A ztable is a standard normal table: it lists numerical values of the cumulative distribution function (CDF) for a standard normal random variable \(Z\sim N(0,1)\).
Two common ztable formats
- Left-tail table: entries are \(\Phi(z)=P(Z\le z)\).
- Mean-to-z table: entries are \(A(z)=P(0\le Z\le z)\) for \(z\ge 0\).
Conversions for \(z\ge 0\):
Symmetry for any \(z\):
How to read a ztable (row and column)
- Write the z-score to two decimal places (or the precision used by the table).
- Use the row for the ones and tenths digits (for example, \(1.2\)).
- Use the column for the hundredths digit (for example, \(0.03\)).
- The table entry is the probability associated with that \(z\) (usually \(\Phi(z)\)).
Mini ztable excerpt (left-tail CDF)
The excerpt below illustrates how the row \(1.2\) and column \(0.03\) locate \(\Phi(1.23)\).
| z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 |
|---|---|---|---|---|---|
| 1.2 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 |
Core probability computations using ztable values
1) Left-tail probability
If the table is a left-tail table, the entry is already the desired probability:
2) Right-tail probability
Right-tail areas use the complement rule:
3) Between two z-scores
Areas between two values use subtraction:
Example with \(a=-0.85\) and \(b=1.23\). Using symmetry, \(\Phi(-0.85)=1-\Phi(0.85)\approx 1-0.8023=0.1977\).
4) Two-tailed probability
Two-tailed areas are often written as:
Example with \(z=1.96\), where a typical ztable gives \(\Phi(1.96)\approx 0.9750\):
Visualization: ztable area as shaded standard normal probability
Practical checklist for ztable use
- