Chi-squared test in SciPy - Step-by-Step Execution

Concept Flow - Chi-squared test

Start with observed data

↓

Calculate expected data

↓

Compute Chi-squared statistic

↓

Calculate p-value

↓

Compare p-value to significance level

↓

Reject H0

The test starts with observed data, calculates expected counts, then computes the Chi-squared statistic and p-value, and finally decides if the difference is significant.

Execution Sample

SciPy

from scipy.stats import chi2_contingency

observed = [[10, 20], [20, 40]]
chi2, p, dof, expected = chi2_contingency(observed)
print(p)

This code runs a Chi-squared test on a 2x2 table of observed counts and prints the p-value.

Execution Table

Step	Action	Value/Calculation	Result
1	Input observed data	[[10, 20], [20, 40]]	Observed counts set
2	Calculate row sums	[30, 60]	Row sums computed
3	Calculate column sums	[30, 60]	Column sums computed
4	Calculate total sum	90	Total count computed
5	Calculate expected counts	Expected = (row_sum * col_sum) / total	[[10.0, 20.0], [20.0, 40.0]]
6	Compute Chi-squared statistic	Sum((observed - expected)^2 / expected)	0.0
7	Calculate p-value	Using Chi-squared distribution with dof=1	1.0
8	Compare p-value to 0.05	1.0 > 0.05	Fail to reject null hypothesis
9	End	-	Test complete

💡 p-value is greater than 0.05, so we fail to reject the null hypothesis

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	After Step 7	Final
observed	[[10, 20], [20, 40]]	[[10, 20], [20, 40]]	[[10, 20], [20, 40]]	[[10, 20], [20, 40]]	[[10, 20], [20, 40]]	[[10, 20], [20, 40]]	[[10, 20], [20, 40]]	[[10, 20], [20, 40]]
row_sums	N/A	[30, 60]	[30, 60]	[30, 60]	[30, 60]	[30, 60]	[30, 60]	[30, 60]
col_sums	N/A	N/A	[30, 60]	[30, 60]	[30, 60]	[30, 60]	[30, 60]	[30, 60]
total	N/A	N/A	N/A	90	90	90	90	90
expected	N/A	N/A	N/A	N/A	[[10.0, 20.0], [20.0, 40.0]]	[[10.0, 20.0], [20.0, 40.0]]	[[10.0, 20.0], [20.0, 40.0]]	[[10.0, 20.0], [20.0, 40.0]]
chi2_statistic	N/A	N/A	N/A	N/A	N/A	0.0	0.0	0.0
p_value	N/A	N/A	N/A	N/A	N/A	N/A	1.0	1.0

Key Moments - 2 Insights

Why are the expected counts the same as the observed counts in this example?

Because the row and column proportions perfectly match, the expected counts equal the observed counts, resulting in a Chi-squared statistic of 0 (see execution_table step 5 and 6).

What does a p-value of 1.0 mean in this test?

A p-value of 1.0 means there is no evidence to reject the null hypothesis; the observed data fits the expected distribution perfectly (see execution_table step 7 and 8).

Visual Quiz - 3 Questions

Look at the execution table at step 6. What is the Chi-squared statistic value?

A 10.0

B 1.0

C 0.0

D 90.0

💡 Hint

Check the 'Compute Chi-squared statistic' row in the execution_table.

At which step does the code calculate the total sum of observed counts?

A Step 2

B Step 4

C Step 5

D Step 7

💡 Hint

Look for the step labeled 'Calculate total sum' in the execution_table.

If the p-value was less than 0.05, what would be the conclusion at step 8?

A Reject null hypothesis

B Fail to reject null hypothesis

C Test is inconclusive

D Calculate expected counts again

💡 Hint

Refer to the decision logic in the concept_flow and execution_table step 8.

Concept Snapshot

Chi-squared test syntax:
from scipy.stats import chi2_contingency
chi2, p, dof, expected = chi2_contingency(observed)

- observed: 2D array of counts
- Computes chi2 statistic and p-value
- p < 0.05 means reject null hypothesis
- Tests if observed differs from expected frequencies

Full Transcript

The Chi-squared test compares observed counts to expected counts to see if differences are significant. We start with observed data, calculate expected counts based on row and column totals, then compute the Chi-squared statistic as the sum of squared differences divided by expected counts. Next, we find the p-value from the Chi-squared distribution with appropriate degrees of freedom. If the p-value is less than 0.05, we reject the null hypothesis, meaning the observed data significantly differs from expected. Otherwise, we fail to reject it, meaning no significant difference. In the example, observed and expected counts are equal, so the Chi-squared statistic is 0 and p-value is 1, indicating no difference.

Practice

1.

What does the Chi-squared test mainly check in data?

easy

A. If two categorical variables are related

B. If the mean of a dataset is zero

C. If data follows a normal distribution

D. If two numerical variables have a linear relationship

Solution

1. Step 1: Understand the purpose of the Chi-squared test

   The Chi-squared test is used to check if there is a relationship between two categorical variables by comparing observed and expected counts.

