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Chi-squared test in SciPy - Time & Space Complexity

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Time Complexity: Chi-squared test
O(n * m)
Understanding Time Complexity

We want to understand how the time needed to run a chi-squared test changes as the data size grows.

Specifically, how does the test's execution time grow when we have more categories or observations?

Scenario Under Consideration

Analyze the time complexity of the following code snippet.


import numpy as np
from scipy.stats import chi2_contingency

# Create a contingency table with n rows and m columns
n, m = 100, 5
data = np.random.randint(1, 10, size=(n, m))

# Perform chi-squared test
chi2, p, dof, expected = chi2_contingency(data)

This code creates a table of counts and runs the chi-squared test to check if rows and columns are independent.

Identify Repeating Operations

Identify the loops, recursion, array traversals that repeat.

  • Primary operation: Calculating expected frequencies and summing over all cells in the contingency table.
  • How many times: Once for each cell in the table, so n times m.
How Execution Grows With Input

As the number of rows or columns increases, the number of cells grows, so the work grows too.

Input Size (n x m)Approx. Operations
10 x 550
100 x 5500
1000 x 55000

Pattern observation: The operations grow roughly in direct proportion to the number of cells in the table.

Final Time Complexity

Time Complexity: O(n * m)

This means the time needed grows linearly with the total number of cells in the contingency table.

Common Mistake

[X] Wrong: "The chi-squared test runs in constant time no matter the data size."

[OK] Correct: The test must look at every cell to calculate expected counts and contributions, so more data means more work.

Interview Connect

Understanding how the chi-squared test scales helps you explain performance when working with larger datasets in real projects.

Self-Check

"What if we changed the contingency table to be square with n rows and n columns? How would the time complexity change?"

Practice

(1/5)
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
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10 questions~5 mineasy hard