The Normal Distribution and the Empirical Rule

The Normal Distribution and Z-Scores

By the end of this lesson, you will be able to calculate and interpret z-scores, apply the 68-95-99.7 rule, use the normal distribution to find probabilities and percentiles, and assess normality.

The Normal Distribution

Normal distribution: a continuous probability distribution that is:

• Bell-shaped and symmetric about the mean
• Described completely by two parameters: mean $\mu$ and standard deviation $\sigma$
• Notation: $N(\mu, \sigma)$

Why the normal distribution matters:

• Many real-world variables are approximately normally distributed
• The Central Limit Theorem makes it central to inference
• It's the foundation of many statistical tests

The 68-95-99.7 Rule (Empirical Rule)

For any Normal distribution $N(\mu, \sigma)$:

• Approximately 68% of observations fall within 1 standard deviation of the mean: $(\mu - \sigma, \mu + \sigma)$
• Approximately 95% of observations fall within 2 standard deviations: $(\mu - 2\sigma, \mu + 2\sigma)$
• Approximately 99.7% of observations fall within 3 standard deviations: $(\mu - 3\sigma, \mu + 3\sigma)$

Example: SAT Math scores are approximately $N(530, 100)$. What percent score between 430 and 630?

$430 = 530 - 100 = \mu - \sigma$; $630 = 530 + 100 = \mu + \sigma$

By the empirical rule: approximately 68%.

Z-Scores (Standardized Scores)

Z-score: measures how many standard deviations an observation is from the mean.

$$z = \frac{x - \mu}{\sigma}$$

• $z > 0$: observation is above the mean
• $z < 0$: observation is below the mean
• $z = 0$: observation equals the mean

Interpretation: "This observation is $z$ standard deviations above/below the mean."

Example: In a distribution with $\mu = 70$ and $\sigma = 15$:

$x = 85 \Rightarrow z = \dfrac{85-70}{15} = 1.0$ → 1 standard deviation above mean