Rounding and Practical Sample Size Planning

From a Formula to a Recruitment Plan

A sample-size calculation can produce a number such as 61.4, but a study cannot collect 0.4 of an observation. The calculation also usually describes the number of usable measurements needed, not the number of people who must be invited. Practical planning therefore involves more than evaluating a formula: round appropriately, account for nonresponse, and compare the resulting recruitment target with available resources.

For planning a confidence interval for a population mean, a common approximation uses a pilot estimate of the population standard deviation. The target margin of error, confidence level, and planning estimate together determine an initial required sample size. Because the population standard deviation is generally unknown, and because a final interval for a mean commonly uses a $t$ critical value, the calculation is a planning estimate rather than a promise that the final interval will have a particular width.

Definition: The required analyzable sample size is the number of observations that must be available for the planned analysis. A recruitment target is the number of eligible people or units to approach so that the study is expected to obtain that many analyzable observations. These numbers can differ when some people do not respond or their measurements cannot be used.

For a two-sided mean interval with target margin of error $E$, confidence level $C$, and planning estimate $s_{\text{plan}}$ for the population standard deviation, a widely used normal-critical-value approximation is

n_0=\left(\frac{z^*s_{\text{plan}}}{E}\right)^2,

where $z^*$ is the standard normal critical value for confidence level $C$. The resulting $n_0$ is generally not an integer, so the initial required sample size is

n_{\text{plan}}=\lceil n_0\rceil.

The ceiling symbol means “round up to the next integer,” even when the decimal part is small. Rounding to the nearest integer can leave the planned sample just below the size needed for the target precision. This calculation assumes that the planning estimate is relevant to the intended population and measurement, and that the sampling design and conditions for using a mean interval are appropriate. It does not correct a poor sampling design or a biased measurement process.

Why the Sample Size Is Rounded Up

The margin of error in the normal approximation is $z^*s_{\text{plan}}/\sqrt{n}$. To achieve a margin of error no greater than $E$, require

\frac{z^*s_{\text{plan}}}{\sqrt{n}}\leq E.

Both sides are nonnegative, so rearranging and squaring gives $n\geq (z^*s_{\text{plan}}/E)^2=n_0$. Since $n$ must be a whole number, the smallest eligible choice is the least integer greater than or equal to $n_0$, namely $\lceil n_0\rceil$. If $n_0$ is already an integer, no additional observation is needed; otherwise rounding down violates the inequality.

Worked Example: Rounding a Pilot-Based Calculation

Suppose a hypothetical team wants a 95% confidence interval for a population mean with a target margin of error of 3 minutes. A pilot study suggests a standard deviation of about 12 minutes. Assume a suitable random sample can be drawn, the pilot variability is a reasonable planning value, and the intended mean interval conditions are plausible.

For the initial normal approximation, use $z^*=1.96$. Substitute the planning values:

n_0=\left(\frac{1.96(12)}{3}\right)^2 =\left(\frac{23.52}{3}\right)^2 =7.84^2 =61.4656.

The required sample size under this approximation is $\lceil61.4656\rceil=62$ usable observations. Rounding down to 61 would not meet the approximate target: $1.96(12)/\sqrt{61}\approx3.011$ minutes. At 62, the approximate margin is $1.96(12)/\sqrt{62}\approx2.987$ minutes.

There is a further planning detail for a final $t$ interval. Its critical value depends on the degrees of freedom, $df=n-1$, and is slightly larger than 1.96 for these sample sizes. For example, with $n=64$, $t^*$ for a two-sided 95% interval and $df=63$ is about 1.998. Using the pilot standard deviation as a planning value gives an estimated margin of $1.998(12)/\sqrt{64}\approx2.997$ minutes. Thus a planner seeking to meet the target under this $t$-based approximation would use 64 rather than stopping at the normal-based result of 62. In an actual study, the observed sample standard deviation may differ from the pilot estimate, so the final margin is not guaranteed to be at most 3 minutes.

Allowing for Nonresponse

A recruitment target must account for the fact that not everyone approached will provide a usable measurement. If the anticipated response rate is $r$, expressed as a proportion between 0 and 1, a basic adjustment is

N_{\text{recruit}}=\left\lceil\frac{n_{\text{usable}}}{r}\right\rceil.

This is based on the expectation that a proportion $r$ of those recruited will provide usable data. It does not ensure that the realized response rate will equal $r$. Response rates can vary, and the number of usable observations can fall short of the expectation. A planner may therefore need an additional buffer when a shortfall would be costly, while recognizing that no fixed buffer eliminates uncertainty.

Worked Example: Adjusting for Expected Nonresponse

A hypothetical study needs 64 usable measurements to plan for its desired precision. Past experience with a similar recruitment method suggests an 80% usable response rate. Assume this rate is a reasonable forecast for the new study and that each recruited person is counted once.

Here $r=0.80$, so the initial recruitment target is

N_{\text{recruit}} =\left\lceil\frac{64}{0.80}\right\rceil =\lceil80\rceil =80.

