# Sampling distribution of the sample mean tutorial

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## Sampling distribution of the sample mean Definition

**Sample mean: **Sample mean is the Arithmetic mean of a Small sample data drawn from a Population.

**Sampling Distribution:**

A sampling distribution acts as a frame of reference for the statistical decision-making process.

The sampling distribution allows us to identify whether, the given variability among all possible sample means, the one we observed is a common out-come or a rare outcome.

## Sampling Distribution for Sample Mean Formula

The Sampling Distribution of the Mean is the mean of the population from where the items are sampled. If the population distribution is normal, then the sampling distribution of the mean is likely to be normal for the samples of all sizes.

Its mean is equal to the population mean, thus,

The population standard deviation divided by the square root of the sample size is equal to the standard deviation of the sampling distribution of the mean, thus:

Where:

**σ** = population standard deviation

**n** = sample size

### Why we need the Sampling distribution of the sample mean?

The sampling distribution of a statistic is a probability distribution based on a large number of samples of size from a given population.

**Sampling distribution of the sample mean Example**

There is n number of athletes participating in the Olympics. We want to know the average height of them. Instead of measuring all of the athletes, we randomly sample twenty athletes and use the sample mean to estimate the population mean.

Denote the sample mean of the twenty athletes as X_{1}.

Suppose we take a separate sample of size twenty from the same.

Denote that sample mean as X_{2}. Would X_{1} and X_{2 }equal? Not necessarily.

What if we took another sample and found the mean? Consider now taking 5000 random samples of size twenty-five and recording all of the sample means.

We can take the 5000 sample means and create a histogram. This would give us a graphical idea of what the distribution of the sample means looks like.

The distribution of all of these sample means is the Sampling Distribution of the Sample Mean.

### How to Calculate the Sampling distribution of the sample mean?

A rowing team consists of four rowers who weigh 154, 158, 162, and 166 pounds. Determine all possible random samples with the replacement of size 2 and compute the sample mean for each one. Use them to determine the probability distribution, the mean, and the standard deviation of the sample mean x̅.

The following table shows all possible samples with the replacement of size two, along with the mean of each:

Sample | Mean |

154,154 | 154 |

154,158 | 156 |

154,162 | 158 |

154,166 | 162 |

Sample | Mean |

158,154 | 156 |

158,158 | 158 |

158,162 | 164 |

158,166 | 164 |

Sample | Mean |

162,154 | 158 |

162,158 | 160 |

162,162 | 162 |

162,166 | 164 |

Sample | Mean |

162,154 | 158 |

162,158 | 160 |

162,162 | 162 |

162,166 | 164 |

Sample | Mean |

166,154 | 160 |

166,158 | 162 |

166,162 | 164 |

166,166 | 166 |

The table shows that there are seven possible values of the sample mean x̅. The value x̅ =154 happens only one way (the rower weighing 154 pounds must be selected both times), as does the value x̅ =166, but the other values happen more than one way, hence are more likely to be observed than 154 and 166 are. Since the 16 samples are equally likely, we obtain the probability distribution of the sample mean just by counting:

“The Mean and Standard Deviation of a Discrete Random Variable” for the mean and standard deviation of a discrete random variable to x̅. For μ_{x} we obtain.

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