Statistical Tests

A telecom company had taken a survey of smartphone owners in a certain town 5 years back and found 73% of the population own a smartphone, and have been since using this data to make their business decisions.

Now a new marketing manager has joined, and he believes this value is not valid anymore. Thus he conducts a survey of 500 people and finds that 420 of them responded with affirmation as to owning a smartphone. Which statistical test would you use to compare these two survey data?

  1. Test of proportions, z-test:

    • Applicability: This is the correct option. The z-test for proportions is suitable when comparing the proportions of two independent samples. In this case, you are comparing the proportion of smartphone owners in the town based on the data from 5 years ago (73%) and the recent survey (where 420 out of 500 respondents own a smartphone).

    • Reasoning: The z-test for proportions allows you to assess whether the observed difference in proportions is statistically significant. It is appropriate when you have a large sample size (which is often the case in surveys) and when the conditions for using a z-test are met.

  2. Test of independence, chi-square test:

    • Applicability: The chi-square test of independence is used when you have categorical data and want to test if there is a significant association between two variables.

    • Reasoning: While the chi-square test is useful in certain scenarios, it is not the best choice for comparing proportions between two independent samples. It is more suitable for analyzing contingency tables with categorical data.

  3. Test of means, t-test:

    • Applicability: The t-test is used when comparing means of two independent samples, not proportions.

    • Reasoning: Since you are interested in comparing the proportion of smartphone owners, the t-test is not the appropriate choice. The t-test is used for continuous data (such as comparing the means of two groups) and is not suitable for proportions.

Kolmogorov–Smirnov Test (KS Test)

This test checks if two sets of data have the same type of distribution

Kruskal Wallis Test

This test does not assume that the data are normal, it does assume that the different groups have the same distribution, and groups with different standard deviations have different distributions



This test checks if data arrays passed to it has equal variance

Shapiro-Wilk Test

This test checks whether data has normal distribution

CHI Square

This is used to check if 2 categorical variable are related

  • Null Hypothesis : 2 groups are independent

  • Alternate Hypothesis: 2 groups are dependent

This means that the expected value table received from following calculations have independent values

from scipy.stats import chi2_contingency

table = [[10, 20, 30],[6,  9,  17]]
stat, p, dof, expected = chi2_contingency(table)

# interpret test-statistic
prob = 0.95
critical = chi2.ppf(prob, dof)
if abs(stat) >= critical:
 print('Dependent (reject H0)')
 print('Independent (fail to reject H0)')
# p-value < alpha, then reject the null hypothesis

Different combinations and corresponding test

ScenarioStatistical TestExample

Numerical vs. Numerical


Examining the relationship between hours of study and exam scores.

Numerical vs. Categorical (Binary)

Logistic Regression

Predicting the likelihood of a student passing an exam based on the number of hours of study.

Numerical vs. Categorical (More than 2 Categories)


Comparing the average test scores of students who studied for different durations across multiple study groups.

Numerical vs. Categorical (Repeated Measures)

Repeated Measures ANOVA

Investigating changes in blood pressure levels across different time points with different treatment groups.

Numerical vs. Categorical (Longitudinal Data)

Mixed Effects Models

Analyzing repeated measurements of cholesterol levels over time for patients receiving different treatments.

Numerical vs. Categorical (Survival Analysis)

Kaplan-Meier, Cox Regression

Assessing the time until relapse for patients with different types of cancer treatments.

Categorical vs. Categorical

Chi-square Test

Examining the association between gender and smoking status.

Categorical vs. Categorical (Association)

Cramér's V

Measuring the strength of association between political affiliation and voting behavior.

Numerical vs. Categorical (Ordinal)

Kruskal-Wallis Test

Comparing the median satisfaction scores for customers across different levels of service quality.

1 sample z test for mean 1 sample t test for mean

1 sample z test for proportion 1 sample t test for proportion

2 sample independent test for mean 2 sample independent test for proportion

Paired test

Cheat sheet for different test

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