A normal z table is a standardized statistical tool that shows the area under the standard normal curve to the left of a given z score. It helps researchers and analysts translate a specific test statistic into a probability value.
This resource explains how to read the table, convert percentiles to z scores, and use it across common applications such as confidence intervals and hypothesis testing.
| z Score | Cumulative Probability | Left Tail Area | Two Tail Critical Values |
|---|---|---|---|
| -2.0 | 0.0228 | 0.0228 | ±1.96 |
| -1.645 | 0.0500 | 0.0500 | ±1.645 |
| 0.0 | 0.5000 | 0.5000 | — |
| 1.0 | 0.8413 | 0.8413 | — |
| 1.96 | 0.9750 | 0.9750 | ±1.96 |
| 2.576 | 0.9950 | 0.9950 | ±1.96 |
Understanding the Standard Normal Distribution
The standard normal distribution is a bell-shaped curve with a mean of 0 and a standard deviation of 1. The normal z table provides exact probabilities for this distribution.
Each z score corresponds to a cumulative probability representing the area under the curve to the left of that value.
How to Read a Normal Z Table
To locate a probability, first identify the row for the z score ones and first decimal, then move across to the column for the second decimal.
For example, a z score of 1.40 is found by going to row 1.4 and column 0.00, giving a cumulative probability around 0.9192.
Using Z Scores and Percentiles
You can convert percentiles to z scores using the inverse normal approach, which is common in setting thresholds for statistical significance.
Knowing the corresponding z score for a percentile lets researchers define cutoffs without recomputing the entire distribution each time.
Applications in Confidence Intervals and Hypothesis Testing
In confidence intervals, the normal z table supplies critical values that determine the margin of error for large samples.
In hypothesis testing, it helps compare a test statistic to a reference distribution to decide whether to reject the null hypothesis.
Best Practices for Accurate Table Lookup
Follow these steps to use the normal z table reliably in your statistical work.
- Round your test statistic to two decimal places before looking it up.
- Use the correct tail based on your hypothesis, one or two sided.
- Verify the table header matches cumulative area from the left.
- Combine table values with software for more demanding precision needs.
FAQ
Reader questions
How do I find the area to the right of a positive z score using the table?
Subtract the cumulative probability from 1 to obtain the right tail area for any positive z score.
Can I use the normal z table for sample means when the population standard deviation is unknown? Use it only with large samples and estimated standard deviation; otherwise prefer the t distribution for better accuracy. What z score corresponds to the 95th percentile in a standard normal distribution?
The 95th percentile is approximately 1.645 for a one-tailed test and 1.96 for a two-tailed 95% confidence interval.
How do I handle negative z scores in the table?
For negative z scores, read the row for the negative value and column for the second decimal to find the left tail area directly.