Sample Size Calculator

Margin of Error (±e)	90% Confidence	95% Confidence (Standard)	99% Confidence
±10.0%p	68 respondents	96 respondents	166 respondents
±5.0%p	270 respondents	383 respondents	660 respondents
±3.1%p	700 respondents	990 respondents	1,697 respondents
±2.0%p	1,664 respondents	2,345 respondents	3,983 respondents
±1.0%p	6,337 respondents	8,763 respondents	14,230 respondents

Sampling Theory & Statistical Inference Guide

Mathematical Architecture of Sample Size Calculation & Cochran's Formula

Conducting a full census of an entire population is prohibitively expensive and time-consuming. Instead, researchers collect data from a representative Sample to estimate population parameters.

If a sample is too small, confidence intervals widen and statistical significance is lost; if a sample is excessively large, financial and operational resources are wasted. Mathematical statistics quantifies this balance through the Confidence Level and Margin of Error.

This guide breaks down William G. Cochran's formula for finite and infinite populations, demonstrating why 1,000 respondents serves as the golden standard for national public opinion polls worldwide.

Cochran's Formula & Finite Population Correction (FPC)

Calculates base infinite population size ( $n_0$ ) and applies finite correction adjustments ( $n$ ) based on exact population size ( $N$ ).

90%, 95%, and 99% Confidence Level Mapping

Maps standardized Z-score critical values from academic standards ( $Z=1.96$ at 95%) to stringent clinical standards ( $Z=2.576$ at 99%).

12-Cell Confidence × Margin Matrix Grid Table

Dynamically generates a comparison matrix showcasing sample sizes needed across $pm 1%$ to $pm 10%$ margins of error.

5 Real-World Research Presets

One-click presets for national presidential election polls, academic theses, B2B SaaS CSAT, and digital marketing A/B tests.

1. 3 Intuitive Everyday Analogies for Sampling

① The Soup Pot Analogy (Why 1,000 people represent 50 million):

- Whether tasting a 1-person bowl of soup or a 100-person military banquet cauldron, tasting one well-stirred spoonful tells you the saltiness of the entire pot.

- In statistics, Random Sampling is the act of stirring the soup. As long as the mix is thoroughly randomized, the size of the spoon (~1,000 people) determines precision, regardless of whether the pot holds 100,000 or 50,000,000 people.

② The Hospital Blood Test Analogy:

- Doctors do not drain all 5 liters of blood in your body to measure cholesterol or blood sugar; testing a single 5cc vial (sample) provides an accurate diagnosis.

③ The Crate of Oranges Analogy:

- You don't need to peel every single orange in a crate to determine sweetness; tasting 3 to 5 sampled oranges from different layers reveals the average quality.

2. 3 Essential Statistical Terms Explained

① Population ( $N$ ): The total group of individuals you want to understand (e.g. 44 million national voters, 15,000 college students, or 500 corporate staff).

② Confidence Level (Typically 95%): How certain you can be. A 95% confidence level means that if the survey were repeated 100 times, 95 of those surveys would capture the true population parameter within the margin of error.

③ Margin of Error (Typically ±3% to ±5%): The allowable range of deviation. If a candidate receives 50% with a

\pm 3\%p

margin, their true support lies between

47\%

and

53\%

3. Recommended Sample Sizes by Research Scenario

① Undergraduate & Master's Dissertations:

- Recommended: 300 to 400 respondents (95% confidence,

pm 5%p

margin). The standard baseline for academic thesis defense committees.

② B2B SaaS / Product Customer Satisfaction (CSAT):

- Recommended: 200 to 350 respondents. Sufficient to identify product roadblocks and feature prioritization trends.

③ Internal Employee Culture Surveys:

- Recommended: 30% to 50% of staff. For a 500-person company, finite population correction requires ~218 respondents for a

pm 5%p

margin.

④ National Media Polls (Presidential / General Elections):

- Recommended: 1,000 respondents. Delivers a

pm 3.1%p

margin with optimal cost-benefit efficiency.

⑤ A/B Testing Conversion Optimization:

- Recommended: 2,000 to 5,000+ users. Necessary to detect small

1%

2%

conversion lift differences with statistical confidence.

4. Mathematical Formulas for Sample Size (Cochran & FPC)

① Cochran's Formula for Infinite Populations:

\displaystyle n_0 = \frac{Z^2 \cdot p \cdot (1 - p)}{e^2}

Z

: Critical value of standard normal distribution (

90\% \to 1.645

95\% \to 1.96

99\% \to 2.576

)

p

: Estimated proportion with the attribute (

p = 0.5

maximizes variance

p(1-p)=0.25

for conservative sizing)

e

: Margin of error (

5\% = 0.05

)

② Finite Population Correction (FPC):

\displaystyle n = \frac{n_0}{1 + \frac{n_0 - 1}{N}} = \frac{N \cdot Z^2 \cdot p \cdot (1 - p)}{e^2 (N - 1) + Z^2 \cdot p \cdot (1 - p)}

N

: Total Population Size

③ Taro Yamane Simplified Formula (

p=0.5, 95\%

Confidence Approximation):

\displaystyle n = \frac{N}{1 + N \cdot e^2}

5. Sample Size Requirements by Margin of Error at 95% Confidence (Infinite Population)

Exponential growth of required sample size as margin of error tightens.

Margin of Error (±e)	Z-Score (95%)	Required Sample (n)	Relative Sample Multiplier	Typical Research Field
±10.0%p	Z = 1.96	97 respondents	1.0x (Baseline)	Exploratory discovery interviews & pilot studies
±5.0%p (Standard)	Z = 1.96	385 respondents	~4.0x	Academic journal articles, master theses, consumer surveys
±3.1%p (Polls)	Z = 1.96	1,000 respondents	~10.3x	National political polls, presidential approval surveys
±2.0%p	Z = 1.96	2,401 respondents	~24.8x	High-volume A/B testing & blind product evaluations
±1.0%p (Clinical)	Z = 1.96	9,604 respondents	~99.0x	National census sub-sampling & clinical trials

Frequently Asked Questions (FAQ)

Q.What should I enter if the population size is unknown?

Leave the Population Size field blank. The calculator will default to an Infinite Population ( $N \to \infty$ ). Statistically, once a population exceeds 50,000-100,000, sample size requirements level off asymptotically.

Q.Why is Expected Proportion (p) set to 50% by default?

The variance term $p(1 - p)$ reaches its maximum of $0.25$ when $p = 0.5$ . Assuming 50% produces the most conservative, robust sample size that ensures your margin of error will never be exceeded.

Q.What do a 95% confidence level and ±5% margin of error mean?

They mean that if the same survey were run 100 times independently, 95 of those iterations would find the true population parameter within $pm 5%p$ of the survey result.

Q.Does increasing the sample size by 10x make the survey 10x more accurate?

No. Statistical error decreases in proportion to the inverse square root of sample size ( $\frac{1}{\sqrt{n}}$ ). To reduce error by 10x, you must collect $10^2 = 100\text{ times}$ more respondents.

Q.Is my research parameter data saved on external servers?

No. All statistical operations run locally within your browser memory.

PARAMETERS

Confidence Level & Margin of Error Matrix Table