Chi-squared test in Data Analysis Python - Step-by-Step Execution

Concept Flow - Chi-squared test

Start with observed data

↓

Calculate expected data

↓

Compute (Observed - Expected)^2 / Expected

↓

Sum all values to get Chi-squared statistic

↓

Compare statistic to Chi-squared distribution

↓

Decide if variables are independent or not

The test compares observed counts to expected counts to check if variables are independent.

Execution Sample

Data Analysis Python

import scipy.stats as stats
observed = [[10, 20], [20, 40]]
chi2, p, dof, expected = stats.chi2_contingency(observed)
print(chi2, p, dof)
print(expected)

This code runs a chi-squared test on a 2x2 table and prints the test statistic, p-value, degrees of freedom, and expected counts.

Execution Table

Step	Action	Calculation	Result
1	Input observed data	Observed = [[10, 20], [20, 40]]	Observed data set
2	Calculate row sums	Row sums = [30, 60]	30, 60
3	Calculate column sums	Column sums = [30, 60]	30, 60
4	Calculate total sum	Total = 90	90
5	Calculate expected counts	Expected[i,j] = (row_sum[i] * col_sum[j]) / total	[[10.0, 20.0], [20.0, 40.0]]
6	Calculate (O - E)^2 / E for each cell	For cell (0,0): (10-10)^2/10=0 For cell (0,1): (20-20)^2/20=0 For cell (1,0): (20-20)^2/20=0 For cell (1,1): (40-40)^2/40=0	[0,0,0,0]
7	Sum all values	Sum = 0 + 0 + 0 + 0	Chi-squared statistic = 0
8	Calculate p-value and degrees of freedom	dof = (rows-1)*(cols-1) = 1	p-value = 1.0, dof = 1
9	Interpret result	p-value > 0.05 means no evidence to reject independence	Variables likely independent

💡 Test ends after calculating p-value and interpreting result

Variable Tracker

Variable	Start	After Step 2	After Step 3	After Step 4	After Step 5	After Step 6	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]]
row_sums	N/A	[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]
total	N/A	N/A	N/A	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]]
chi_squared_components	N/A	N/A	N/A	N/A	N/A	[0,0,0,0]	[0,0,0,0]
chi_squared_statistic	N/A	N/A	N/A	N/A	N/A	N/A	0
p_value	N/A	N/A	N/A	N/A	N/A	N/A	1.0
degrees_of_freedom	N/A	N/A	N/A	N/A	N/A	N/A	1

Key Moments - 3 Insights

Why do we calculate expected counts using row and column sums?

Expected counts show what we would expect if variables were independent. We use row and column sums to find proportions assuming no relationship, as shown in step 5 of the execution_table.

Why can the chi-squared statistic be zero?

If observed counts exactly match expected counts, the differences are zero, so the statistic sums to zero. This is shown in step 7 where all components are zero.

What does a high p-value mean in this test?

A high p-value means we do not have enough evidence to say variables are related. Step 9 explains that p-value > 0.05 means likely independence.

Visual Quiz - 3 Questions

Look at the execution_table at step 5. What is the expected count for cell (1,1)?

A 40

B 20

C 30

D 60

💡 Hint

Check the expected counts calculated in step 5 of the execution_table.

At which step does the chi-squared statistic get calculated?

A Step 6

B Step 7

C Step 8

D Step 9

💡 Hint

Look for the step where all components are summed to get the statistic in the execution_table.

If the observed counts were very different from expected, how would the chi_squared_components change at step 6?

A They would be mostly zero

B They would be negative

C They would be larger values

D They would be equal to expected counts

💡 Hint

Step 6 shows how differences contribute to components; bigger differences mean bigger values.

Concept Snapshot

Chi-squared test checks if two categorical variables are independent.
Input: observed counts in a contingency table.
Calculate expected counts assuming independence.
Compute sum of (Observed - Expected)^2 / Expected.
Compare statistic to chi-squared distribution for p-value.
Low p-value (<0.05) means variables likely related.

Full Transcript

The chi-squared test compares observed data counts to expected counts calculated assuming variables are independent. We start with observed data, calculate row and column sums, then total sum. Expected counts are found by multiplying row and column sums divided by total. For each cell, we compute (Observed - Expected)^2 divided by Expected, then sum these values to get the chi-squared statistic. We find the p-value using the chi-squared distribution with degrees of freedom based on table size. A high p-value means no evidence to reject independence. This process helps us decide if two categorical variables are related or not.

