SciPy Statistical Tests

Statistical tests help you decide if a difference in your data is real or just due to chance.

The Idea of a Hypothesis Test

1. Start with the null hypothesis: "there is no real difference"
2. Collect data and run a test
3. The test gives a p-value: how likely data like yours would be if the null hypothesis were true
4. If the p-value is small (usually below 0.05), you reject the null hypothesis and call the result significant

One-Sample t-Test

A juice factory claims each bottle holds 500 ml. You measure ten bottles. Is the average really 500?

import numpy as np
from scipy import stats

bottles = np.array([498, 495, 501, 497, 494, 499, 496, 500, 493, 497])
result = stats.ttest_1samp(bottles, popmean=500)
print("Mean:", bottles.mean())
print("p-value:", round(result.pvalue, 4))

Mean: 497.0
p-value: 0.0051

The p-value is far below 0.05, so the bottles really do hold less than 500 ml on average.

Two-Sample t-Test

Two teachers used different methods. Did their classes score differently?

import numpy as np
from scipy import stats

class_a = np.array([72, 78, 81, 69, 75, 80, 77, 74])
class_b = np.array([80, 85, 79, 88, 84, 82, 86, 81])
result = stats.ttest_ind(class_a, class_b)
print(round(class_a.mean(), 2), round(class_b.mean(), 2))
print("p-value:", round(result.pvalue, 5))

75.75 83.12
p-value: 0.00116

Class B scored higher, and the very small p-value says this difference is unlikely to be luck.

Which t-Test?

Paired t-Test: Before and After

Six students took a test before and after a revision course:

from scipy import stats

before = [55, 60, 48, 70, 65, 52]
after = [62, 66, 55, 74, 70, 58]
result = stats.ttest_rel(before, after)
print("p-value:", round(result.pvalue, 6))

p-value: 6.5e-05

The tiny p-value shows the course helped. 6.5e-05 means 0.000065.

Correlation

Correlation measures how strongly two things move together, from -1 to 1.

• Close to 1: when one goes up, the other goes up
• Close to -1: when one goes up, the other goes down
• Close to 0: no clear straight-line link

from scipy import stats

study_hours = [1, 2, 3, 4, 5, 6, 7, 8]
marks = [52, 55, 61, 64, 70, 72, 79, 85]
r, p = stats.pearsonr(study_hours, marks)
print("Pearson r:", round(r, 3))
print("p-value:", round(p, 7))

rho, p2 = stats.spearmanr(study_hours, marks)
print("Spearman:", rho)

Pearson r: 0.994
p-value: 4e-07
Spearman: 1.0

More study hours go with higher marks almost perfectly. Spearman is 1.0 because the marks always rise in the same order as the hours.

Chi-Square Test

The chi-square test checks if two categories are related. Does the choice of course depend on gender? Here is a table of counts:

from scipy.stats import chi2_contingency

#            Python  Design
table = [[30, 10],   # male
         [20, 25]]   # female
result = chi2_contingency(table)
print("p-value:", round(result.pvalue, 4))

p-value: 0.0084

The p-value is below 0.05, so in this sample course choice and gender are related.

Exercise