# Sample size calculator

Calculates the sample size for a survey (proportion) or calculates the sample size for a normal confidence interval.

**proportion**,

**or**the

**Standard deviation (σ)**.

**Estimate value ± MOE**

**Leave empty for an infinite population.**When the sample size is less than 5% of the entire population you can assume an infinite population.

## What is the sample size?

The sample size is the number of observations in research, study, experiment, or survey. For example, the number of subjects participating in the research

A sample is only a subset of subjects from the entire population.

## Sample size formula

##### Sample size formula when using the population standard deviation (S)

n = ( | Z_{1-α/2} * σ | ) | ^{2} |

MOE |

Since Z is symmetrical, you may use Z_{α/2} or Z_{1 - α/2}.

##### Sample size formula when using the sample standard deviation (σ)

n = ( | t(n-1)_{1-α/2} * S | ) | ^{2} |

MOE |

Since n appears also in t(n-1), we run several iterations until finding the smaller sample size that results in MOE that is smaller or equal to the defined MOE:

MOE = | t(n-1)_{1-α/2} * S |

√n |

**n**- sample size.

**Z**- Z score from the standard normal distribution.

_{1-α/2}**t**- t score from the t distribution.

_{1-α/2}**σ**- the population standard deviation, you use it when you know it before the research.

**S**- the sample standard deviation, you use it when you don't know it before the research.

**MOE**- Margin Of Error, half-width of the confidence interval, for a smaller MOE mean, narrower confidence interval, you need a larger sample size.

**CL**- the Confidence Level is the required degree of certainty that the population parameter will be in the confidence interval.

**α**the error: α = 1 - CL.

### Is a larger sample size better?

From a statistical point of view, larger sample size is better, with a smaller margin of error.

Usually, a larger sample size costs more and takes more time to gather. Sometimes you even need to destroy each unit in the sample to get the result.

__Example__What sample size should you use for a survey in a city with a population of 120,000 people?

You want to have a confidence level of 0.95 (95%).

The confidence interval should be ± 0.04.

Survey question: "what party will you vote?"

__Solution__**Confidence level = 0.95**.

If you don't know what proportions to expect you should assume the

**worth case**meaning the largest standard deviation.

**Proportion = 0.5**

The largest standard deviation for a proportion is for proportion =

**0.5**, which means that one party will gain 50% of the votes.

**Population standard deviation (σ)**

- leave empty, the standard deviation will be calculated from the proportion.

**Margin of error (MOE) = 0.04 .**

**Population (N) = 120,000 .**