The Normal Distribution and Empirical Rule
Understand the bell-shaped normal distribution, its properties (symmetric, mean = median = mode), and the 68-95-99.7 empirical rule. Apply to real-world data and identify when a normal model is appropriate.
A lesson of Algebra 2, in Statistics and Data Analysis, about 10 minutes.
By the end of this lesson, you will be able to describe properties of the normal distribution, apply the 68-95-99.7 Empirical Rule, calculate probabilities using symmetry and the normal curve, and recognize normally distributed data.
The Big Idea
The normal distribution is the most important distribution in statistics. It describes scores, heights, measurement errors, and countless natural phenomena. Its bell-shaped symmetry means that most data clusters near the mean, with the frequency decreasing symmetrically as values move further from the center.
Properties of the Normal Distribution
- Symmetric about the mean $\mu$
- Bell-shaped — unimodal (one peak)
- Mean = Median = Mode (all equal at the center)
- Asymptotic to the $x$-axis — never touches it
- Total area under the curve = 1 (represents 100% of data)
- Determined by two parameters: mean $\mu$ and standard deviation $\sigma$
The Empirical Rule (68-95-99.7 Rule)
For normally distributed data:
$$68%\text{ of data lies within } 1\sigma \text{ of } \mu$$ $$95%\text{ of data lies within } 2\sigma \text{ of } \mu$$ $$99.7%\text{ of data lies within } 3\sigma \text{ of } \mu$$
Breaking it down (using symmetry):
- Within $\pm 1\sigma$: 68% → 34% on each side
- Between $1\sigma$ and $2\sigma$: 13.5% on each side
- Between $2\sigma$ and $3\sigma$: 2.35% on each side
Example 1 — Applying the Empirical Rule
Test scores are normally distributed with $\mu = 75$ and $\sigma = 8$.
Within 1σ: $75 \pm 8$ → scores 67 to 83 → 68% of students
Within 2σ: $75 \pm 16$ → scores 59 to 91 → 95% of students
What percent scored above 91?
$91 = \mu + 2\sigma$ → 2.5% above (half of the 5% outside 2σ, since symmetric)
Example 2 — Finding the Range
Heights of adult males: $\mu = 70$ inches, $\sigma = 3$ inches.
99.7% of males are between $70 - 9 = 61$ and $70 + 9 = 79$ inches.
What percent are between 67 and 76 inches?
67 = 70 - 3 = $\mu - \sigma$; 76 = 70 + 2 = ... wait, $\mu + 2\sigma = 76$. So this spans from $-1\sigma$ to $+2\sigma$.