Z-Table (Standard Normal)

How to read the z-table

1. Work out your z-score: z = (x − μ) / σ, the number of standard deviations your value lies above or below the mean.
2. Split z into two parts: the ones digit and first decimal (the row, e.g. 1.2) and the second decimal (the column, e.g. .05).
3. The cell where they meet is Φ(z), the proportion of the distribution below z.
4. Convert to what you need: the area above z is 1 − Φ(z), and the area between two z-scores is the difference of their two table values.

Worked example: what percentile is a score of 82?

Exam scores are normally distributed with mean 70 and standard deviation 8. A student scores 82, so z = (82 − 70) / 8 = 1.50. Row 1.5, column .00 gives 0.9332: the student scored higher than about 93.3% of the class, the 93rd percentile. The share of students scoring above 82 is 1 − 0.9332 = 0.0668, about 6.7%.

For a range, subtract: the proportion scoring between 74 (z = 0.5) and 82 (z = 1.5) is 0.9332 − 0.6915 = 0.2417, about 24%.

Negative z-scores

The table lists positive z only, because the normal curve is symmetric. For a negative z, use Φ(−z) = 1 − Φ(z). For example, Φ(−1.25) = 1 − 0.8944 = 0.1056: about 10.6% of values lie more than 1.25 standard deviations below the mean.

Reading the table backwards: z from a percentile

To find the z-score for a given percentile, look for the probability inside the table and read off its row and column. For the 90th percentile, the closest values are 0.8997 (z = 1.28) and 0.9015 (z = 1.29), so z ≈ 1.28 (the exact value is 1.2816). The familiar critical values come from this: z = 1.645 leaves 5% in the upper tail, and z = 1.96 leaves 2.5% in each tail of a 95% interval.

Common mistakes

• Using the left-tail value for a right-tail question. “What proportion scores above” needs 1 − Φ(z), not Φ(z).
• Using the z-table for small samples with an unknown standard deviation. When σ is estimated from fewer than about 30 observations, critical values come from the t-table instead.
• Forgetting to double a one-tail area for a two-sided p-value. If z = 2.33, the upper tail is 1 − 0.9901 = 0.0099, and the two-sided p-value is about 0.0198.

Need more precision than a table gives? The p-value calculator returns the area for any z-score, and how to calculate a p-value shows the method.

See also: Critical value calculator · The empirical rule

Frequently asked questions

How do you read a z-table?

Split your z-score into the row (ones digit and first decimal) and the column (second decimal). The cell where they meet is the proportion of the standard normal distribution below that z-score.

What does 0.9750 mean for z = 1.96?

97.5% of a normal distribution lies below 1.96 standard deviations above the mean, leaving 2.5% in the upper tail. With 2.5% in each tail, that is why 1.96 is the critical value for a 95% confidence interval.

How do I find the area to the right of a z-score?

Subtract the table value from 1. For z = 1.5 the table gives 0.9332, so the area to the right is 0.0668.

How do I use the z-table for negative z-scores?

Use symmetry: the area below -z equals 1 minus the area below z. For z = -1.25 the area below is 1 - 0.8944 = 0.1056.