# 3.4 Relationship between Sample Size and Margin of Error

As discussed in the previous section, the margin of error for sample estimates will shrink with the square root of the sample size. For example, a typical margin of error for sample percents for different sample sizes is given in **Table 3.1** and plotted in **Figure 3.2**.

*Table 3.1. Calculated Margins of Error for Selected Sample Sizes*

Sample Size (n) | Margin of Error (M.E.) |
---|---|

200 | 7.1% |

400 | 5.0% |

700 | 3.8% |

1000 | 3.2% |

1200 | 2.9% |

1500 | 2.6% |

2000 | 2.2% |

3000 | 1.8% |

4000 | 1.6% |

5000 | 1.4% |

Let's look at the implications of this square root relationship. To cut the margin of error in half, like from 3.2% down to 1.6%, you need four times as big of a sample, like going from 1000 to 4000 respondents. To cut the margin of error by a factor of five, you need 25 times as big of a sample, like having the margin of error go from 7.1% down to 1.4% when the sample size moves from n = 200 up to n = 5000.

*Figure 3.2 Relationship Between Sample Size and Margin of Error*

In **Figure 3.2**, you again find that as the sample size increases, the margin of error decreases. However, you should also notice that there is a diminishing return from taking larger and larger samples. in the table and graph, the amount by which the margin of error decreases is most substantial between samples sizes of 200 and 1500. This implies that the reliability of the estimate is more strongly affected by the size of the sample in that range. In contrast, the margin of error does not substantially decrease at sample sizes above 1500 (since it is already below 3%). It is rarely worth it for pollsters to spend additional time and money to bring the margin of error down below 3% or so. After that point, it is probably better to spend additional resources on reducing sources of bias that might be on the same order as the margin of error. An obvious exception would be in a government survey, like the one used to estimate the unemployment rate, where even tenths of a percent matter.