Z-table values provide the cumulative probability for a standard normal distribution at specific z-scores. These values are essential for calculating confidence intervals, p-values, and critical thresholds in classical hypothesis testing.
Understanding how to read the z-table helps analysts translate a test statistic into actionable insights about where an observation sits within a normal curve.
| Z-Score | Cumulative Probability | Left-Tail Area | Right-Tail Area |
|---|---|---|---|
| -2.00 | 0.0228 | 0.0228 | 0.9772 |
| -1.00 | 0.1587 | 0.1587 | 0.8413 |
| 0.00 | 0.5000 | 0.5000 | 0.5000 |
| 1.00 | 0.8413 | 0.8413 | 0.1587 |
| 2.00 | 0.9772 | 0.9772 | 0.0228 |
How to Interpret Z-Table Values in Practice
Interpreting z-table values starts with locating the row for the ones and first decimal of your z-score, then reading across to the column for the second decimal. The intersecting cell gives the area to the left under the standard normal curve.
For example, a z-score of 1.96 corresponds to a cumulative probability of about 0.9750, meaning 97.5% of the data fall below that point in a standard normal distribution.
Using Z-Scores for Confidence Intervals
Z-table values are foundational for building confidence intervals around sample means when the population standard deviation is known or the sample size is large.
Selecting a 95% confidence level uses the critical z-values of approximately ±1.96, which capture the central 95% of the distribution between the two tails.
To calculate the interval, multiply the standard error by the z-score and add and subtract this margin from the sample mean to obtain lower and upper bounds.
Z-Table Values in Hypothesis Testing
In hypothesis testing, z-table values help determine whether an observed statistic is statistically significant by comparing it to a critical value or by computing a p-value.
For a two-tailed test at the 0.05 significance level, you compare your test statistic to ±1.96; if it falls outside this range, you reject the null hypothesis.
Common Z-Score Regions and Probabilities
Memorizing key regions associated with common z-scores speeds up analysis and reduces lookup errors in day-to-day work.
- A z-score between -1 and 1 covers approximately 68% of the data in a normal distribution.
- Z-scores between -2 and 2 capture about 95% of the data.
- Z-scores between -3 and 3 encompass roughly 99.7% of the observations.
- For a one-tailed test at alpha 0.05, the critical z-value is about 1.645.
- For a two-tailed test at alpha 0.05, the critical z-values are approximately ±1.96.
Best Practices for Working with Z-Table Values
Developing consistent habits when using z-table values improves accuracy and speed in statistical work.
- Always sketch a normal curve and shade the relevant area before looking up values.
- Memorize key z-scores such as ±1.645, ±1.96, and ±2.58 for common confidence levels.
- Double-check whether your test is one-tailed or two-tailed to select the correct critical region.
- Round z-scores to two decimal places when using printed standard normal tables.
- Use software or calculator functions for higher precision and to handle non-standard normal cases.
FAQ
Reader questions
How do I find the cumulative probability for a negative z-score in the z-table?
Locate the row for the first two digits of the negative z-score, then move across to the column for the second decimal. The table typically reports left-tail cumulative probability directly, so the value you read is the proportion of observations below that z-score.
What does a z-table value of 0.975 mean for my test statistic?
A cumulative probability of 0.975 corresponds to a z-score of roughly 1.96, indicating that your test statistic is 1.96 standard deviations above the mean, which is the critical value for a two-tailed 0.05 significance level.
Can I use z-table values for sample sizes smaller than 30?
Z-table values assume an approximate normal shape, which is reasonable for large samples or when the population standard deviation is known. For small samples with unknown population standard deviation, use the t-distribution instead of the standard normal z-table.
How are z-table values applied to calculate p-values in real-world testing?
For a given test statistic, convert it to a z-score, then use the z-table to find the corresponding cumulative probability. The p-value is derived from the tail area beyond the observed statistic, with one-tailed and two-tailed tests adjusting the tail area accordingly.