Probability Distributions > *Helmert’s Distribution*

## What is Helmert’s distribution?

Helmert’s distribution is another name for the chi-square distribution. It’s named after F.R. Helmert, who proved the general reproductive property of chi-square distributions [1].

Helmert’s most noted contribution was establishing that if we have a set of independent, normally distributed random variables, the sum of the squared standardized deviations follows a chi-square distribution; The chi-square variable is distributed as a chi-square variable with n – 1 degrees of freedom. This chi-square variable is statistically independent of the sample mean. Helmert also proved that the sample mean and variance are independent. This allows us to obtain a set of n – 1 new independent and identically distributed normally distributed samples with mean = 0 and the same variance of the original distribution.

“Squared standardized deviations” are the squares of the deviations of each data point from the mean, divided by the sample standard deviation. This process makes the deviations comparable across different datasets with varying scales.

## A Summary of Helmert’s Theorem

Helmert’s theorem states that [2]:

- Given a set of independent, normally distributed random variables, the sum of squared standardized deviations follows a chi-square distribution with n – 1 degrees of freedom.
- The chi-square variable is independent of the sample mean.
- The sample mean and variance are also independent.
- From a sample of size n, we can obtain a new set of n – 1 new iid distributed normally distributed samples with a mean of zero and the same variance as the original distribution.

Helmert’s theorem allows us to more effectively study normally distributed variables. It is a fundamental result in inferential statistics and has many applications, especially in hypothesis testing and construction of confidence intervals.

## References

- Helmert, F. R. (1876). Die Genauigkeit der Formel von Peters zue Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit, Astronomische Nachrichten, 88, columns 113-120
- Kruskal, W. Helmert’s Distribution. The American Mathematical Monthly. Vol. 53, No. 8 (Oct., 1946), pp. 435-438 (4 pages). Published By: Taylor & Francis, Ltd