Practice

1. What does the Chi-squared test primarily check in data analysis?

easy

A. If two numerical variables have a linear relationship

B. If two categorical variables are related or independent

C. If a dataset follows a normal distribution

D. If the mean of a dataset is greater than zero

Solution

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

   The Chi-squared test is used to check relationships between categorical variables, not numerical data.

2. Step 2: Identify the correct relationship tested

   It tests if two categories are independent or related based on observed counts.

3. Final Answer:

   If two categorical variables are related or independent -> Option B

4. Quick Check:

   Chi-squared tests category relationships [OK]

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

Common Mistakes:

• Confusing Chi-squared with correlation tests
• Thinking it tests numerical data relationships
• Assuming it checks data distribution shape

2. Which Python library and function is commonly used to perform a Chi-squared test on a contingency table?

easy

A. scipy.stats.chi2_contingency

B. numpy.chisquare_test

C. pandas.chi_squared

D. matplotlib.chi2_test

Solution

1. Step 1: Recall common Python libraries for statistics

   Scipy is the standard library for statistical tests in Python.

2. Step 2: Identify the correct function for Chi-squared test

   The function to perform Chi-squared test on contingency tables is chi2_contingency in scipy.stats.

3. Final Answer:

   scipy.stats.chi2_contingency -> Option A

4. Quick Check:

   Use scipy.stats.chi2_contingency for Chi-squared [OK]

Hint: Remember: scipy.stats has chi2_contingency for Chi-squared [OK]

Common Mistakes:

• Using numpy or pandas which don't have this function
• Confusing function names or libraries
• Trying to use plotting libraries for tests

3. What will be the output of this Python code snippet?

import numpy as np
from scipy.stats import chi2_contingency

observed = np.array([[10, 20], [20, 40]])
chi2, p, dof, expected = chi2_contingency(observed, correction=False)
print('{:.3f}'.format(p))

medium

A. 1.000

B. 0.000

C. 0.500

D. 0.250

Solution

1. Step 1: Understand the observed data and test

   The observed table has proportional counts (10:20 and 20:40), suggesting independence.

2. Step 2: Interpret the p-value from chi2_contingency

   The p-value is approximately 0.5, indicating no significant relationship; the test finds data consistent with independence.

3. Final Answer:

   0.500 -> Option C

4. Quick Check:

   Proportional counts give moderate p-value around 0.5 [OK]

Hint: Proportional tables yield moderate p-values near 0.5 [OK]

Common Mistakes:

• Assuming low p-value means independence
• Confusing chi2 statistic with p-value
• Rounding errors changing p-value interpretation

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

import scipy.stats as stats
observed = [[15, 25], [35, 25]]
chi2, p = stats.chi2_contingency(observed)
print(p)

medium

A. Incorrect unpacking of chi2_contingency output

B. Missing import of numpy for array input

C. Observed data must be a pandas DataFrame

D. No error; code runs correctly

Solution

1. Step 1: Check the output of chi2_contingency

   The function returns four values: chi2, p, dof, expected.

2. Step 2: Verify unpacking matches returned values

   The code unpacks only two values, causing a ValueError due to mismatch.

3. Final Answer:

   Incorrect unpacking of chi2_contingency output -> Option A

4. Quick Check:

   chi2_contingency returns 4 values, unpack all [OK]

Hint: Unpack all 4 outputs from chi2_contingency [OK]

Common Mistakes:

• Unpacking fewer than 4 values
• Assuming input must be numpy array
• Thinking pandas DataFrame is required

5. You have a 3x3 contingency table of observed counts for three categories of two variables. After running chi2_contingency, you get a p-value of 0.03. What is the best interpretation if your significance level is 0.05?

hard

A. The variables are independent

B. You need a larger sample size to decide

C. The test is inconclusive; p-value is too high

D. There is a significant relationship between the variables

Solution

1. Step 1: Compare p-value with significance level

   The p-value 0.03 is less than 0.05, indicating statistical significance.

2. Step 2: Interpret significance in context of independence

   A significant result means we reject the null hypothesis of independence; variables are related.

3. Final Answer:

   There is a significant relationship between the variables -> Option D

4. Quick Check:

   p-value < 0.05 means reject independence [OK]

Hint: p-value < alpha means variables related, not independent [OK]

Common Mistakes:

• Thinking p-value < 0.05 means independence
• Ignoring significance level in decision
• Assuming inconclusive without checking p-value