The arithmetic says that 80 recruits would yield 64 usable observations on average if the 80% rate applied. It does not mean that exactly 64 responses are certain. If only 60 usable measurements are obtained, the study has not reached its planned analyzable sample size, even though it approached 80 people. The team would need to decide in advance whether it can recruit additional people, extend recruitment, or accept less precision.

Nonresponse also creates a separate concern from sample size. If people who do not respond differ systematically from those who do, increasing the number invited may increase the number of responses without removing that difference. Inflating a recruitment target addresses a likely shortage of observations; it does not, by itself, make respondents representative of nonrespondents.

Checking Cost and Feasibility

A target can be statistically clear yet impractical. The recruitment target should be compared with the available budget, the number of eligible people, staff capacity, time, and any limits on collecting measurements. Be explicit about which costs are fixed and which apply per person. If a per-recruit cost includes unsuccessful contacts, use the recruitment total—not the desired number of completed measurements—to estimate that expense.

Worked Example: When the Budget Falls Short

Continue with the hypothetical plan requiring 64 usable measurements and expecting an 80% usable response rate. Suppose the study has a budget of $1,800, including $300 in fixed costs, and each person approached costs $20. Assume these costs are accurate for planning and that there are enough eligible people to recruit if funding allows.

After fixed costs, the budget available for recruitment is $1800-300=\$1{,}500$. The maximum affordable number of recruits is

\left\lfloor\frac{1500}{20}\right\rfloor =\lfloor75\rfloor =75.

At an 80% expected response rate, 75 recruits correspond to $75(0.80)=60$ expected usable measurements, fewer than the required 64. Recruiting the full target of 80 would cost $\$300+80(\$20)=\$1{,}900$, which exceeds the budget by $100. Under these assumptions, the desired plan is not attainable within the stated budget.

If exactly 60 usable measurements were obtained and the planning standard deviation remained 12 minutes, a rough $t$-interval margin estimate would be $2.001(12)/\sqrt{60}\approx3.10$ minutes, using $df=59$. This is larger than the 3-minute target. It is only a planning estimate: the final interval uses the observed standard deviation, and the actual number of responses may differ from 60.

The team can now make a transparent choice: seek more funding or lower-cost recruitment, improve the response process, revise the target margin or confidence level with a clear justification, or accept that the planned precision is unlikely. Simply reporting 60 expected responses as though they met the 64-observation requirement would conceal the shortfall.

What to Do When the Target Is Not Attainable

If resources or the eligible population impose a firm limit, calculate the precision that the attainable sample is expected to provide. Under the normal approximation, this is

E_{\text{achievable}}\approx\frac{z^*s_{\text{plan}}}{\sqrt{n_{\text{usable}}}}.

For a $t$ interval, use $t^*s_{\text{plan}}/\sqrt{n_{\text{usable}}}$ as a planning estimate instead, with $df=n_{\text{usable}}-1$. Report the assumptions and the fact that the estimate depends on the planning standard deviation. This calculation makes the consequence of a resource limit visible; it does not convert a shortfall into successful attainment of the original target.

Possible responses include expanding the eligible recruitment pool, reducing costs per recruit, improving response procedures, or revising the study plan. A smaller margin of error requires more observations, all else equal. A lower confidence level can reduce the required sample size, but it also changes the coverage goal and should not be selected merely to make a budget work. A larger allowable margin accepts less precision. Any change should be explained in terms of the research purpose, not disguised as a calculation adjustment.

Common Mistakes

Rounding to the nearest integer. If the calculation gives 61.46, rounding to 61 falls below the required threshold. Use the ceiling, 62, then check the margin at the rounded value.
Treating invited people as analyzable observations. The sample-size formula concerns usable measurements. Divide by the anticipated usable response rate to estimate how many people to recruit.
Assuming the response-rate adjustment guarantees the target. An expected rate is not a certainty. Plan for possible shortfalls and monitor the number of usable observations.
Assuming more recruitment removes nonresponse bias. A larger respondent count may improve precision but does not automatically correct systematic differences between respondents and nonrespondents.
Presenting a planning estimate as a guaranteed final margin. The actual sample standard deviation, response count, and—when using a $t$ interval—critical value affect the final interval width.

Key takeaway: Round the calculated analyzable sample size upward, adjust the recruitment target for expected nonresponse, and compare that target with actual resources. State clearly which quantities are required, which are expected, and which remain uncertain.

Check Your Understanding

Use the planning ideas above to answer these questions.

A calculation gives $n_0=47.08$. What is the required integer sample size, and why is rounding to 47 inappropriate?
A study needs 90 usable observations and expects a 75% usable response rate. What recruitment target follows from the basic adjustment?
Why does a recruitment target based on an expected response rate not guarantee the needed number of usable observations?
A project can afford fewer recruits than its response-adjusted target. What can the planner calculate to describe the likely consequence for precision?
Explain why increasing the number of people invited may fail to remove bias caused by systematic nonresponse.