s$, in minutes, to describe the spread of observed waits. Without it, the reader cannot tell whether the 25 waits were tightly grouped around 32 or highly variable.

The report should specify how the patients were selected and confirm that the target is all clinic patients during that month. For instance, if the sample consists only of patients who chose to complete an online survey, it may differ from the full group of clinic patients. The mean of those respondents is still calculable, but representativeness is not guaranteed by the calculation.

Finally, the report should name the measurement clearly: wait from which event to which event, in minutes. These details do not change the arithmetic that produced 32, but they determine what that number represents and how appropriately it can be used to estimate the target population mean.

What to Check Before Interpreting an Estimate

For a single mean, start by stating the target population and the quantitative variable. Then identify the sample mean, sample size, sample standard deviation, and sampling method. These are distinct pieces of information, not interchangeable summaries. A confidence interval for a mean will use sample information to represent uncertainty, but a numerical interval also requires a chosen confidence level and an appropriate method. The method's assumptions must be checked for the data and sampling design before its result is interpreted.

Name the target.
State whose mean is being estimated and define the variable, including its units.

Report the observed centre and size.
Give $\bar{x}$ and $n$, and distinguish this sample result from the unknown population parameter $\mu$.

Describe variability and selection.
Give $s$ and explain how the sample was selected; both affect how uncertainty can be assessed.

Check whether an inferential method is appropriate.
Consider whether the sampling design and the observed data meet the requirements of the method being used.

Common Mistakes

Treating the sample mean as the population mean. The reasoning “the sample average is 32 minutes, so the population average is exactly 32 minutes” confuses an estimate with the unknown parameter. Instead, say that the sample mean is 32 minutes and is being used to estimate the target population mean.
Assuming a large sample automatically represents the target. A very large group of volunteers can still differ systematically from the population of interest. The sampling method and target must be considered, not just $n$.
Calling $s$ the uncertainty in the mean. The sample standard deviation describes spread among individual observations. It is relevant to assessing uncertainty in $\bar{x}$, but it is not itself a measure of that uncertainty.
Reporting a mean without units or a defined population. “The average is 32” is incomplete. State what was measured, in what units, and for which population the estimate is intended.

Key takeaway: The sample mean $\bar{x}$ is an estimate of the population mean $\mu$, not a guarantee of its exact value. To assess uncertainty for one mean, identify the target and units, report $n$ and the sample spread $s$, and describe how the sample was obtained.

Check Your Understanding

Use the definitions and examples above to answer these questions.

A random sample of 12 devices has a mean battery life of 9.4 hours. What does 9.4 hours describe, and what population quantity might it estimate?
Why are $n$ and $s$ useful information in addition to $\bar{x}$ when assessing uncertainty about a mean?
Two samples have the same sample mean but different sample standard deviations. What does the difference in their standard deviations tell you about the observed data?
A report uses 500 online volunteers to estimate the mean daily screen time of all adults in a country. Why does the large sample size alone not establish that the estimate represents the target population?
For an estimate of the mean weight of patients at a clinic, name two details about the target or measurement that should be made clear in the report.

What You'll Learn