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# Estimated Mean of a Population

This online calculator allows you to estimate mean of a population using given sample

Let's suppose you have number of values, randomly drawn from some source population (these values are usually referred to as a sample). For given sample you can calculate the mean and the standard deviation of the sample. But the question is - what is the mean and the standard deviation of the source population. Intuitively, you feel that, of course, the sample mean isn't equal to the source mean, but they should be somewhat close, or, in the vicinity of each other.

Calculator below estimates mean of the population using the sample. Vicinity is found for different confidence levels using Student's t-distribution.

For this to work, the following assumptions should be met:

1. The scale of measurement has the properties of an equal interval scale.
2. The sample is randomly drawn from the source population.
3. The source population can be reasonably supposed to have a normal distribution.

The formula for estimating mean of a population based on the sample is
$est.\mu_{source}=M_x\pm (t*est.\sigma_M)$, where

$M_x$ - mean of the sample

$t$ - t-ratio for the p value which corresponds to chosen confidence level for non-directional test.

It is calculated from the inverse of the cdf for the Student's T distribution with degrees of freedom equals to N-1, where N is the number of values in the sample. For example, to get t-ratio for 0.05 level of significance, or 95% confidence level, you need to take absolute value of the inverse at 0.025.

$est.\sigma_M$ - estimate of the standard deviation of the sampling distribution of sample means (or standard error of the mean)

It is calculated as $est.\sigma_M=\sqrt{\frac{\frac{\sum{(X_i-M_x)^2}}{N-1}}{N}}$

If you care about how these formulas are derived, you can read excellent explanation here, starting from Chapter 9.

### Estimated Mean of a Population

Digits after the decimal point: 2
Estimated Mean

Lower limit

Mean of the sample

Upper limit

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