# Sample Size Calculator: Population Proportion

 A sample size calculator is a tool that helps entities to decide the best number of participants for a test. It calculates the minimum and maximum sample sizes in order to make sure that there will be enough data. It is also useful in determining how big the study should be. The calculation process is fairly simple, but it requires some knowledge of statistics and math. The sample size calculator has become essential in many industries today because it helps to save time, money, and effort when conducting marketing research or other types of studies. This calculator can be used to estimate the proportion of your population who, within a specified margin of error, possess specific characteristics (e.g., they like your product, own a car, speak a second language, etc.). The largest sample size across all questions should be used if more than one question is to be asked. When more than two valid answers to a question are available (for example, yes or no), then you will need a different sample size calculator (e.g., do not know).

# Calculator

5% is a common choice

%

The margin of error is the level of precision you require. This is the range in which the true proportion is estimated to be and should be expressed in percentage points (e.g., ±2%).

A lower margin of error requires a larger sample size.

Typical choices are 90%, 95%, or 99%

%

The confidence level specifies the amount of uncertainty associated with your estimate. This is the chance that the margin of error will contain the true proportion.

A higher confidence level requires a larger sample size.

If you don't know, use 100,000

How many people are there in the population from which you are sampling? The sample size doesn't change much for populations larger than 100,000.

If you're not sure, leave this as 50%

%

What do you expect the sample proportion to be? This can often be determined by using the results from a previous survey, or by running a small pilot study.

383

This is the minimum sample size you need to estimate the true population proportion with the required margin of error and confidence level.

# Alternative Scenarios

With a sample size of
Your margin of error would be
9.79%
3.08%
0.93%
With a margin of error of % % %
8763
2345
383
With a confidence level of % % %
270
383
660
With a population size of
80
278
370
With a sample proportion of
139
288
246

### Worked Example

If a retailer would like to estimate the proportion of their customers who bought an item after viewing their website on a certain day with a 95% confidence level and 5% margin of error, how many customers do they have to monitor?   Given that their website has on average 10,000 views per day and they are uncertain of their current conversion rate, then they would need to sample 370 customers.  If, however they know from previous studies that they would expect a conversion rate of 5%, then a sample size of 73 would be sufficient.

### Formula

This calculator uses the following formula for the sample size n:

n = N*X / (X + N - 1),

where,

X = Zα/22 ­*p*(1-p) / MOE2,

and Zα/2 is the critical value of the Normal distribution at α/2 (e.g. for a confidence level of 95%, α is 0.05 and the critical value is 1.96), MOE is the margin of error, p is the sample proportion, and N is the population size.  Note that a Finite Population Correction has been applied to the sample size formula.

### Discussion

The above sample size calculator provides you with the recommended number of samples required to estimate the true proportion mean with the required margin of error and confidence level.

You can use the Alternative Scenarios to see how changing the four inputs (the margin of error, confidence level, population size and sample proportion) affect the sample size.  By watching what happens to the alternative scenarios you can see how each input is related to the sample size and what would happen if you didn't use the recommended sample size. The larger the sample size, the more certain you can be that the estimates reflect the population, so the narrower the confidence interval. However, the relationship is not linear, e.g., doubling the sample size does not halve the confidence interval.

### Definitions

#### Margin of error

The margin of error is the the level of precision you require. This is the plus or minus number that is often reported with an estimated proportion and is also called the confidence interval. It is the range in which the true population proportion is estimated to be and is often expressed in percentage points (e.g., ±2%).  Note that the actual precision achieved after you collect your data will be more or less than this target amount, because it will be based on the proportion estimated from the data and not your expected sample proportion.

#### Confidence level

The level of confidence indicates the likelihood that the true proportion is within the margin of error. It is reasonable to assume that the true value will fall within this range on 95% of the occasions if the study is repeated and the range is calculated each time. You can be more confident that the interval contains the true proportion if your confidence level is higher.

#### Population size

This is the total number of distinct individuals in your population.  To account for sampling from small populations, we use a finite population correction in this formula. If you're not sure how big your population is, you can use 100,000 as a conservative estimate. If a population is over 100,000 people, the sample size doesn't change all that much.

#### Sample proportion

The proportion of the population that you expect to see in the results is known as the sample proportion. The results of a previous survey or a small pilot study can often be used to make this determination. Whenever you aren't sure, go with a conservative estimate like 50%, which will give you the largest possible sample size. Note that the Normal approximation to the Binomial distribution is used to calculate the sample size. In cases where the proportion of the sample is close to 0 or 1, this approximation is not valid and you should use a different method for calculating sample size.

#### Sample size

With the required margin of error and level of confidence, this is the bare minimum sample size you require. Be aware that if some respondents choose not to respond, they will be excluded from your sample. If this is a concern, you will need to increase your sample size. Non-response will often lead to biases in your estimate, so the higher the response rate, the better your estimate will be.