Cesar Chew at 18:07, 25 November 2014

2014-11-25T18:07:25Z

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{{Cat|Chi Square| 01}}

Prerequisites
* Distributions, Standard Normal Distribution, Degrees of Freedom

== Define the Chi Square distribution in terms of squared normal deviates ==
* The Chi Square Distribution is the distribution of the sum of squared standard normal deviates
* The degrees of freedom of the distribution is equal to the number of standard normal deviates being summed
* Therefore, Chi Square with one degree of freedom, written as χ<sup>2</sup>(1), is simply the distribution of a single normal deviate squared
* The area of a Chi Square distribution below 4 is the same as the area of a standard normal distribution below 2 since 4 is 2<sup>2</sup>.

=== Example ===
* You sample two scores from a standard normal distribution, square each score, and sum the squares.
* What is the probability that the sum of these two squares will be six or higher?
* Since two scores are sampled, the answer can be found using the Chi Square distribution with two degrees of freedom
* A Chi Square calculator can be used to find that the probability of a Chi Square (with 2 df) of being six or higher is 0.05

== How does the shape of the Chi Square distribution change its degrees of freedom increase? ==
* The mean of a Chi Square distribution is its degrees of freedom.
* Chi Square distributions are positively skewed, with the degree of skew decreasing with increasing degrees of freedom
* As the degrees of freedom increase, the Chi Square Distribution approaches a normal distribution
* Notice how the skew decreases as the degrees of freedom increases.
[[File:Chi_squared.gif|300x300px]]

=== Where can we use Chi Square distribution ? ===
* The Chi Square distribution is very important because many test statistics are approximately distributed as Chi Square
* Two of the more commonly tests using the Chi Square distribution are:
** tests of deviations of differences between theoretically expected and observed frequencies (one-way tables)
** the relationship between categorical variables (contingency tables)
* Numerous other tests beyond the scope of this work are based on the Chi Square distribution.

== Questions ==

<quiz display=simple >

{Imagine that you sample 12 scores from a standard normal distribution, square each score, and sum the squares. How many degrees of freedom does the Chi Square distribution that corresponds to this sum have?
|type="{}"}
{ 12 _10 }

{
{{Show Answer|
The degrees of freedom of the Chi Square distribution are equal to the number of standard normal deviates being summed (which is 12 in this case).
}}
}

{What is the mean of a Chi Square distribution with 8 degrees of freedom?
|type="{}"}
{ 8 _10 }

{
{{Show Answer|
The mean of a Chi Square distribution is its degrees of freedom.
}}
}

{Which Chi Square distribution looks the most like a normal distribution?
|type="()"}
- A Chi Square distribution with 0 df
- A Chi Square distribution with 1 df
- A Chi Square distribution with 2 df
+ A Chi Square distribution with 10 df

{
{{Show Answer|
As the degrees of freedom of a Chi Square distribution increase, the Chi Square distribution begins to look more and more like a normal distribution. Thus, out of these choices, a Chi Square distribution with 10 df would look the most similar to a normal distribution.
}}
}

{Imagine that you sample 3 scores from a standard normal distribution, square each score, and sum the squares. What is the probability that the sum of these 3 squares will be 9 or higher?
|type="{}"}
{ 0.0293 _10 }

{
{{Show Answer|
Because three scores are sampled, the answer can be found using the Chi Square distribution with three degrees of freedom. A Chi Square calculator can be used to find that the probability of a Chi Square (with 3 df) being 9 or higher is .0293.
}}
}
</quiz>

[[:Category:Chi Square|Chi Square]] | [[One-Way Tables]] >

Chi Square Distribution - Revision history

Cesar Chew at 18:07, 25 November 2014