MPH 560 Module 3 Sampling and Confidence Intervals Example

How Sure Are We? Sampling, Standard Errors and Confidence Intervals in Public Health

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Master of Public Health Program, Aspen University

MPH 560: Applied Biostatistics for Public Health

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How Sure Are We? Sampling, Standard Errors and Confidence Intervals in Public Health

Public health rarely measures entire populations. Surveys, screenings and studies sample some people and use them to estimate values for everyone. Every such estimate carries uncertainty, and a good analyst reports how much. This paper explains sampling, the standard error and confidence intervals, working through two composite examples: a mean blood pressure and a hypertension prevalence from a county survey.

Populations and Samples

A population is the whole group of interest; a sample is the part actually measured. A random sample gives every member a known chance of selection, allowing probability theory to describe how sample results vary. Convenience samples, such as people who attend a screening event, may differ systematically from the population, and no statistical formula corrects that bias (Whitley & Ball, 2002).

Sampling Distributions

If many random samples of the same size were drawn, their means would vary, forming a sampling distribution. For reasonably large samples, this distribution is approximately normal and centered on the population mean, even when individual values are skewed. Its spread is measured by the standard error. This result, known as the central limit theorem, is what allows confidence intervals for means to be calculated even when individual values are skewed, provided the sample is not too small.

The Standard Error

To get the standard error of a mean, take the spread of individual values and scale it down by the root of how many people were measured. It measures the precision of the estimate, not the variability of individuals, and it shrinks as samples grow (Altman & Bland, 2005). Quadrupling the sample size halves the standard error.

Worked Examples

The table shows two calculations.

Interpreting the Intervals

The prevalence interval of 35.6% to 42.4% means that, if the survey were repeated many times with random samples, about 95% of intervals calculated this way would contain the true county prevalence. It does not mean there is a 95% probability that this particular interval contains the true value, though that informal reading is common. The interval shows a plausible range for the population value given the data.

Why Use t for Small Samples

For the blood pressure sample of 20, the multiplier comes from the t distribution rather than the normal distribution, because the standard deviation is itself estimated from a small sample. The t multiplier of 2.093 is larger than 1.96, producing a wider interval that reflects the additional uncertainty. As samples grow, t approaches 1.96.

Sample Size and Precision

If the county survey had included 3,200 adults instead of 800, with the same prevalence, the standard error would halve and the interval would narrow to roughly 37.3% to 40.7%. Planning sample size in advance ensures estimates will be precise enough for decisions, such as whether prevalence exceeds a program threshold.

Estimation Rather Than Testing Alone

An interval tells readers how big an effect might be and how uncertain that estimate is; a lone P value says nothing about either. Medical statisticians have long urged authors to report confidence intervals rather than relying only on hypothesis tests (Gardner & Altman, 1986). An interval for a difference that excludes zero also indicates statistical significance, so intervals give more information with no loss.

What Intervals Do Not Cover

Confidence intervals reflect random sampling error only. They do not account for bias from non-response, measurement error or unrepresentative samples. A county survey with a 40% response rate might produce a narrow interval around a biased estimate. Analysts should report response rates and consider weighting to adjust for known differences between respondents and the population.

Reporting

Good reports give the estimate, the confidence interval, the sample size and how the sample was drawn, for example: hypertension prevalence was 39.0% (95% CI, 35.6% to 42.4%) among 800 randomly selected adults. Rounding should match the precision of the data. When several estimates are presented together, such as prevalence by county, intervals should be shown for each so readers can see which differences are likely real.

Confidence Intervals for Differences

Intervals can also be calculated for differences between groups, such as the difference in prevalence between two counties. If the interval for a difference includes zero, the data are compatible with no difference; if it excludes zero, they suggest a real difference in the populations.

Survey Weights and Design Effects

Large health surveys often use complex designs with clusters and strata and weight responses to represent the population. These features change standard errors, usually widening them, so analysts must use methods that account for the design rather than simple formulas.

Practical Use

A county health department can use confidence intervals to judge whether a change in a rate from one year to the next is likely real or within the range of random variation. Overlapping intervals do not always mean no difference, so formal comparison of the two estimates is better.

Common Errors

Frequent errors include reporting a standard deviation where a standard error is needed, or the reverse; using 1.96 with very small samples; and treating a confidence interval as covering bias. Another is claiming precision that the sample size cannot support, such as reporting prevalence to two decimal places from 50 respondents.

Communicating Uncertainty

Decision-makers need uncertainty explained in plain language. Saying that between about 36% and 42% of adults are estimated to have hypertension conveys the range clearly. Charts with error bars or shaded intervals help audiences see uncertainty at a glance.

Conclusion

Sampling allows public health to estimate population values, and standard errors and confidence intervals express how precise those estimates are. In the composite examples, a mean systolic pressure of 137.9 had a 95% interval of 129.9 to 145.9, and a prevalence of 39.0% had an interval of 35.6% to 42.4%. Intervals are more informative than P values alone, but they address only random error, so sample design remains essential.

References

Altman, D. G., & Bland, J. M. (2005). Standard deviations and standard errors. BMJ, 331(7521), 903. https://doi.org/10.1136/bmj.331.7521.903

Gardner, M. J., & Altman, D. G. (1986). Confidence intervals rather than P values: Estimation rather than hypothesis testing. BMJ, 292(6522), 746-750. https://doi.org/10.1136/bmj.292.6522.746

Whitley, E., & Ball, J. (2002). Statistics review 2: Samples and populations. Critical Care, 6(2), 143-148. https://doi.org/10.1186/cc1473