# Sample Size Calculator

You can use this free sample size calculator to determine the sample size of a given survey per the sample proportion, margin of error, and required confidence level.

You can calculate the sample size in five simple steps:

- Choose the required confidence level from the dropdown menu
- Input the margin of error
- Input the proportion of the total population (%)
- If required, specify the population size
- Click on the "Calculate" button to generate the results.

## What is Sample Size?

The sample size of a survey is the total number of complete responses that were received during the survey process. It is referred to as a sample because it does not include the full target population; it represents a selection of that population. For example, many studies involve random sampling by which a selection of a target population is randomly asked to complete a survey.

Some basic terms are of interest when calculating sample size. These are as follows:

**Confidence level:** The level of confidence of a sample is expressed as a percentage and describes the extent to which you can be sure it is representative of the target population; that is, how frequently the true percentage of the population who would select a response lies within the confidence interval. For example, if you have a confidence level of 90%, if you were to conduct the survey 100 times, the survey would yield the exact same results 90 times out of those 100 times.

**Margin of Error:** Margin of error is also measured in percentage terms. It indicates the extent to which the outputs of the sample population are reflective of the overall population. The lower the margin of error, the nearer the researcher is to having an accurate response at a given confidence level. To determine the margin of error, take a look at our margin of error calculator.

**Percentage of population selecting a given choice:** The accuracy of the research outputs also varies according to the percentage of the sample that chooses a given response. If 98% of the population select "Yes" and 2% select "No," there is a low chance of error. However, if 35% of the population select "Yes" and 65% select "No", there is a higher chance an error will be made, regardless of the sample size. When selecting the sample size required for a given level of accuracy, researchers should use the worst-case percentage; i.e., 50%.

**Population Size:** The population size is the total number of people in the target population. For example, if you were performing research that was based on the people living in the UK, the full population would be approximately 66 million. Likewise, if you were conducting research on an organization, the total size of the population would be the number of employees who work for that organization.

## Sample Size Formula

The Sample Size Calculator uses the following formulas:

1. n = z^{2} * p * (1 - p) / e^{2}

2. n (with finite population correction) = [z^{2} * p * (1 - p) / e^{2}] / [1 + (z^{2} * p * (1 - p) / (e^{2} * N))]

*Where:*

**n** is the sample size,

**z** is the z-score associated with a level of confidence,

**p** is the sample proportion, expressed as a decimal,

**e** is the margin of error, expressed as a decimal,

**N** is the population size.

**Example of a Sample Size Calculation: ** Let's say we want to calculate the proportion of patients who have been discharged from a given hospital who are happy with the level of care they received while hospitalized at a 90% confidence level of the proportion within 4%. What sample size would we require?

The sample size (n) can be calculated using the following formula:

n = z^{2} * p * (1 - p) / e^{2}

where z = 1.645 for a confidence level (α) of 90%, p = proportion (expressed as a decimal), e = margin of error.

z = 1.645, p = 0.5, e = 0.04

n = 1.645^{2} * 0.5 * (1 - 0.5) / 0.04^{2}

n = 0.6765 / 0.0016 = 422.816

n ≈ 423 patients.

Desired Confidence Level | Z-Score |
---|---|

70% | 1.04 |

75% | 1.15 |

80% | 1.28 |

85% | 1.44 |

90% | 1.645 |

91% | 1.70 |

92% | 1.75 |

93% | 1.81 |

94% | 1.88 |

95% | 1.96 |

96% | 2.05 |

97% | 2.17 |

98% | 2.33 |

99% | 2.576 |

99.5% | 2.807 |

99.9% | 3.29 |

99.99% | 3.89 |

**Reference:*** Daniel WW (1999). Biostatistics: A Foundation for Analysis in the Health Sciences. 7th edition. New York: John Wiley & Sons.*

You may also be interested in our Effect Size (Cohen's d) Calculator or Relative Risk Calculator