## 6. Standard error and confidence intervals

If we wished to find the overall mean value for a population, we could take several subsamples and calculate an independent value for the mean of each. It can be shown that the standard deviation of the means of N measurements from a population with an overall standard deviation of σ (Greek lowercase 'sigma') is given by:

σ/√N

This quantity is termed the '**standard error**' of the population mean, and defines a range either side of the estimated population mean that is likely to contain the true value.

Where a population has been sampled several times, and the samples are normally distributed, the standard error of the mean of the sample (s_{xmean}) can be estimated from the standard deviation calculated in section 5:

s_{xmean} = s/√N

The standard error of a sample mean can be multiplied by a value, usually denoted by the letter t, to provide what is termed a '**confidence interval**' (CI). This assigns a probability that the interval between the sample mean minus the CI and the sample mean plus the CI will include the value of the population mean.

The confidence interval is calculated as:

CI = mean ± t . s_{xmean}

For example, if a sample size of 15 has a mean value of 100 and a standard error of 10, then the value of t for 15-1 = 14 degrees of freedom and a probability of 0.95 is 2.145. The upper CI is given by 100 plus (10 × 2.145) equals 121.45, and the lower CI is given by 100 minus (10 × 2.145) equals 78.55, more conventionally written as:

95% confidence interval = 100 ± 21.45

This indicates that the true population mean has a 95% probability of being within the specified confidence interval around the sample mean (or a 5% chance of falling outside this range).

The value of t decreases as sample size increases, so that the ability to predict the population mean from the sample mean improves with larger sample sizes for a given standard deviation. As an illustration, for the same mean of 100 and standard error of 10:

Number of samples (df = n-1)* |
90% CI† | 95% CI | 99% CI |

α = 0.10† | α = 0.05 | α = 0.01 | |

5 | 79.68 - 121.32 | 72.24 - 127.76 | 53.96 - 146.04 |

10 | 81.67 – 118.33 | 77.38 – 122.62 | 67.50 – 132.50 |

15 | 82.39 – 117.61 | 78.55 – 121.45 | 70.23 – 129.77 |

20 | 82.71 – 117.29 | 79.07 – 120.93 | 71.39 – 128.61 |

30 | 83.01 – 116.99 | 79.55 – 120.45 | 72.44 – 127.46 |

* df denotes degrees of freedom for critical values of t,

† a confidence interval (CI) of 90% equates to a significance level, *α*, of 0.10, i.e. a probability of 0.1 (10%) of the mean falling outside the CI.