Standard Normal Distribution Table — Definition, Formula & Examples

A standard normal distribution table (z-table) is a reference chart that shows the probability that a value from the standard normal distribution falls at or below a given z-score. You look up a z-score in the table and read off the corresponding cumulative probability.

The standard normal distribution table lists values of $\Phi(z) = P(Z \leq z)$ , where $Z \sim N(0,1)$ is a standard normal random variable with mean 0 and standard deviation 1. Each entry gives the area under the standard normal curve to the left of the specified z-value.

Key Formula

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

Where:

$x$ = The observed data value
$\mu$ = The population mean
$\sigma$ = The population standard deviation
$z$ = The z-score used to look up probability in the table

How It Works

To use the table, first convert your raw data value to a z-score using

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

. Then find the row matching the ones and tenths digit of your z-score (e.g., 1.4) and the column matching the hundredths digit (e.g., 0.05). The intersection gives

P(Z \leq 1.45)

. For right-tail probabilities, compute

P(Z > z) = 1 - \Phi(z)

. For probabilities between two z-scores, subtract:

P(a < Z < b) = \Phi(b) - \Phi(a)

Worked Example

Problem: Exam scores are normally distributed with a mean of 70 and a standard deviation of 10. What is the probability a student scores below 85?

Convert to a z-score: Subtract the mean and divide by the standard deviation.

z = \frac{85 - 70}{10} = 1.50

Look up z = 1.50 in the table: Find row 1.5 and column 0.00. The table entry is 0.9332.

P(Z \leq 1.50) = 0.9332

Interpret the result: This means about 93.32% of the area under the curve lies to the left of z = 1.50.

Answer: The probability a student scores below 85 is approximately 0.9332, or 93.32%.

Why It Matters

The z-table is one of the most frequently used tools in AP Statistics and introductory college statistics courses. Hypothesis testing, confidence intervals, and quality control calculations all rely on looking up probabilities from the standard normal distribution.

Common Mistakes

Mistake: Forgetting that the table gives the left-tail probability and using the table value directly for right-tail questions.

Correction: For P(Z > z), subtract the table value from 1. For example, if the table gives 0.9332 for z = 1.50, then P(Z > 1.50) = 1 − 0.9332 = 0.0668.