Understanding Z-Scores and P-Values
A Z-score (also called a standard score) measures how many standard deviations a data point is from the mean of a distribution. In hypothesis testing and confidence interval calculations, Z-scores correspond to specific probability values (P-values) that represent confidence levels.
This calculator provides a quick lookup table for converting commonly used P-values (confidence levels) to their corresponding one-tailed Z-scores.
What is a P-Value?
The P-value represents the probability that a random variable will have a value less than or equal to a given value, assuming a normal distribution. In confidence interval terminology, it represents the area under the standard normal curve.
For example:
- A 95% confidence level corresponds to a Z-score of 1.645 (one-tailed)
- A 99% confidence level corresponds to a Z-score of 2.326 (one-tailed)
One-Tailed vs. Two-Tailed Z-Scores
This calculator provides one-tailed Z-scores, which are used when you're testing for a deviation in one specific direction (either greater than or less than the mean).
Two-tailed Z-scores would be different values, as they account for deviations in both directions. For example:
- 95% one-tailed = 1.645
- 95% two-tailed = 1.96
Common Z-Score Lookup Table
| P-Value (Confidence Level) | One-Tailed Z-Score |
|---|---|
| 70% | 0.524 |
| 75% | 0.674 |
| 80% | 0.842 |
| 85% | 1.036 |
| 90% | 1.282 |
| 95% | 1.645 |
| 98% | 2.054 |
| 99% | 2.326 |
| 99.5% | 2.576 |
| 99.9% | 3.291 |
Applications
Z-scores and P-values are fundamental in:
- Hypothesis testing: Determining statistical significance
- Confidence intervals: Establishing ranges for population parameters
- Quality control: Setting control limits in manufacturing
- Research analysis: Evaluating experimental results
- Risk assessment: Calculating probabilities in finance and insurance
Example
If you're conducting a one-tailed hypothesis test at a 95% confidence level:
- Select 95% from the dropdown
- Click Calculate
- The Z-score is 1.65 (rounded from 1.645)
This means that 95% of the area under the standard normal curve falls below a Z-score of 1.645. In hypothesis testing, this value is often compared to your calculated test statistic to determine statistical significance.
Important Notes
- These values are for the standard normal distribution (mean = 0, standard deviation = 1)
- For two-tailed tests, you would use different critical values
- More precise Z-scores can be obtained from complete Z-tables or statistical software
- The relationship between P-values and Z-scores is based on the cumulative distribution function of the normal distribution