How to Use the Normal Distribution to Find Probabilities

The normal distribution helps you calculate how likely values or ranges are when a variable is modeled by a symmetric, bell-shaped curve. The key move is to convert the values you care about into z-scores, then use the standard normal cumulative distribution function to get probabilities.

This method answers questions such as “What proportion of scores fall between 80 and 90?” or “What is the chance a measurement exceeds a threshold?” It does not, by itself, show that your data are normally distributed; that assumption needs to be reasonable for the situation.

The normal model and z-scores

A normal random variable $X$ is specified by its mean $\mu$ and standard deviation $\sigma$, written $X \sim N(\mu,\sigma^2)$. The mean is the center of the distribution, and the standard deviation describes its spread. About 68% of values lie within one standard deviation of the mean, about 95% within two, and about 99.7% within three.

To compare a value $x$ with this distribution, standardize it:

The z-score tells you how many standard deviations $x$ is above or below the mean. For example, $z=1.5$ means the value is 1.5 standard deviations above the mean; $z=-1.5$ means it is 1.5 standard deviations below it.

The standard normal cumulative distribution function, written $\Phi(z)$, gives the probability of being at or below a z-score:

A calculator or statistical software can evaluate $\Phi$. A traditional z-table usually gives the same cumulative probabilities, though you should check what convention its entries use.

Worked example: probability of a score in a range

Suppose exam scores are modeled as normally distributed with mean 70 and standard deviation 10. Find the probability that a randomly selected score is between 80 and 90.

First, standardize both endpoints:

The probability between the endpoints is the cumulative probability at the upper endpoint minus the cumulative probability at the lower endpoint:

So about 13.6% of scores fall between 80 and 90 under this model. For a continuous distribution, including or excluding either exact endpoint makes no difference: the probability of one exact value is zero.

Finding a tail probability

For a value below a cutoff $x$, use $P(X \le x)=\Phi(z)$. For a value above it, take the complement:

In the exam-score example, the probability of a score above 80 is $1-\Phi(1)\approx 1-0.84134=0.15866$, or about 15.9%. The probability below 80 is approximately 84.1%. These two tail probabilities add to 1.

If you know a probability and need the corresponding cutoff instead, use the inverse normal function, often labeled invNorm or a quantile function. For example, the 90th percentile corresponds to $z=\Phi^{-1}(0.90)$; convert that z-score back to the original units with $x=\mu+z\sigma$.

Common mistakes

• Using the standard deviation incorrectly: Divide by $\sigma$, not by the variance $\sigma^2$.
• Forgetting the upper-tail complement: $\Phi(z)$ is the probability below $z$, not above it. For an upper tail, calculate $1-\Phi(z)$.
• Subtracting in the wrong order: For an interval from $a$ to $b$, with $a<b$, use $\Phi(z_b)-\Phi(z_a)$.
• Treating the model as a guarantee: The normal distribution is an approximation to real data, not proof that every observation follows a bell curve. Check for strong skew, outliers, or multiple clusters, and consider a graph or subject-matter knowledge.
• Mixing up sample and population quantities: A model may use a known population standard deviation, while a practical analysis may estimate it from a sample. If you are making inference about a mean with an estimated standard deviation, a t-distribution may be more appropriate, especially for small samples.

When the normal distribution is an approximation

Some discrete distributions, such as a binomial distribution with suitable parameters, can be approximated by a normal distribution. Since a discrete variable takes integer values but a normal variable is continuous, a continuity correction may improve the approximation. For example, when approximating $P(X\le 12)$ for an integer-valued variable, use a normal cutoff near 12.5 rather than 12. The adequacy of the approximation depends on the distribution’s parameters; do not apply it automatically.