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Normal Distribution and Others

We meet various distributions when we learn statistics .

In my experience, the knowledge of distribution is used when I decide that whether I can use normal distribution or not. And the knowledge that " Hypothesis testing of two averages uses t-distribution." is also important when I test by MS-EXCEL function.

Today, software of statistics helps complicated process of analysis. The ideas of this page are that I need in these days.

Probability Density Function

"What is the distribution?" is the first wall of difficulties to learn.

X-axis of the graphs of distributions is statistical value. And Y-axis is the probability density.

Probability density function is used to calculate area. The total area of distribution is 1. It means that total probability is 1.

Additionally speaking, other statistical values are also calculated by probability density and integration. Monte Carlo method is used for integration of complicated figures.

Normal distribution

Normal distribution is also called "Gauss distribution". It depends on average and standard deviation.

Robustness of the Theory of Normal distribution

The methods using normal distribution are robust because it can be used for many cases even if the distribution is not normal distribution. In my experience, if the data has a range, I can use the methods.

One of the reasons of robustness is that many distributions can be approximated by normal distribution if data (sample number) is not small.

So the information of average and standard deviation is useful to study the outline of the data even if the data is not normal distribution.

Non-Normal distribution


If population variance is unknown, we use t-distribution. It is used for hypothesis testing of two averages. And it is also used paired testing.

ANOVA is hypothesis testing of average. But it uses F-distribution.

Chi-squared distribution

Chi-squared distribution is used to study whether the variance is included the distribution or not. If we want to compare two variances, F-distribution is used.


F-distribution is used to study whether two variances are from one population.

It is used in ANOVA. ANOVA is often used in Design of Experiments .

Binomial Distribution

Front side of a coin appears at the probability of 1/2. After two tosses, front side appears two times at the probability of 1/4. Binomial distribution is used for the analysis of two-value data.

Go and No-Go data is often wrote by yield. So the background of yield data is binominal distribution.

Non-parametric testing used this distribution to study the bias of the number of plus and minus.

NEXT Made by Normal Distribution But Not Normal Distribution