Chi-squared test in R Programming - Step-by-Step Execution

Concept Flow - Chi-squared test

Start with observed data

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Calculate expected data

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Compute (Observed - Expected)^2 / Expected

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Sum all values to get Chi-squared statistic

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Compare statistic to Chi-squared distribution

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Decide if difference is significant

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End

The test compares observed counts to expected counts, calculates a statistic, then checks if the difference is significant.

Execution Sample

R Programming

observed <- c(50, 30, 20)
expected <- c(40, 40, 20)
chisq_stat <- sum((observed - expected)^2 / expected)
p_value <- 1 - pchisq(chisq_stat, df=2)
p_value

This code calculates the chi-squared statistic and p-value for observed vs expected counts.

Execution Table

Step	Calculation	Value	Explanation
1	Observed counts	50, 30, 20	Given data from experiment
2	Expected counts	40, 40, 20	Hypothesized distribution
3	(O - E)	10, -10, 0	Difference between observed and expected
4	(O - E)^2	100, 100, 0	Square differences
5	(O - E)^2 / E	2.5, 2.5, 0	Divide squared differences by expected
6	Sum statistic	5.0	Sum of all values from step 5
7	Degrees of freedom	2	Number of categories minus 1
8	p-value	0.0821	Probability of observing this or more extreme
9	Decision	Not significant	p-value > 0.05, fail to reject null hypothesis

💡 Test ends after calculating p-value and making decision based on significance level

Variable Tracker

Variable	Start	After Step 3	After Step 5	Final
observed	c(50,30,20)	c(50,30,20)	c(50,30,20)	c(50,30,20)
expected	c(40,40,20)	c(40,40,20)	c(40,40,20)	c(40,40,20)
difference	NA	c(10,-10,0)	c(10,-10,0)	c(10,-10,0)
squared_diff	NA	c(100,100,0)	c(2.5,2.5,0)	c(2.5,2.5,0)
chisq_stat	NA	NA	NA	5.0
p_value	NA	NA	NA	0.0821

Key Moments - 3 Insights

Why do we square the difference (O - E) before dividing by E?

Squaring makes all differences positive and emphasizes larger differences, as shown in step 4 of the execution_table.

What does the p-value tell us in this test?

The p-value (step 8) tells us how likely it is to see the observed data if the null hypothesis is true. A high p-value means no strong evidence against the null.

Why do we subtract 1 from the number of categories to get degrees of freedom?

Degrees of freedom (step 7) reflect the number of independent comparisons. Since total counts are fixed, one category is dependent on others.

Visual Quiz - 3 Questions

Look at the execution_table at step 5, what is the value of (O - E)^2 / E for the second category?

A 100

B 2.5

C 10

D -10

💡 Hint

Check the row labeled '5' under the 'Value' column in execution_table.

At which step does the test calculate the total chi-squared statistic?

A Step 6

B Step 4

C Step 8

D Step 2

💡 Hint

Look for the step where the sum of all values from step 5 is computed.

If the observed counts were exactly equal to expected counts, what would be the chi-squared statistic?

A Equal to number of categories

B 1

C 0

D Cannot be determined

💡 Hint

Refer to the formula in execution_sample and consider when (O - E) is zero.

Concept Snapshot

Chi-squared test compares observed and expected counts.
Calculate sum of (O - E)^2 / E.
Degrees of freedom = categories - 1.
Use p-value to decide significance.
If p < 0.05, difference is significant.

Full Transcript

The Chi-squared test starts with observed and expected counts. We find the difference between them, square it, and divide by expected counts. Summing these values gives the chi-squared statistic. We then compare this statistic to a chi-squared distribution with degrees of freedom equal to the number of categories minus one. The p-value tells us if the observed differences are likely due to chance. If the p-value is small (usually less than 0.05), we say the difference is significant and reject the null hypothesis. Otherwise, we do not reject it.