One Population, Many Possible Samples
Suppose a health department wants to describe the proportion of residents in a town who received a particular vaccine. The town and its residents are the population of interest. If the department surveys a random sample, the resulting sample proportion depends on which residents happen to be selected. A different random sample could give a different proportion, even though the population has not changed.
This distinction is central to sampling: a population parameter describes the population, while a sample statistic is calculated from sample data. A parameter is fixed for a specified population and characteristic, though it may be unknown. A statistic is determined by the sample actually observed, and its value can vary from sample to sample.
“Fixed” does not mean “known.” The vaccine proportion \(p\) is fixed once the population, characteristic, and time period are specified, even if no one has measured it exactly. Uncertainty about its value is not the same thing as the parameter itself changing from sample to sample. By contrast, the sample statistic changes when a different sample produces different data.
The phrase repeated sampling describes a thought experiment: keep the population and sampling design the same, repeatedly draw a sample according to that design, and calculate the statistic each time. This lets us consider the different outcomes the statistic could have. It does not mean that one actual sample is repeatedly altered after it has been collected.
Why a Statistic Has Random Outcomes
A sampling design determines how individuals can be selected. If the design uses random selection, then before selection we do not know which eligible individuals will be included. Because the statistic is calculated from those selected individuals, its value can depend on the random selection. Once the sample is fixed, however, calculating the statistic is no longer random: the observed data determine its value.
For a sample mean, it is useful to distinguish the random sample from the observed numbers. Let \(X_i\) denote the value for the \(i\)th selected unit before the sample is observed. The statistic \(\bar{X}\) is then random under the sampling design. Once the sample values \(x_1,\ldots,x_n\) are observed, their mean is written \(\bar{x}\). Thus \(\bar{X}\) describes the statistic across possible samples, while \(\bar{x}\) is its realized value for one sample.
The arrow represents observing a particular sample, not a change to the definition of the mean. For a sample proportion, the same distinction is usually made using \(\hat{p}\): before selection it represents the proportion statistic under the sampling process; after selection its numerical value is calculated from the sample.
This fixes the parameter being discussed, such as the proportion of residents who received a vaccine.
The design determines which samples are possible and how selection takes place. Random selection means the sample is not known in advance.
Each possible sample supplies data to the same calculation, such as a mean or proportion. Different samples can give different results.
After one sample is selected and measured, its statistic is a particular observed number.
Worked Examples
Worked Example: A Sample Proportion from a Fixed Population
Imagine a hypothetical town with eight residents. Three received a vaccine, and five did not. The town is the population, and the characteristic is whether a resident received the vaccine. Find the population proportion, then compare the sample proportions from two possible random samples of four residents.
Population parameter. The total number of residents and their vaccination statuses are fixed in this scenario. The population proportion is
Thus 37.5% of the town’s residents received the vaccine. This parameter is unchanged when a sample is drawn.
First possible sample. Suppose a random sample includes two residents who received the vaccine and two who did not. The observed sample proportion is
Second possible sample. Another possible sample of four residents might include one vaccinated resident and three who were not. Its sample proportion is
Both samples come from the same population, whose proportion remains 0.375. The sample proportions differ because the samples contain different residents. These two outcomes illustrate variation; they do not, by themselves, list every possible sample or describe a complete sampling distribution.
Worked Example: A Sample Mean from Fixed Population Values
Consider a hypothetical population of four patients’ waiting times, measured in minutes: 8, 10, 12, and 18. Suppose two patients are selected at random without replacement. Find the population mean and the sample means for two possible selections.
Population parameter. The population mean waiting time is
The population values are fixed here, so \(\mu=12\) minutes remains the same for either sample.
First possible sample. If the selected waiting times are 8 and 10 minutes, the observed sample mean is
Second possible sample. If the selected waiting times are 12 and 18 minutes, the observed sample mean is
The sample means differ by 6 minutes, although both samples are drawn from the same four patients. The difference is a consequence of which waiting times were included. The example assumes the population values are recorded accurately and that selection is random without replacement; the two pairs are possible samples under that design.
