14: Population Estimation

14.1 From Sample to Population

Population estimation refers to the process of estimating unknown population parameters, such as the mean, based on data collected from a sample. Because surveying an entire population is often impractical or impossible, researchers rely on representative samples to draw conclusions about the larger group. While estimation is a key part of inferential statistics, it begins with descriptive statistics, which summarize sample data to support generalizations. In this way, population estimation serves as a bridge between descriptive and inferential methods. It plays a vital role in making evidence-based predictions about a population and in quantifying the confidence we can place in those results.

14.2 Estimating Population Parameters

Researchers use sample statistics to estimate unknown population parameters, most commonly the population mean. Because collecting data from an entire population is often unrealistic, a well-chosen sample allows us to make educated estimates about the larger group.

The sample mean, or average, is the most common method for estimating the population mean. When the sample is randomly selected and represents the population well, the sample mean can provide a reasonably accurate estimate of the true average in the population. The larger and more representative the sample, the more reliable the estimate.

14.3 Sampling Error

Sampling error refers to the difference between a sample statistic (such as the sample mean) and the true population parameter. Because a sample includes only part of the population, it may not perfectly reflect the population’s characteristics. This error is natural and expected, but its impact can be minimized through random sampling and larger sample sizes, both of which increase representativeness and precision.

Understanding sampling error is essential for interpreting results accurately. It helps researchers judge how much confidence they can place in an estimate and how well the sample reflects the population. Larger, unbiased samples typically produce more accurate estimates and result in smaller sampling error. A related concept, standard error, describes the expected variability of a sample statistic across repeated samples and plays a central role in constructing confidence intervals.

Because sampling error introduces uncertainty into every estimate, researchers use confidence intervals to express the likely range in which the true population parameter falls. These intervals provide a practical way to account for the fact that no sample perfectly mirrors its population.

Power Analysis

One strategy for managing sampling error is to plan your sample size in advance using a technique called power analysis. Power analysis helps researchers determine the minimum number of participants needed to detect an effect if one truly exists. In general, the smaller the effect, the larger the sample required to detect it reliably. Larger effects are easier to identify and typically require fewer participants. Power analysis ensures that a study includes enough participants to produce reliable estimates without collecting more data than necessary. While detailed power calculations often require specialized tools, the core idea is to align sample size with the expected effect size and the desired level of precision. Researchers commonly use software such as G*Power to conduct power analyses and calculate appropriate sample sizes for their studies.

14.4 Standard Error

Standard error is a measure of how much the sample mean is expected to vary from one sample to another due to random chance. It reflects the precision of the estimate and provides insight into the stability of your results. If many samples were drawn from the same population, the standard error describes how much the resulting sample means would typically differ from one another.

Standard error plays a key role in constructing confidence intervals and assessing the reliability of sample-based estimates. A smaller standard error indicates a more precise estimate, while a larger standard error reflects greater uncertainty. Because standard error decreases as sample size increases, collecting more data generally improves the quality and trustworthiness of population estimates.

14.5 Confidence Intervals

A confidence interval (CI) is a range of values that is likely to contain the true population parameter, based on your sample data. It provides a way to express the uncertainty around an estimate. Rather than reporting a single value, such as a sample mean, a confidence interval gives a range of plausible values for the population mean. This helps researchers understand both the precision and reliability of their estimates.

For example, a 95% confidence interval means that if you were to take 100 different random samples from the same population, about 95 of those intervals would contain the true population parameter. The confidence level reflects how certain you want to be that the interval includes the true value. The width of the interval reflects how precise the estimate is. Higher confidence levels increase certainty but require wider intervals, while lower confidence levels offer more precision (narrower intervals) but less certainty.

Common Confidence Levels

Confidence levels represent a trade-off between certainty and precision when estimating population parameters. A 90% confidence level offers less certainty but greater precision, resulting in a narrower interval. A 95% confidence level, which is most commonly used, strikes a balance between certainty and precision. A 99% confidence level provides the most certainty but at the cost of reduced precision, producing a wider interval.

Increasing the confidence level results in a wider interval. This is because achieving greater certainty requires covering a broader range of plausible values. Conversely, accepting a lower level of certainty allows for a narrower, more precise interval.

Interpreting Confidence Intervals

Confidence intervals are central to making inferences about populations. They communicate both the estimate and the degree of uncertainty. For instance, suppose your sample mean is 50, and the 95% confidence interval is (47, 53). This means you can be 95% confident that the true population mean lies between 47 and 53.

It’s important to understand that this does not mean there is a 95% chance that the true mean lies within that specific interval. Rather, the interpretation is about the process: if the same sampling method were repeated many times, 95% of the confidence intervals generated would include the true population parameter.

14.6 Population Estimation in Jamovi

Jamovi makes it easy to estimate population and construct confidence intervals, allowing you to see the range within which the true population parameter will likely fall.

14.7 Choosing the Right Population Estimation

Selecting the appropriate approach to population estimation depends on the type of data you’re working with and the goals of your analysis. In quantitative research, the most common task is estimating the population mean using the sample mean, particularly when working with continuous, interval, or ratio-level variables.

If your variable is continuous and measured on a consistent scale, the sample mean can be used to estimate the population mean. In these cases, it’s important to also consider the standard error, which tells you how much the sample mean might vary from one sample to another, and to use confidence intervals to express the uncertainty around your estimate.

The choice of confidence level (such as 90%, 95%, or 99%) depends on how much certainty you require. Higher confidence levels offer greater certainty but less precision, while lower confidence levels provide more precise estimates but less certainty. When reporting estimates, it’s good practice to include both the sample mean and the confidence interval to give readers a clear understanding of both the central estimate and the range of plausible values for the population mean.