Confidence intervals on parameters in SciPy

Introduction

Confidence intervals show a range where a parameter likely lies. They help us understand how sure we are about our estimates.

When you want to know the range of possible values for a mean from sample data.

When estimating the effect size in an experiment and want to express uncertainty.

When comparing two groups and want to see if their means differ significantly.

When reporting results in a way that shows reliability, not just a single number.

Syntax

SciPy

from scipy import stats

# Example: Calculate confidence interval for the mean
mean = sample_data.mean()
sem = stats.sem(sample_data)  # standard error of the mean
confidence = 0.95
interval = stats.t.interval(confidence, len(sample_data)-1, loc=mean, scale=sem)

stats.sem calculates the standard error of the mean.

stats.t.interval returns the confidence interval using the t-distribution.

Examples

This example calculates a 95% confidence interval for the mean of a small dataset.

SciPy

import numpy as np
from scipy import stats

data = np.array([5, 7, 8, 9, 10])
mean = np.mean(data)
sem = stats.sem(data)
ci = stats.t.interval(0.95, len(data)-1, loc=mean, scale=sem)
print(ci)

This example calculates a 99% confidence interval, which is wider because we want more certainty.

SciPy

import numpy as np
from scipy import stats

# 99% confidence interval
data = np.array([12, 15, 14, 16, 13, 15])
mean = np.mean(data)
sem = stats.sem(data)
ci = stats.t.interval(0.99, len(data)-1, loc=mean, scale=sem)
print(ci)

Sample Program

This program calculates the average test score and the 95% confidence interval around that average. It shows the range where the true average likely falls.

SciPy

import numpy as np
from scipy import stats

# Sample data: test scores
scores = np.array([88, 92, 85, 91, 87, 90, 93])

# Calculate mean and standard error
mean_score = np.mean(scores)
sem_score = stats.sem(scores)

# Calculate 95% confidence interval for the mean
confidence_level = 0.95
ci_lower, ci_upper = stats.t.interval(confidence_level, len(scores)-1, loc=mean_score, scale=sem_score)

print(f"Mean score: {mean_score:.2f}")
print(f"95% confidence interval: ({ci_lower:.2f}, {ci_upper:.2f})")

Important Notes

Confidence intervals depend on sample size; bigger samples give narrower intervals.

The t-distribution is used when the sample size is small and population standard deviation is unknown.

Always check assumptions like normality when interpreting confidence intervals.

Summary

Confidence intervals give a range for parameter estimates, showing uncertainty.

Use stats.t.interval with sample mean and standard error to calculate them.

Higher confidence levels mean wider intervals.