Sample size worksheet: plan a human share or a two-group test

Use the one-proportion worksheet to plan the precision of one human survey share, or the comparison worksheet to plan one independent binary difference. At 95% confidence, expected 50% and ±5 points, the large-population requirement is 385 valid answers. A 50%-versus-60% comparison at 80% power needs 388 per group. Choose the decision and assumptions before treating either number as a recruitment target.

Start with the decision, then choose the calculation

“How many responses?” is incomplete until the study says what it must estimate or compare. One overall share needs a precision target. A difference between two independent randomized groups needs anticipated proportions, a decision-relevant effect and power. A paired task, several concept comparisons or a small segment decision needs a different plan.

If your decision concerns the share of buyers who understand instructions, choose how much uncertainty the recommendation can tolerate. If you are comparing offers, choose the smallest difference worth acting on before collection. Entering the observed gap afterward describes a different exercise and cannot turn an underplanned study into a prospectively powered one.

Use the spreadsheet worksheets below to calculate either target, then keep the inputs and result with the study brief. The formulas are copyable; this page presents the method and worked results rather than an embedded calculator.

Set the one-share inputs deliberately

1. Choose confidence: 90%, 95% or 99%.
2. Enter the desired half-width in percentage points: 5 means ±5 points, rather than 5% of the estimated value. The input is greater than 0 and less than 100.
3. Enter an expected proportion strictly between 0% and 100%. Use 50% when no defensible prior estimate exists; it maximizes the variance term in this formula.
4. Leave population size blank for the large-population approximation, or enter a positive whole number for the entire eligible population sampled without replacement.
5. Calculate the rounded-up completed-response requirement and save its inputs in the sheet.

A CRM list size is the appropriate population only when that list is the population for your claim. A list of 1,000 customers cannot shrink the sample requirement for a broader prospect population it does not cover.

Confidence describes the long-run performance of the interval procedure under the sampling assumptions; it is not the probability that the fixed population share lies in this particular completed interval. The worksheet uses a normal-approximation planning formula, so the achieved data and design still matter.

A five-point half-width means an interval extending about five percentage points on either side of a share estimate. At an estimate of 40%, that describes roughly 35% to 45%, rather than 38% to 42%. Save the units with the recorded inputs so another analyst can reproduce the plan.

The expected share controls the variance term p(1−p). Fifty percent is conservative within this formula because that term reaches its maximum at 0.5. A lower requirement based on a confident prior share needs evidence relevant to this population, stimulus and measure. An attractive prior from a different task can create false economy.

Precision becomes expensive quickly

All four examples use 95% confidence and an expected share of 50%:

Moving from a five-point margin to three points changes the unrounded requirement by (5/3)², about 2.78 times. Halving the margin changes it by four times. The inverse-square relationship explains why slightly narrower reporting can require much more fieldwork.

The 278 example is different: it uses a finite population of 1,000 sampled without replacement. Use that adjustment only when those 1,000 units are the entire eligible population for the intended claim. A CRM list cannot reduce uncertainty for a larger prospect population it does not cover.

Reproduce the one-proportion calculation

For a large population, n₀=z²×p×(1−p)/e². With z approximately 1.959964, p=0.50 and e=0.05, n₀ is approximately 384.146. Round upward to 385 completed valid answers. Use fractions for proportions and half-widths.

For an entire finite population N sampled without replacement, n=n₀/[1+(n₀−1)/N]. With N=1,000, the unrounded result is approximately 277.73; round the final adjusted requirement upward to 278. Applying the correction after prematurely rounding is unnecessary.

The Penn State sampling lesson derives the one-share formulas. They are conditional normal approximations, not exact interval guarantees for every possible proportion and sample. The worksheet’s worked outputs can be checked from the formula independently of its interface.

Plan the effect you need to distinguish

The comparison worksheet plans one binary outcome in two different independent groups with equal allocation. At 50% versus 60%, two-sided alpha 0.05 and 80% power, the approximation gives about 387.34 per group. Round up to 388 per group before doubling to 776 total. The overall one-share target of 385 would not plan this comparison.

Alpha is the selected false-positive rate for the planned test under its assumptions. Power is the chance of detecting the specified difference when those anticipated proportions apply. They answer different questions from the half-width of one estimate. A study can estimate an overall share reasonably while lacking the ability to distinguish a small between-group effect.

Independent-group formula and its approximation

For p̄=(pA+pB)/2, n per group = [z(1−α/2)√(2p̄(1−p̄)) + z(power)√(pA(1−pA)+pB(1−pB))]² ÷ (pA−pB)². Proportions are fractions.

