Standard normal distribution

This lesson focuses on standard normal distribution. Mathematics provides tools for modeling and analyzing complex systems.

Advanced statistics provides tools for making inferences from data. Normal distribution: bell-shaped, symmetric; described by mean (μ) and standard deviation (σ). The 68-95-99.7 rule: ~68% of data falls within 1σ of mean, ~95% within 2σ, ~99.7% within 3σ. Z-score: number of standard deviations from the mean: z = (x − μ)/σ. Linear regression finds the best-fit line (ŷ = a + bx) to predict values; the correlation coefficient r measures strength and direction of linear relationship (r near ±1 = strong, near 0 = weak). Correlation ≠ causation—two variables can be correlated without one causing the other. Sampling distribution: distribution of a statistic (like the mean) across many samples. Confidence interval: a range of plausible values for a population parameter; a 95% CI means 95% of all such intervals would contain the true parameter. Statistical significance (p-value < 0.05) means results are unlikely to occur by random chance alone.

As you engage with this material, consider both the theoretical foundations and practical applications. Think critically about how this concept builds on prior knowledge and where you might apply it beyond the classroom.

Challenge yourself to go beyond memorization—seek to understand the "why" behind the processes and principles.