Statistics / Unit 3: The Normal Distribution
3.4 Normal Calculations with a Table
Notes
A standard normal table gives the area to the left of a z-score, which is the proportion of values below it.
- Standardize first: $\displaystyle z = \frac{x - \mu}{\sigma}$. Then use the table.
- Area to the right $= 1 -$ area to the left.
- Area between two values $=$ larger left area $-$ smaller left area.
- Example: If the area left of $z = 1.5$ is $0.9332$, the area to the right is $1 - 0.9332 = 0.0668$.
Practice
The area to the left of $z = 1.5$ is $0.9332$. What proportion of values are above $z = 1.5$?
$1 - 0.9332 = 0.0668$.
Scores are normal with $\mu = 70$ and $\sigma = 10$. The area left of $z = 1.0$ is $0.8413$. What proportion of scores are below 80?
$\displaystyle z = \frac{80 - 70}{10} = 1.0$, so the proportion below is $0.8413$.
IQ scores are normal with $\mu = 100$ and $\sigma = 15$. The area left of $z = 0.5$ is $0.6915$ and the area left of $z = -0.5$ is $0.3085$. What proportion of people have IQ scores between 92.5 and 107.5?
$z = \pm 0.5$ for those scores. $0.6915 - 0.3085 = 0.3830$.
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