The Normal Distribution and Its Areas: Quick Revision

A brief note on The Normal Distribution and Its Areas. Read it before a test. Try each question at the end before you open its answer.

The curve

• Density curve: total area $1$. The area between two values is the share of values there.
• Normal curve: a symmetric bell, fixed by the mean $\mu$ and the standard deviation $\sigma$.
• The rule: about $68\%$, $95\%$ and $99.7\%$ within $1$, $2$ and $3$ standard deviations (exactly $0.6827$, $0.9545$ and $0.9973$).

Areas

• z-score: $z = \dfrac{x - \mu}{\sigma}$.
• Below: $\Phi(z)$. Above: $1 - \Phi(z)$. Between: $\Phi(z_2) - \Phi(z_1)$.
• Symmetry: $\Phi(-z) = 1 - \Phi(z)$.
• Percentile: find $z$ from the share, then $x = \mu + z\sigma$.
• Python: NormalDist(mu, sigma).cdf(x) and .inv_cdf(share).

When it fits

• Binomial counts: close to normal when $np \ge 10$ and $n(1 - p) \ge 10$.
• Not normal: skewed data such as waiting times and incomes, bounded data, small counts.
• Checks: a histogram, the mean against the median, the 68-95 shares.
• Totals and averages of many independent parts tend to be normal.

A worked example

Masses of made-up pumpkins follow a normal curve with mean $6.0\ \text{kg}$ and standard deviation $1.5\ \text{kg}$.

1. Heavier than $8.25\ \text{kg}$. $z = \dfrac{8.25 - 6.0}{1.5} = 1.5$, so the share is $1 - 0.9332 = 0.0668$.
2. Between $4.5\ \text{kg}$ and $7.5\ \text{kg}$. That is one standard deviation each way: $0.6827$.
3. The lightest $5\%$. $z \approx -1.6449$, so they weigh less than $6.0 - 1.6449 \times 1.5 \approx 3.53\ \text{kg}$.

Practice with new numbers

A program to try

The program checks the worked example with statistics.NormalDist.

Mistakes to avoid

• The height of the curve read as a probability.
• The area to the left used for "above".
• The sign of $z$ dropped for a value below the mean.
• A normal model used for skewed or bounded data.

Check yourself

1. What share of a normal distribution lies below $\mu - \sigma$?

2. A normal distribution has $\mu = 30$ and $\sigma = 5$. Find the z-score of $22$.

3. Use the table value $\Phi(2) = 0.9772$ to find $P(Z < -2)$.

4. Is $B(40, 0.5)$ close to normal by the rule of $10$?

Sources

• OpenStax, Introductory Statistics, Chapter 6 on the normal distribution, and Chapter 7 on the central limit theorem.