The standard normal distribution table, commonly called the z table, is a foundational tool for interpreting test scores, survey results, and statistical models. Professionals use it to translate a specific observation into a probability that reflects how typical or extreme that result is within a defined population.
Below is a detailed summary of the z table, core terminology, sample values, and common use cases that help you quickly reference and compare key characteristics.
| z value | Area to the left | Two-tailed p value | Common use case |
|---|---|---|---|
| -1.0 | 0.1587 | 0.3173 | Introductory examples |
| -0.67 | 0.2514 | 0.5026 | Borderline significance |
| 0.00 | 0.5000 | 1.0000 | Reference baseline |
| 1.645 | 0.9500 | 0.1000 | 90% confidence critical value |
| 1.960 | 0.9750 | 0.0500 | 95% confidence critical value |
| 2.33 | 0.9901 | 0.0198 | One-tailed 1% significance |
| 2.58 | 0.9951 | 0.0100 | Two-tailed 1% significance |
Core definition and interpretation of the z table
The z table shows the cumulative area under the standard normal curve to the left of a given z score. A z score indicates how many standard deviations an observation is from the mean, assuming a normal distribution with mean zero and standard deviation one. By mapping z scores to areas, the table lets analysts convert test statistics into probabilities, critical values, or confidence levels.
For example, a z score of 1.96 corresponds to an area of 0.9750, meaning 97.5% of the distribution falls below this value. This relationship supports hypothesis testing, quality control, and survey analysis where decisions rely on probabilistic thresholds rather than raw numbers alone.
How to read and look up values in the z table
Reading the z table requires understanding rows and columns that represent z scores to one and two decimal places. The left column provides the z score up to one decimal, and the top row adds the second decimal. The intersecting cell gives the cumulative probability to the left.
- Locate the row for the first one or two digits of the z score.
- Move across to the column for the second decimal place.
- Find the intersection to read the area to the left of that z score.
- For negative z scores, use symmetry and subtract the area from 1 when needed.
Common applications and practical usage of the z table
In practice, the z table supports decisions in education, healthcare, finance, and social science by quantifying how unusual an observed result is. Analysts use it to set critical values for significance tests, construct confidence intervals, and communicate uncertainty in a standardized way.
For instance, a researcher might compare an individual survey score to population norms, a quality engineer might monitor whether a production process remains within control limits, or a policy analyst might evaluate whether an intervention shifts key outcomes beyond random variation.
Relationship to confidence intervals and significance testing
The z table directly informs confidence intervals and hypothesis testing by identifying critical z values that correspond to desired confidence levels or significance thresholds. A 95% confidence interval commonly uses a z value of 1.96 for two-tailed intervals, capturing the central 95% of the normal distribution.
When testing hypotheses, analysts compare a calculated test statistic to critical z values from the table. If the test statistic exceeds the critical value, the result is considered statistically significant, prompting further investigation or action based on the finding.
Best practices and next steps with the z table
- Verify whether you are working with a one-tailed or two-tailed test to choose the correct area from the table.
- Use the z table for quick reference, but pair it with software for precise calculations and large datasets.
- Confirm that the normality assumption is reasonable for your data before relying on z based inference.
- Document the critical z values and confidence levels used so your analysis remains transparent and reproducible.
- Combine the z table with other tools, such as p charts, control charts, and effect size measures, for a complete interpretation.
FAQ
Reader questions
How do I find the area to the right of a positive z score using the table?
First, look up the area to the left of the positive z score in the table, then subtract that value from 1 to get the area to the right.
What do I do if my z score is negative and I need the left tail area?
Use the symmetry of the normal distribution to find the area for the corresponding positive z score, then subtract that area from 1, or directly consult the negative z score section of the table.
Can the z table be used for sample sizes that are not very large?
The z table applies when the sampling distribution of the statistic is approximately normal, which is often reasonable for large samples or when the underlying population is normal and the population standard deviation is known.
How do I convert a probability back to a z score using the table?
Scan the body of the table for the probability closest to your target area, then read the corresponding z score in the margins to identify the critical value associated with that probability.