Hypothesis Tests
Yamane's Formula Calculator
Yamane's formula starts from the same base as a worst-case proportion, then inflates the sample for nonresponse and for the design effect.
Worksheet
Press Calculate to see the result.
Yamane's formula starts from the same base as a worst-case proportion, then inflates the sample for nonresponse and for the design effect.
What this calculator answers
Yamane's planning formula begins where a worst-case proportion sample ends, then it asks two field questions Slovin's page refuses. How many invitations must you send if only a fraction come back, and how much larger must the sample be if the design is less efficient than a simple random draw? The base quotient is N divided by 1 plus N times e squared. Dividing by the response rate lifts that base, and multiplying by the design effect lifts it again. The worksheet rounds up only after both adjustments, so a fractional invitation is not dropped in the middle.
You see the base quotient, the response-rate uplift, the design-effect product, and the final whole number of invitations. The page does not stop at the base the way Slovin's page does. Supporting counts and intermediate terms sit under the headline so a reader can see what was divided by what. When a checkbox or a tail setting changes the divisor or the rejection region, that choice appears in the steps rather than vanishing into a silent default.
The formula, written out
Yamane sample size with uplift uses this relationship.
base = N / (1 + N e^2), then invitations = base / response rate * DEFF, rounded up
- N. Population size.
- e. Margin as a proportion.
- response rate. Share of invitations you expect to get back, such as 0.8.
- DEFF. Design effect. Values above 1 inflate the sample.
The base is the finite-population size at p equal to 0.5, where p(1-p) hits its maximum of 0.25, with the normal cutoff absorbed near 2. Dividing by the response rate converts completed interviews into invitations, and the design effect then scales those invitations for clustering or weighting. The algebra is the named textbook identity for this worksheet. Intermediate products and quotients appear in order so a calculator and a pencil can agree to the displayed rounding. If a neighbouring page uses a different divisor or a continuity correction, that page says so on its own ticket.
A full worked example
Example loaded in the worksheet
The loaded plan uses N of 1000, a margin of 0.05, a response rate of 0.8, and a design effect of 1.
Population N is 1,000 and the margin e is 0.05. The P equal to 0.5 bound is why this formula never asks for a guessed proportion: p times (1 minus p) is largest at one half, where it equals 0.25, and the base sizes the sample for that worst case. The base is N divided by 1 plus N e squared, which equals 285.7143.
Divide the base by the response rate 0.8 to get 357.1429 invitations before the design effect. Multiply by the design effect 1 to get 357.1429. Rounding up 357.1429 gives 358 invitations to send.
A design effect of 1 would leave the post-response figure unchanged. The value typed here is 1. Slovin's page stops at the base and does not apply the 0.8 response division or this design effect.
| Invitations | 358 |
|---|---|
| Unrounded final n | 357.1429 |
| Base n | 285.7143 |
| After response uplift | 357.1429 |
A second situation
The same base plan with a design effect of 1.5
Population N is 1,000 and the margin e is 0.05. The P equal to 0.5 bound is why this formula never asks for a guessed proportion: p times (1 minus p) is largest at one half, where it equals 0.25, and the base sizes the sample for that worst case. The base is N divided by 1 plus N e squared, which equals 285.7143.
Divide the base by the response rate 0.8 to get 357.1429 invitations before the design effect. Multiply by the design effect 1.5 to get 535.7143. Rounding up 535.7143 gives 536 invitations to send.
A design effect of 1 would leave the post-response figure unchanged. The value typed here is 1.5. Slovin's page stops at the base and does not apply the 0.8 response division or this design effect.
| Invitations | 536 |
|---|---|
| Unrounded final n | 535.7143 |
| Base n | 285.7143 |
| After response uplift | 357.1429 |
Raising the design effect from 1 to 1.5 multiplies the post-response size by 1.5 and changes the invitation count. The population and the margin stay at the loaded example, so the movement is only the design penalty. The second example changes the feature that this formula is sensitive to, and the ticket shows the new figure beside the same definition. That contrast is the fastest way to see which input owns the movement.
What moves when the input moves
The expected response rate fell from 0.8 to 0.6. N, e, and DEFF stayed put. The headline result went from 358 to 477.
The nudged inputs
Population N is 1,000 and the margin e is 0.05. The P equal to 0.5 bound is why this formula never asks for a guessed proportion: p times (1 minus p) is largest at one half, where it equals 0.25, and the base sizes the sample for that worst case. The base is N divided by 1 plus N e squared, which equals 285.7143.
Divide the base by the response rate 0.6 to get 476.1905 invitations before the design effect. Multiply by the design effect 1 to get 476.1905. Rounding up 476.1905 gives 477 invitations to send.
A design effect of 1 would leave the post-response figure unchanged. The value typed here is 1. Slovin's page stops at the base and does not apply the 0.6 response division or this design effect.
| Invitations | 477 |
|---|---|
| Unrounded final n | 476.1905 |
| Base n | 285.7143 |
| After response uplift | 476.1905 |
How to read the result
Read the final integer as invitations to send, not as completed forms. The base, before the response division, is the completed-interview target under a simple random sample. Say the unit, the sample size or contract terms, and the formula name in the same sentence as the figure. A number without those three pieces invites a later reader to invent a definition that this page never used.