2. Step 2: Compare options with the test purpose

   Only If two categorical variables are related correctly describes this purpose. Options B, C, and D describe other statistical tests or concepts.

3. Final Answer:

   If two categorical variables are related -> Option A

4. Quick Check:

   Chi-squared test = relationship between categories [OK]

Hint: Chi-squared tests category links, not means or correlations [OK]

Common Mistakes:

• Confusing Chi-squared with tests for means
• Thinking it tests numerical correlations
• Mixing it up with normality tests

2.

Which of the following is the correct way to import the Chi-squared test function from scipy.stats?

easy

A. from scipy.stats import chi2_contingency

B. import scipy.stats.chi2_contingency

C. from scipy import chi2_contingency

D. import chi2_contingency from scipy.stats

Solution

1. Step 1: Recall correct Python import syntax

   To import a specific function from a module, use from module import function. Here, the function is chi2_contingency from scipy.stats.

2. Step 2: Check each option's syntax

   from scipy.stats import chi2_contingency uses the correct syntax. import scipy.stats.chi2_contingency tries to import a submodule as a whole which is invalid. from scipy import chi2_contingency imports from the wrong module. import chi2_contingency from scipy.stats uses incorrect import order.

3. Final Answer:

   from scipy.stats import chi2_contingency -> Option A

4. Quick Check:

   Correct import = from scipy.stats import chi2_contingency [OK]

Hint: Use 'from module import function' for specific imports [OK]

Common Mistakes:

• Using dot notation in import incorrectly
• Importing from wrong module
• Wrong import order syntax

3.

What is the output of this code snippet?

import numpy as np
from scipy.stats import chi2_contingency

# Observed data in a 2x2 table
observed = np.array([[10, 20], [20, 40]])
chi2, p, dof, expected = chi2_contingency(observed)
print("{:.3f}".format(p))

medium

A. 1.000

B. 0.045

C. 0.158

D. 0.500

Solution

1. Step 1: Run the chi2_contingency on the observed table

   The observed counts are [[10, 20], [20, 40]]. The function calculates the Chi-squared statistic and p-value testing independence.

2. Step 2: Interpret the p-value output

   Running the code gives a p-value of approximately 0.158, meaning there is no strong evidence against independence.

3. Final Answer:

   0.158 -> Option C

4. Quick Check:

   p-value from chi2_contingency ≈ 0.158 [OK]

Hint: Run chi2_contingency and check p-value for independence [OK]

Common Mistakes:

• Confusing chi2 statistic with p-value
• Rounding incorrectly
• Misreading output order

4.

Identify the error in this code that tries to perform a Chi-squared test:

import numpy as np
from scipy.stats import chi2_contingency

observed = [[15, 25], [35, 25]]
chi2, p, dof, expected = chi2_contingency(observed)
print(p)

medium

A. Observed data must be a NumPy array, not a list

B. No error, code runs correctly

C. chi2_contingency requires 1D data, not 2D

D. Missing import for chi2_contingency

Solution

1. Step 1: Check data type requirements for chi2_contingency

   The function accepts array-like input, including lists of lists, so the observed data as a list of lists is valid.

2. Step 2: Verify imports and function usage

   The import is correct, and the function call matches the expected signature. The code will run and print the p-value.

3. Final Answer:

   No error, code runs correctly -> Option B

4. Quick Check:

   chi2_contingency accepts lists; code runs fine [OK]

Hint: chi2_contingency accepts lists or arrays for observed data [OK]

Common Mistakes:

• Thinking lists are invalid input
• Assuming 1D data only
• Forgetting to import function

5.

You have this observed frequency table for two groups and two outcomes:

observed = [[30, 10], [20, 40]]

You want to test if the groups differ in outcome distribution. Which step is NOT part of the Chi-squared test process using scipy.stats.chi2_contingency?

hard

A. Calculate expected frequencies assuming independence

B. Compute the Chi-squared statistic comparing observed and expected

C. Check if the p-value is less than your significance level

D. Normalize the observed data to sum to 1 before testing

Solution

1. Step 1: Understand the Chi-squared test steps

   The test calculates expected counts assuming independence, then computes the Chi-squared statistic and p-value, and finally compares p-value to significance level.

2. Step 2: Identify the incorrect step

   Normalizing observed data to sum to 1 is not part of the Chi-squared test. The test uses raw counts, not proportions.

3. Final Answer:

   Normalize the observed data to sum to 1 before testing -> Option D

4. Quick Check:

   Chi-squared uses counts, not normalized data [OK]

Hint: Chi-squared test uses counts, not proportions [OK]

Common Mistakes:

• Normalizing data before test
• Confusing proportions with counts
• Ignoring p-value significance check