Chi Square Formula

In statistics, various measuring methods are widely used. A chi-square test is required for many experimental studies to conclude. It is a very useful non-parametric statistic. The chi Square test formula is used to collect data from people who are divided into several categories. And to know if the distribution differs from what one may predict.

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What is Chi Square?

The difference between actual and predicted data values are calculated using the chi square test formula, which is used in statistics. It's used to see how closely real and predicted data correlate. The chi-square value will help us in determining the statistical significance of the discrepancy between predicted and actual data. A small chi-square value indicates that any differences between the actual and expected data are due to chance.

As a result, the data is not statistically significant. A large value also indicates that the data is statistically significant and that something is causing the data differences. A statistician can then look at the reasons that might be generating the differences.

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Chi Square Formula

As we can see in the chi square distribution formula, Chi is a Greek symbol that looks like the letter x. To calculating chi square, multiply the square of the difference between the observed O and expected values E then divide it by the expected value E. We end up with two or more values depending on the number of categories in the input. The sum of these numbers is the Chi-square. We don't need to take the square root because the number we're searching for is chi-square.

A small Chi square test equation indicates that the mean data fits the expected data very well statistically. A large Chi square test formula indicates that the data does not fit well statistically. We must reject the null hypothesis if the chi-square value is very large. One way to show the relationship between two categorical variables is using the Chi square formula. In statistics, there are two types of variables: numerical variables and non-numerical variables.

The chi square test statistic formula is as follows,

χ2 = \[\sum\frac{(O-E){2}}{E}\]

Where,

O: Observed frequency

E:  Expected frequency

∑ : Summation

χ2 : Chi Square Value

Expected Frequency for Chi Square Equation

In contingency table calculations, including the chi-square test, the expected frequency is a probability count. Standardized residuals are calculated using expected frequencies, in which the expected count is subtracted from the observed count in the numerator. 

• Observed Frequencies: Counts of experimental data are called observed frequencies. Or look at it another way, you actually observe the data and take measurements.

• Expected Frequency: Expected Frequencies are a type of frequency that may be calculated using probability theory.

The expected frequency formula is as follows,

E_ij = \[\frac{T_{i} x T_{j}}{N}\]

Where, 

E_ij = Expected frequency for the i^th row/j^th column.

T_i = Total in the i^th row

T_j = Total in the j^th column

N = Net Total

Solved Examples

Example.1. Which Pet Will You Prefer?

Solution:

Firstly, Make a Table Using the Given Data,

Now, Add Rows and Columns, We Get,

Now, calculate the expected value for each entry by using the expected frequency formula,

E_ij = \[\frac{T_{i} x T_{j}}{N}\]

Now calculating chi square value, subtract expected from actual value, square it, then divide by expected value by using chi square statistic formula,

The chi square formula is as follows,

χ2 = \[\sum\frac{(O-E){2}}{E}\]

Now, Add All Chi-Squared Values,

0.0029 + 0.0028 + 0.0028 + 0.0028 = 0.0113

Hence the chi-square value is very small; its mean data fits the expected data very well statistically.