The standard error of the mean is also called the standard deviation of the mean. It is a method used to estimate the standard deviation of a sampling distribution. In the context of statistical data analysis, the mean and standard deviation of sample population data is used to estimate the degree of dispersion of the individual data within the sample but the standard error of the mean (SEM) is used to estimate the sample mean dispersion from the population mean. The standard error (SE) along with the sample mean is used to estimate the approximate confidence intervals for the mean. The estimation with a lower standard error of the mean (SEM) indicates that it has a more precise measurement. The standard error of mean is always smaller than the standard deviation, SEM gives the accuracy of the sample mean by measuring the sample-to-sample variability of the sample means. It describes how precise the mean of the sample is an estimate of the true mean of the population. The standard error of mean is calculated by using the following formula:

where,

- σ = Standard deviation of the population
- n = Size of the sample

**Example 1:**

Calculate the standard error of the mean of tablets not passing the test of dissolution which are collected from 10 different batches out of 20 tablets.

Number of tablets 3, 4, 4, 6, 5, 3, 3, 4, 3, 5

(Not passes the test)

**Solution:**

First, we have to calculate the standard deviation.

X | (X- X̄) | (X- X̄)^{2}^{} |

3 | -1 | 1 |

4 | 0 | 0 |

4 | 0 | 0 |

6 | 2 | 4 |

5 | 1 | 1 |

3 | -1 | 1 |

3 | -1 | 1 |

4 | 0 | 0 |

3 | -1 | 1 |

5 | 1 | 1 |

Σx=50 | Σ(X- X̄)^{2 }=10 |

**Example 2:**

If the standard deviation of the population is 15, the population means is 40, the sample mean is 30, and the confidence level of 95%. Then calculate the sample size.

**Solution:**

- z = Specific level of confidence (95%) i.e. z = 1.96
- σ = Standard deviation of population i.e. σ = 15
- d = Difference between population mean and sample mean i.e. d = 40 – 30 = 10

Make sure you also check our other amazing Article on : Methods of Sampling