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Descriptive Statistics
arithmetic mean weighted mean median mode range variance standard deviation interquartile range
Probability
sample space and events addition rule multiplication rule conditional probability Bayes theorem independence expected value law of total probability
Distributions
Bernoulli distribution binomial distribution Poisson distribution uniform distribution normal distribution exponential distribution z-score standardization central limit theorem simulation
Sampling & Estimation
simple random sampling sampling distribution of mean standard error point estimate confidence interval for mean confidence interval for proportion margin of error bootstrap estimate
Hypothesis Testing
null and alternative hypotheses p-value one-sample z test one-sample t test two-sample t test paired t test chi-square independence test Type I and Type II errors
Relationships & Models
covariance Pearson correlation Spearman rank correlation simple linear regression coefficient of determination R² residual analysis multiple regression concept confounding variables
Statistics / Hypothesis Testing

paired t test

Learn paired t test in Statistics with a dedicated explanation, a concept-specific example, practical cautions, and a mastery check.

12 min read • Advanced
statistics-038.py
python 
from scipy.stats import ttest_rel
before=[120,118,130,125,127]; after=[115,112,121,119,120]
print(ttest_rel(before,after))
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