#### Data Science Day 3: **Chi-square Test**

Learning Objectives

1. **Define** the Chi-Square distribution

2. **Explain** the 3 Chi-square test applications scenario

Learning Objectives

The **Chi- Square distribution** is the **sum of variance** (squared standard normal deviates). The following equation represents a Chi-Square distribution with *m degrees of freedom.
*

where X1, X2, … Xm are **independent random variables** having the **standard normal distribution.** The higher the degree of freedom, the more it approaches to a normal distribution.

**The Chi-Square distribution has 3 basic properties:**

- Not symmetric, Skewed to the right
- No Negative Values
- Total area under the curve=1

**Three primary Chi-square test applications:
**

**1.Test independence of two categorical variables:**

*Whether the two categorical variables have a strong association, or whether the two categorical variables are independently distributed in one sample space.*

**Null hypothesis:**Two categorical variables are independent.

*Note:*There are two categorical variables from one sample space

Mini E.g. The Sex Frequency (Boys and Girls number) and Nationality distribution in a class

**2*.Test the Goodness of Fit (P earson):
**

*Whether the sample categorical data are consistent with a hypothesized distribution.*

**Null hypothesis**:

*Sample data are consistent with a specified distribution*

*Note:*It is one Categorical variable from one sample space

Mini E.g. The Sex Ratio (Girls : Boys) in one class is the 50%

**3.Test of Homogeneity:**

*Whether frequency counts of the categorical variable have the same distribution for different sample spaces.*

**Null hypothesis**: The proportion of the categorical variable is the same in all sample space.

*Note:* It is one categorical variable from two or more different sample space.

Mini E.g. Whether the Sex Frequency is the same in all classes.

* In Clinical Trials, we use Chi-square log-rank test in survival analysis.

We will show the application examples next time!

Thanks very much to* Renee Wu, Ali Motamedi~ *

**Happy learning!**

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