Worked Example: An Unknown Mean and Two Observed Samples
A hypothetical laboratory process produces a large population of measurement values. Its population mean \(\mu\) is not known. Two different random samples of four measurements are collected. The first sample is 248, 252, 250, and 260 grams; the second is 246, 250, 254, and 258 grams. Calculate the observed sample means and identify which quantities are fixed or varying.
For the first sample,
For the second sample,
The observed means differ by \(252.5-252.0=0.5\) grams. For the specified population and measurement characteristic, \(\mu\) is fixed, even though its value is unknown. Each sample mean is a statistic, and the two calculated values are its outcomes for these two samples. The measurements are in grams, and the interpretation assumes both samples were selected using the same random sampling design and the measurement process was consistent.
A Short Justification: The Statistic Depends on the Sample
The distinction can be stated directly using fixed population values. Suppose a population contains values \(y_1,\ldots,y_N\), and a sample of size \(n\) is selected. Let \(I_i\) identify the population member selected for sample position \(i\). Under random sampling, which members occupy those positions is not known in advance. The random sample mean can be written as
The values \(y_1,\ldots,y_N\) are fixed, but the selected indices \(I_1,\ldots,I_n\) can vary across samples. Therefore the sum, and thus \(\bar{X}\), can vary. For instance, if one possible sample has values 8 and 10, its mean is \(9\); if another has values 12 and 18, its mean is \(15\). The calculation is deterministic for each chosen sample, but the sample selection makes the statistic’s outcome uncertain before selection. If every possible sample happened to give the same statistic, that statistic would not vary under that design; variation is possible, not guaranteed.
This argument depends on having a specified population and sampling design. Change the population, the characteristic measured, or the selection design, and the parameter or the statistic’s possible outcomes may change. When discussing repeated samples, keep those ingredients fixed unless a comparison explicitly changes one of them.
Parameter and Statistic: A Practical Comparison
| Feature | Population parameter | Sample statistic |
|---|---|---|
| Describes | A specified population | A particular sample, or the statistic across possible samples |
| Examples | \(p\), the population proportion; \(\mu\), the population mean | \(\hat{p}\), the sample proportion; \(\bar{x}\), the sample mean |
| Across repeated samples from the same population | Stays fixed | Can have different values |
| Known from the sample alone? | Not necessarily; it may be unknown | Yes, once the sample data are observed and the statistic is calculated |
Whether a number is a parameter or a statistic depends on what population or sample it describes, not on its numerical size. For example, a proportion of 0.500 could be a population parameter in one investigation or an observed sample proportion in another. Always name the population, sample, and characteristic in context.
Common Mistake: Treating an Unknown Parameter as a Random Statistic
A common mistake is to say, “The population proportion is random because we do not know its value,” or to say, “The sample proportion is fixed because I have calculated it.” These statements confuse uncertainty about a value with variation across the sampling process.
For repeated sampling from one specified population, the population parameter stays fixed. Before a sample is drawn, the statistic can have different outcomes because the selected units are not yet known. After a particular sample is observed, the statistic’s numerical value is fixed for that sample. A new sample can produce a new value, but it does not retroactively change the value calculated from the first sample.
It is also important not to confuse differences among individual measurements with differences among sample statistics. Individual measurements can vary within the population. A statistic, such as a sample mean, summarizes a selected group of those measurements; it can vary because different groups are selected. The two kinds of variation refer to different quantities.
Check Your Understanding
For each question, identify the relevant population, parameter, statistic, or source of variation as appropriate.
- A factory’s full production for a specified day has a fixed mean part length, \(\mu\). Is \(\mu\) fixed or does it change when different random samples of parts are measured? Explain.
- A random sample of 30 residents gives \(\hat{p}=0.40\) for the proportion who use public transit. What does 0.40 describe: the population parameter or the observed statistic?
- Explain why a statistic can be uncertain before a random sample is selected but have a definite value after the data are observed.
- Two samples from the same population give different sample means. Does this fact alone show that the population mean changed? Why or why not?
- Can a statistic have the same value in two different samples? Explain how this is consistent with saying that a statistic can vary across samples.