# Standard error of the mean tutorial

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## Standard error of the mean Definition

The standard error is a method used to estimate the approximate standard deviation of a sampling distribution**.**

**Sampling Distribution**

The Sampling Distribution 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.

Suppose there are 1000 girls in a school and we are supposed to find the average height of these girls.

If we take random a set of 20 girls from the population measure their heights and find its mean as x̅_{1. }Then we take another random set of 20 girls from the population and measure their heights and find its mean as x̅_{2}.

Similarly, if took 20 such sets randomly and finds their means as

x̅_{1} x̅_{2} x̅_{3 }– – – – x̅_{20}

### Standard error of the mean formula

_{Where:} σ = standard deviation of the original distribution

N =Sample Size

σ_{m= }Standard Error.

### Why we need Standard error of the mean?

Standard error plays a very important role in the large sample theory. It forms the basis for the testing of a hypothesis.

The statistical inference involved in the creation of confidence interval is mainly on the basis of standard error.

The magnitude of standard error indicates an index of the precision of the estimate of the parameter. Standard Error is inversely proportional to the sample size, i.e. that a smaller sample size tends to produce greater standard errors.

### How to Calculate the Standard error of the mean?

### Standard error of the mean Example

_{Name} |
_{Jack} |
_{John} |
_{Jonny} |
_{Jaaz} |

_{Weight (in kg)} |
_{72} |
_{74} |
_{78} |
_{84} |

Now find the Deviation

_{Name} |
_{Jack} |
_{John} |
_{Jonny} |
_{Jazz} |

_{Weight (in kg)} |
_{72} |
_{74} |
_{78} |
_{84} |

_{Deviation} |
_{72-77 = -5} |
_{74-77 = -3} |
_{78-77 = 1} |
_{84-77 = 7} |

Now perform squares of all the Deviations

_{Name} |
_{Jack} |
_{John} |
_{Jonny} |
_{Jazz} |

_{Weight (in kg)} |
_{72} |
_{74} |
_{78} |
_{84} |

_{Deviation} |
_{72-77 = -5} |
_{74-77 = -3} |
_{78-77 = 1} |
_{84-77 = 7} |

_{Square of Deviation} |
_{25} |
_{9} |
_{1} |
_{49} |

Now find Standard Deviation by taking a mean of Square of Deviations:

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