This is the dominant-tail normal approximation described in the statsmodels documentation. It ignores the negligible far tail in the common two-sided setting; it is not an exact binomial calculation. The formula adds neither a continuity nor a finite-population correction to this comparison.

Record the calculation, both planned proportions, power, alpha, per-group requirement and total in your study brief. Keep them beside our monadic price-test plan. Unequal allocation, paired answers and multiple comparisons require separate design work.

Copy the sample size worksheets

Copy each tab-separated block into A1 of its own blank spreadsheet sheet. The label goes in column A and the input or formula in column B. Use Excel or Google Sheets functions with decimal points and comma-separated arguments; adapt separators if your spreadsheet locale requires them. Keep the cell positions unchanged.

One-share worksheet

Enter confidence as 0.90, 0.95 or 0.99; use fractions for expected share and half-width. Both must be strictly between 0 and 1. Leave B4 blank for a large population, or enter the positive whole-number population for the eligible units sampled without replacement. The starting values produce 385 valid answers; setting B4 to 1000 produces 278.

Confidence	0.95
Expected share	0.5
Half-width	0.05
Population	
z	=NORM.S.INV((1+B1)/2)
Large-population n	=B5^2*B2*(1-B2)/B3^2
Adjusted n	=IF(B4="",B6,B6/(1+(B6-1)/B4))
Valid answers	=MAX(1,ROUNDUP(B7,0))

Two-group worksheet

On a separate sheet, enter confidence as 0.90, 0.95 or 0.99 and power as 0.80, 0.90 or 0.95. Both planned shares must be strictly between 0 and 1 and must differ. The starting values produce 388 per group and 776 total. Equal shares give a zero denominator and have no finite detection target under this formula. Do not substitute a different comparison design into these cells.

Confidence	0.95
Power	0.8
Share A	0.5
Share B	0.6
Mean share	=(B3+B4)/2
z alpha	=NORM.S.INV((1+B1)/2)
z power	=NORM.S.INV(B2)
Unrounded per group	=(B6*SQRT(2*B5*(1-B5))+B7*SQRT(B3*(1-B3)+B4*(1-B4)))^2/(B3-B4)^2
Valid per group	=ROUNDUP(B8,0)
Valid total	=2*B9

Before relying on either result, check the anticipated success and failure counts in each relevant cell. Small expected counts, often fewer than ten, call for a more suitable exact or simulation-based design. The spreadsheet arithmetic does not check coverage, dependence, response bias or a missing subgroup plan for you.

Keep the planning record

Save a copy of the sheet for each agreed plan. The formulas let a colleague reproduce the conditional target; the record explains what that target is intended to support.

Plan the unit and the segment that drive the decision

The one-share calculation assumes the stated simple random sampling setting. The comparison calculation assumes independent observations, equal cells and one planned binary difference. Neither automatically accounts for clustering, weighting, coverage, nonresponse or small subgroup decisions.

If a study needs separate precise answers for two mutually exclusive segments, plan each relevant segment base. A large overall count can still leave the decision-critical segment small. If several people belong to the same purchasing account or household, decide whether the unit of inference is the person or that shared unit before treating them as independent observations.

An opt-in panel or convenience sample does not inherit simple-random-sample coverage merely because the arithmetic produces a target. Increasing responses can reduce some variability while leaving systematic participation or measurement error. Keep the achieved sampling design and limitations beside the number.

Convert a valid-answer target into a recruitment funnel

A hypothetical study needing 385 valid answers at 50% completion requires about 770 eligible invitations if every completed questionnaire supplies a usable primary answer. That is an arithmetic illustration, not a forecast of your response rate. Ineligibility and item missingness introduce additional losses.

Keep invited, eligible, completed and valid primary-answer counts separate. Base anticipated rates on suitable prior evidence and monitor important groups during collection. More invitations can address volume while unequal participation still changes who contributes to the result. Do not cure a shortfall by silently relaxing eligibility or dropping an inconvenient segment.

Save the planning sheet, then record achieved bases and exclusions against it. If the study stops below target, report the actual uncertainty or analysis limit rather than presenting the original target as completed fieldwork.

Use validation for modeled evidence

Creating 10,000 generated records does not create 10,000 independently sampled humans. Shared model, prompt and grounding choices can produce error that neither calculator mode measures. A human survey margin of error cannot be transferred to a simulated population by inserting its generated row count.

For modeled comparisons, inspect task-relevant validation against appropriate human evidence and sensitivity to justified construction changes. For human studies, finish the sampling and measurement plan before recruiting. The useful output from this worksheet is a reproducible conditional target attached to the right research question.

Further reading

• our research sampling guide
• our quantitative research guide

Bring the decision, audience and evidence needed to a scoped Sapien study discussion.