Welcome back, folks!
This is going to be a short lecture where we introduce to you the Chi-squared Distribution.
For starters, we define a denote a Chi-Squared distribution with the capital Greek letter
Chi, squared followed by a parameter “k” depicting the degrees of freedom.
Therefore, we read the following as “Variable “Y” follows a Chi-Square distribution
with 3 degrees of freedom”.
Alright!
Let’s get started!
Very few events in real life follow such a distribution.
In fact, Chi-Squared is mostly featured in statistical analysis when doing hypothesis
testing and computing confidence intervals.
In particular, we most commonly find it when determining the goodness of fit of categorical
values.
That is why any example we can give you would feel extremely convoluted to anyone not familiar
with statistics.
Alright!
Now, let’s explore the graph of the Chi-Squared distribution.
Just by looking at it, you can tell the distribution is not symmetric, but rather – asymmetric.
Its graph is highly-skewed to the right.
Furthermore, the values depicted on the X-axis start form 0, rather than some negative number.
This, by the way, shows you yet another transformation.
Elevating the Student’s T distribution to the second power gives us the Chi-squared
and vice versa: finding the square root of the Chi-squared distribution gives us the
Student’s T.
Great!
So, a convenient feature of the Chi-Squared distribution is that it also contains a table
of known values, just like the Normal or Students’–T distributions.
The expected value for any Chi-squared distribution is equal to its associated degrees of freedom,
k.
Its variance is equal to two times the degrees of freedom, or simply 2 times k.