Z Table – Cumulative from Left
P(Z ≤ z) — Area to the left of z
| z |
|---|
How to Read the Z Table
The Z table shows the cumulative probability P(Z ≤ z) — the area under the standard normal curve to the left of the given z-score.
Steps to Use the Z Table
- Find the row corresponding to the integer part and first decimal of your z-score (e.g., for z = 1.65, look in the row for 1.6)
- Find the column corresponding to the second decimal digit (e.g., for z = 1.65, look in the 0.05 column)
- The intersection gives you P(Z ≤ 1.65) = 0.9505
Example: Finding P(Z ≤ 1.96)
1. Row: z = 1.9
2. Column: 0.06
3. Value: P(Z ≤ 1.96) = 0.9750 → This means 97.5% of values fall below z = 1.96
Common Z Values to Remember
| Purpose | Z-Score | P(Z ≤ z) |
|---|---|---|
| 90% confidence interval | ±1.645 | 0.9500 |
| 95% confidence interval | ±1.960 | 0.9750 |
| 99% confidence interval | ±2.576 | 0.9950 |
| 1 standard deviation above mean | +1.000 | 0.8413 |
| 2 standard deviations above mean | +2.000 | 0.9772 |
FAQ
What is a z-score?
A z-score tells you how many standard deviations a value is from the mean of a standard normal distribution (mean = 0, std dev = 1). A z-score of 1.96 means the value is 1.96 standard deviations above the mean.
How do I find the area between two z-scores?
Subtract the two z-table values. For example, P(−1 < Z < 1) = P(Z ≤ 1) − P(Z ≤ −1) = 0.8413 − 0.1587 = 0.6826, which is the famous 68% rule.
Why is z = 1.96 used for 95% confidence intervals?
For a 95% CI, we need 95% of the distribution to fall within our interval. Since the normal distribution is symmetric, we split the remaining 5% equally: 2.5% in each tail. P(Z ≤ 1.96) = 0.9750, which leaves exactly 2.5% in the right tail. So z = ±1.96 captures 95% of the distribution.