Implementing t-Tests in Python with SciPy — One-Sample, Two-Sample (Student's·Welch's), Paired

The theory is in the previous post — this one turns it into SciPy code for all three t-tests. Checking that the hand-computed t and p match SciPy's output makes the concepts stick.

Testing Means — the t-test (One / Two: Student's·Welch's / Paired)

Is a difference in two group means a real difference or just variability — the rationale for the t-test, its assumptions (normality, equal variance), and worked calculations for each type.

taystudios.com/blog

1. Python statistics libraries

The two main statistics libraries in the Python ecosystem are SciPy and Statsmodels. (For machine learning, it's scikit-learn.)

SciPy

• Optimized for quickly producing p-values and basic statistics.
• Good for large datasets or fast decisions.
• Install: pip install scipy

Statsmodels

• Provides extensive features for statistical model diagnosis and evaluation.
• Detailed statistics for each variable (t-statistic), the model (F-statistic), R², and more.
• Install: pip install statsmodels

This post implements all three t-tests with SciPy.

2. One-sample t-test — stats.ttest_1samp

Example — Does the blood pressure of 5 patients after taking a drug differ from the usual mean (120)?

from scipy import stats
import numpy as np

x1 = np.array([118, 121, 119, 117, 120])
mu0 = 120.0  # population mean to compare against

result = stats.ttest_1samp(x1, popmean=mu0, alternative='two-sided', axis=0)
# TtestResult(statistic=-1.4142135623730951, pvalue=0.23019964108049873, df=4)

Key arguments

• x1: the sample to test (numpy array recommended)
• popmean: the population mean to compare against
• alternative: 'two-sided' (default) / 'less' / 'greater'
• axis: 0 (columns, default) / 1 (rows) / None (flatten)

axis option on a 2D array

arr = np.array([[1.5, 2.3, 3.7],
                [4.1, 5.2, 6.8],
                [7.4, 8.5, 9.1],
                [10.2, 11.4, 12.9],
                [13.3, 14.1, 15.6]])
mu0 = 5

# axis=0 — column-wise (each column as a sample)
t_stat, p_val = stats.ttest_1samp(arr, popmean=mu0, axis=0)
# T-statistic: [1.0946 1.5652 2.1805]
# P-value:     [0.3352 0.1926 0.0947]

# axis=1 — row-wise (each row as a sample)
t_stat, p_val = stats.ttest_1samp(arr, popmean=mu0, axis=1)
# T-statistic: [-3.8886  0.4678  6.6965  8.3224 13.8451]
# P-value:     [ 0.0602  0.6860  0.0216  0.0141  0.0052]

# axis=None — flatten (whole array as one sample)
t_stat, p_val = stats.ttest_1samp(arr, popmean=mu0, axis=None)
# T-statistic: 2.9475
# P-value:     0.0106

3. Two-sample t-test — stats.ttest_ind

Example — Compare blood pressure between drug and no-drug groups.

3.1 Student's t-test — equal-variance assumption OK

from scipy import stats
import numpy as np

x1 = np.array([115, 118, 116])  # drug
x2 = np.array([122, 124])       # no drug

t_stat, p_val = stats.ttest_ind(x1, x2, equal_var=True, alternative='two-sided')
# T-statistic: -4.898979485566359
# P-value:      0.016276603459428517

Key arguments

• x1, x2: the two samples
• equal_var:
• True: equal-variance assumption holds → Student's t-test
• False: equal-variance assumption fails → automatically switches to Welch's t-test
• alternative and axis are the same as ttest_1samp

axis on 2D arrays

data  = np.array([[115, 118, 116],
                  [122, 124, 123]])
data1 = np.array([[231, 123, 132],
                  [223, 321, 421]])

# axis=0 — column-wise
t_stat, p_val = stats.ttest_ind(data, data1, axis=0, equal_var=True)
# T-statistic: [-20.4136 -1.0197 -1.0862]
# P-value:     [  0.0024  0.4151  0.3909]

# axis=1 — row-wise
t_stat, p_val = stats.ttest_ind(data, data1, axis=1, equal_var=True)
# T-statistic: [-1.3195 -3.4755]
# P-value:     [ 0.2575  0.0255]

3.2 Welch's t-test — unequal variances

t_stat, p_val = stats.ttest_ind(x1, x2, equal_var=False, alternative='two-sided')
# T-statistic: -5.0000
# P-value:      0.02505

Same method, just toggle equal_var=False for Welch's. Use this when the equal-variance assumption doesn't hold.

4. Paired-samples t-test — stats.ttest_rel

Example — Same patients, blood pressure before vs. after. Paired data.

import numpy as np
from scipy import stats

before = np.array([130, 128, 135, 132, 129])
after  = np.array([120, 119, 125, 123, 121])

t_stat, p_val = stats.ttest_rel(before, after, alternative='less')
# T-statistic: 24.58803425594304
# P-value:      0.9999918819424217

alternative='less' here sets "before < after" as the alternative hypothesis. Since before is actually larger, p is near 1. If you want to test "is BP lower after the dose," call it the other way: stats.ttest_rel(after, before, alternative='less') — then p comes out tiny.

Caveat: the two arrays must have the same length. If not:

ValueError: unequal length arrays

Wrap-up

With SciPy's three functions, each t-test is a one-liner:

• One-sample → stats.ttest_1samp(x, popmean)
• Two-sample → stats.ttest_ind(x1, x2, equal_var=True/False) — Student's vs. Welch's
• Paired → stats.ttest_rel(before, after)

Just keep alternative (two-sided / one-sided) and axis (1D/2D) in mind. Compare the output against the hand calculation from the theory post — t and p match.

References

• SciPy · ttest_1samp · ttest_ind · ttest_rel
• Statsmodels

📦 Migrated from my own Korean blog (my own writing). Original: taehyuklee.tistory.com/26