Quote the base, the response rate, the design effect, and the rounded invitations together. An invitation count without the response rate looks like a completed sample. Write the result with its unit, the inputs that produced it, and the name of the formula. If the ticket shows a decision sentence, quote the alpha and the tail with it. If the ticket shows a money or dose figure, quote the rate and the schedule that produced it.
Assumptions that have to stay true
- The margin is a proportion and the unknown rate may be as high-variance as one half.
- The response rate is the share you expect to complete, written as 0.8 rather than 80.
- The design effect is a multiplier from the sampling plan, with 1 meaning simple random sampling.
- Nonresponse is missing completely at random. A biased refusal pattern is not repaired by sending more of the same invitations.
If an assumption above is false, do not average the discomfort away. Change the worksheet or change the question. A correct arithmetic step on the wrong model is still the wrong answer for the decision you are trying to make.
Mistakes that change the answer
- Stopping after the base and calling that the mailing list. The uplift is the point of this page.
- Typing the response rate as a percent above 1.
- Putting the design effect on Slovin's page and expecting it to stick. That page leaves DEFF out on purpose.
- Rounding up before the response division, which can miss an extra invitation.
Where a different calculator is the honest choice
Slovin's formula calculator stops at the base and will not divide by a response rate. Cochran's formula calculator is the right move when you know z and a planning proportion other than the worst case, or when you are sizing a mean. Related links under the essay open pages that keep their own formulas instead of hiding a mode switch inside this ticket.
When the job changes, move to a page built for that job: Slovin's Formula Calculator, Cochran's Formula Calculator, One Proportion Z-Test Calculator.
Check yourself
Population 1000, margin 0.05, response rate 0.8, and DEFF equal to 1. How many invitations does the page round up to?
The base is 285.7143, the response uplift reaches 357.1429, and rounding up after a design effect of 1 gives 358 invitations.
Using the result in a sentence
Mail surveys of a known customer file and household samples drawn in clusters are the usual homes for the two uplifts. A simple random poll with full response can stay on Slovin's page. Keep the count, the unit, and the assumption list beside any copy of the number that leaves this browser tab.
A wrong response rate scales the whole plan, and the page cannot see the refusals you have not received yet. DEFF below 1 shrinks the plan, which is rarely what a cluster sample should do. Pair the headline with the steps before you staff a route, set a grade boundary, or compare two contracts from a single number.
Questions people ask next
01What definition does the Yamane's Formula Calculator commit to?
This page uses Yamane sample size with uplift: base = N / (1 + N e^2), then invitations = base / response rate * DEFF, rounded up. The base is the finite-population size at p equal to 0.5, where p(1-p) hits its maximum of 0.25, with the normal cutoff absorbed near 2. Dividing by the response rate converts completed interviews into invitations, and the design effect then scales those invitations for clustering or weighting. If a textbook uses a different divisor, a different continuity correction, or a different rounding rule, the last digits can move, so the definition is printed above the result. The worksheet asks for Population N, Margin e, Response rate, Design effect DEFF, and it refuses a blank that the formula cannot interpret. Keep that printed definition in the sentence you write next to the figure so a later reader can rebuild the arithmetic without guessing which convention you chose.
02How should the worked example be read out loud?
In the loaded example the invitations is 358. Read the final integer as invitations to send, not as completed forms. The base, before the response division, is the completed-interview target under a simple random sample. Say the unit and the count before you say whether the number is large or small. Read the steps in order rather than jumping to the headline, because the intermediate terms are where a swapped percent and proportion usually hide. If the ticket shows both a primary figure and a comparison figure, name which one you are quoting.
03What is the fastest way to get a misleading result?
Stopping after the base and calling that the mailing list. The uplift is the point of this page. A wrong response rate scales the whole plan, and the page cannot see the refusals you have not received yet. DEFF below 1 shrinks the plan, which is rarely what a cluster sample should do. A second common failure is leaving a checkbox or tail setting in the wrong state and then treating the number as if the default had matched your plan. Reset the example, change one input, and confirm the ticket moves in the direction the formula predicts before you trust a live run.
04When should I leave this page for a different calculator?
Slovin's formula calculator stops at the base and will not divide by a response rate. Cochran's formula calculator is the right move when you know z and a planning proportion other than the worst case, or when you are sizing a mean. The related links under the essay go to those worksheets, which keep their own formulas instead of hiding them in a mode switch. If the question changed from a summary to a test, from a payment to a refinance, or from a raw score to a percentile, switch pages rather than forcing this ticket to answer a job it was not built for.
05What belongs in the sentence next to the number?
The base is 285.7143, the response uplift reaches 357.1429, and rounding up after a design effect of 1 gives 358 invitations. Quote the base, the response rate, the design effect, and the rounded invitations together. An invitation count without the response rate looks like a completed sample. A bare number without those three pieces cannot be audited when a score, a rate, or a lab value is challenged later. Store the inputs beside the result whenever the figure will travel into a report.