In statistics, **Slovin’s formula** is used to calculate the minimum sample sized needed to estimate a statistic based on an acceptable margin of error.

Slovin’s formula is calculated as:

**n = N / (1 + Ne ^{2})**

where:

**n**: Sample size needed**N**: Population size**e**: Acceptable margin of error

The following examples show how to use Slovin’s formula in practice.

**Example 1: Using Slovin’s Formula to Estimate Population Proportion**

Suppose a lawyer wants to estimate the proportion of individuals in a certain neighborhood that are in favor of a new law.

Suppose he knows there are 10,000 individuals in this neighborhood and it would take far too long to survey each individual, so he would instead like to take a random sample of individuals.

Assume that he would like to estimate this proportion with a margin of error of .05 or less.

He can use Slovin’s formula to figure out the minimum number of individuals he must include in his sample:

- n = N / (1 + Ne
^{2}) - n = 10,000 / (1 + 10,000(.05)
^{2}) - n = 384.615

To be conservative, the lawyer should round up to the nearest integer and include **385** individuals in his sample.

**Example 2: Using Slovin’s Formula to Estimate Population Mean**

Suppose a botanist wants to estimate the mean height of a certain species of plant in some region.

Suppose she knows there are 500 of these plants in the region and it would take far too long to measure each individual plant, so she would instead like to take a random sample of plants.

Assume that she would like to estimate this mean with a margin of error of .02 or less.

She can use Slovin’s formula to figure out the minimum number of plants she must include in her sample:

- n = N / (1 + Ne
^{2}) - n = 500 / (1 + 500(.02)
^{2}) - n = 416.667

To be conservative, the botanist should round up to the nearest integer and include **417 **plants in her sample.

**Slovin’s Formula: The Relationship Between Sample Size & Margin of Error**

There is a simple relationship between sample size and margin of error: **The lower the margin of error, the larger the sample size needed**.

To illustrate this, consider the example from earlier when the lawyer wanted to estimate the proportion of individuals in a neighborhood who were in favor of a new law using a margin of error of **0.05**.

Since the total number of individuals in the neighborhood was 10,000, he used the following formula to calculate the minimum sample size needed for his survey:

- n = N / (1 + Ne
^{2}) - n = 10,000 / (1 + 10,000(.05)
^{2}) - n =
**384.615**

However, suppose the lawyer instead wanted a margin of error of **0.01**.

Here is how he would use Slovin’s formula to calculate the minimum sample size for this survey:

- n = N / (1 + Ne
^{2}) - n = 10,000 / (1 + 10,000(.01)
^{2}) - n =
**5,000**

Since the lawyer decreased his margin of error, his sample size increased.

This should make sense intuitively.

If you want a lower margin of error (i.e. a more accurate estimate) then you must include far more individuals in your sample.

**Bonus:** Feel free to use this Slovin’s Formula Calculator to automatically calculate a minimum sample size based on a population size and acceptable margin of error.

**Additional Resources**

The following tutorials provide additional information about sampling in statistics:

An Introduction to Types of Sampling Methods

Population vs. Sample: What’s the Difference?

The Relationship Between Sample Size and Margin of Error

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