Skill: Calculate probabilities using normal distributions

A normal distribution is a symmetric, bell-shaped model described by its:

• Mean $\mu$: the center of the distribution
• Standard deviation $\sigma$: how spread out the values are

To find a probability, first convert each value to a standard score:

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

A $z$-score tells you how many standard deviations a value is from the mean. Then use a standard-normal table or technology to find the area to the left of that $z$-score.

Worked example

Suppose test scores are normally distributed with mean $70$ and standard deviation $8$. Find the probability that a randomly selected score is between $62$ and $82$.

1. Write the probability

$$P(62<X<82)$$

2. Standardize both values

For $62$:

$$z=\frac{62-70}{8}=-1$$

For $82$:

$$z=\frac{82-70}{8}=1.5$$

So the probability becomes:

$$P(-1<Z<1.5)$$

3. Use the standard-normal table

The table gives the area to the left of each $z$-score:

$$P(Z<1.5)=0.9332$$ $$P(Z<-1)=0.1587$$

4. Subtract the two areas

The area between the two values is:

$$P(-1<Z<1.5)=0.9332-0.1587=0.7745$$

Therefore,

$$\boxed{P(62<X<82)=0.7745}$$

There is a 77.45% probability that a randomly selected score is between $62$ and $82$.

For a value above a cutoff, use the complement:

$$P(X>a)=1-P(X<a)$$

For a continuous normal variable, the probability of one exact value is zero:

$$P(X=70)=0$$

Probabilities refer to areas over intervals, such as being below, above, or between values.