When to use finite vs infinite population?

Use finite population when you know the exact size of your population and it's less than 100,000 users. Cochran's formula includes a correction factor that optimizes resources. Use infinite population for populations greater than 100,000, where the simplified formula is mathematically equivalent (less than 1% difference).

The 5% rule states that if your sample represents less than 5% of the total population, you don't need the finite correction factor. Between 5-10% the correction is recommended, and greater than 10% it's essential to maintain statistical precision without wasting resources.

What confidence level should I use?

The 95% confidence level (Z=1.96) is the industry and academic standard, ideal for publications and critical decisions. Use 99% (Z=2.576) for mission-critical studies, 90% (Z=1.645) for agile business decisions, or 80% (Z=1.282) for rapid prototypes.

What margin of error is appropriate?

A ±5% margin of error offers an optimal balance between precision and resources. Reducing to ±3% can double or triple the required sample size. Increasing to ±7% may be acceptable for non-critical business decisions, significantly reducing required resources.

How many participants do I need for quantitative usability?

Quantitative usability requires a minimum of 40 participants to obtain statistically significant results. This allows quantifying usability problems with metrics like Task Success Rate (average TSR: 78%) and System Usability Scale (average SUS: 68 points).

What is the no-show adjustment?

The no-show adjustment compensates for the typical non-attendance rate in research studies. The calculator automatically applies a correction factor (typically 10-20%) to ensure you reach your target sample considering cancellations and absences.

What is Cochran's formula for sample size?

Cochran's formula calculates sample size for finite populations: n = (N × Z² × p × q) / ((N−1) × E² + Z² × p × q). Where N is the total population, Z the confidence value (1.96 for 95%), p the expected proportion, q its complement (1−p), and E the margin of error. It includes a correction factor that optimizes the sample when you know the exact population size.

What is the infinite population formula?

The simplified formula for infinite or unknown populations is: n = (Z² × p × q) / E². It does not require knowing the population size and is used when it exceeds 100,000 individuals or is unknown. With 95% confidence, 50% proportion, and ±5% margin, the result is always 385 participants.

How to choose between finite and infinite population formula?

If you know the exact size of your population and it's less than 100,000, use the finite formula (Cochran's) to optimize resources. If the population exceeds 100,000 or is unknown, use the infinite formula. The 5% rule indicates that if your sample is less than 5% of the population, both formulas give practically identical results.

What is the finite population correction factor (FPC)?

The finite population correction factor (FPC) is a mathematical adjustment included in Cochran's formula that reduces the required sample size when you know the total population. The greater the sample/population ratio, the greater the reduction. For example, for a population of 2,000 you need 323 instead of 385 participants (16% savings).

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Cochran's Formula: Finite Population

Cochran's formula is the statistical standard for calculating sample size when you know the exact size of your population. It includes a finite population correction factor (FPC) that reduces the required sample, optimizing resources without losing precision.

Cochran's formula with finite correctionn = (N × Z² × p × q) / ((N−1) × E² + Z² × p × q)

When to use this formula?

You know the exact size of your population (employees, registered users, clients)
The population is less than 100,000 individuals
Your sample represents more than 5% of the total population (5% rule)
You need statistical rigor for publications or critical decisions

Parameters

N: Total population size
Z: Z-value for confidence level (1.96 for 95%)
p: Expected proportion (0.5 if unknown)
q: 1 − p (complement of the proportion)
E: Acceptable margin of error (e.g., 0.05 for ±5%)

Practical example

You have 2,000 registered users and want to survey them with 95% confidence and ±5% margin of error.

With Cochran's formula: n = (2,000 × 3.84 × 0.25) / (1,999 × 0.0025 + 3.84 × 0.25) = 323 participants. Without the finite correction you'd need 385.

Simplified Formula: Infinite Population

When the population is unknown or exceeds 100,000 individuals, the finite correction factor has less than 1% impact. In these cases, the simplified formula is used, which does not require knowing the population size.

Simplified formula for infinite populationn = (Z² × p × q) / E²

When to use this formula?

You don't know the exact size of your population
The population exceeds 100,000 individuals
Your sample represents less than 5% of the total population
Market research with broad or public audiences

Parameters

Z: Z-value for confidence level (1.96 for 95%)
p: Expected proportion (0.5 if unknown)
q: 1 − p (complement of the proportion)
E: Acceptable margin of error (e.g., 0.05 for ±5%)

Practical example

You want to survey users of an app with millions of downloads, with 95% confidence and ±5% margin.

With the simplified formula: n = (3.84 × 0.25) / 0.0025 = 385 participants. This result is independent of the population size.

When to use each formula?

The choice between finite and infinite formula depends on two factors: whether you know your population size and what proportion of it your sample will represent.

Do you know the exact size of your population? If not → use the infinite formula.
Does the population exceed 100,000? If yes → use the infinite formula (the result is practically identical).
Will your sample be more than 5% of the population? If yes → use the finite formula (Cochran) to optimize resources.
When in doubt → use the infinite formula. It always gives an equal or larger sample, which is more conservative.

Quick comparison

Criterion	Finite (Cochran)	Infinite
Known population	Yes, required	Not necessary
Population size	< 100,000	> 100,000 or unknown
Sample vs population	> 5% of population	< 5% of population
Correction factor	Yes (reduces sample)	Not applicable
Typical result	Smaller sample (optimized)	Conservative sample
Use case	Companies, closed communities	General market, mass apps

Quantitative Sample Size Calculator UX

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Cochran's Formula: Finite Population

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Quantitative Sample Size Calculator UX

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Cochran's Formula: Finite Population

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Practical example

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When to use each formula?

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