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title: "Example: Fitting a second order Normal distribution to data"
canonical: "https://modelassist.epixanalytics.com/space/EA/26575324/Example%3A%20Fitting%20a%20second%20order%20Normal%20distribution%20to%20data"
format: markdown
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The Normal distribution is easy to fit to data since its two parameters, the mean and standard deviation, are independent of each other. The first is purely a location parameter and the second is purely a shape parameter. Knowledge about one parameter (e.g. the mean, telling us where the distribution is located) tells us nothing about the other parameter (the spread) and vice versa, so the two parameters can be estimated separately and the distributions of uncertainty for these parameters are uncorrelated.
### *Statistical method*
Classical statistics and Bayesian statistics both tell us that the uncertainty distributions for the mean and standard deviation of the Normal distribution are given by:
| | |
| --- | --- |
| [Mean](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574925/): | 
> Macro (mathinline) |
| [Standard deviation](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26574927/): | 
> Macro (mathinline) |
where:
- *m*, *s* are the mean and standard deviation of the population distribution,
- *x* and > Macro (inline-external-image)
are the sample mean and [sample standard deviation](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575351/) of the *n* data points being fitted,
- *t*(*n-*1) is a [Standard Student-t distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575254/) with *n-*1 degrees of freedom and c2(*n-*1) is a [Chi-squared distribution](https://epixanalytics.atlassian.net/wiki/spaces/EA/pages/26575221/) with *n-*1 degrees of freedom.
##### Example
Imagine we have 25 data values that have a mean *x* and standard deviation > Macro (inline-media-image)
of 85 and 11 respectively. The uncertainty distributions for the Normal distribution parameters are thus (see also equations above):
Mean *m*: =Student(25-1)*11/SQRT(25)+85
Standard deviation *s* =11*SQRT((25-1)/Chisquared(25-1))
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