How do you use a z-score?

To calculate a z-score, you need three pieces of information: your specific data point ($x$), the population mean ($\mu$), and the population standard deviation ($\sigma$). The formula is: $z = \frac{x - \mu}{\sigma}$

You subtract the mean from your data point, and then divide that difference by the standard deviation. This tells you the distance from the average, measured in steps of standard deviation.

Once you have a z-score, you can use it to find probabilities. In a normal distribution (a bell curve), the total area under the curve is exactly 1 (or 100%). A z-score lets you use a standard normal table (often called a z-table) to find the area to the left of your score.

Because the curve is standardized, a z-score of 1.0 always has about 84% of the data to its left. By looking up your calculated z-score in the table, you instantly know the probability of a random value falling below your specific data point.

A very common mistake is confusing the area to the left with the area to the right. A standard z-table always gives you the probability that a value is less than your data point. If a question asks for the probability of scoring higher than a certain value, you must look up the z-table value and subtract it from 1.

Another common slip-up is using the variance instead of the standard deviation in the denominator of the formula. Always double-check that you are using standard deviation ($\sigma$), not variance ($\sigma^2$).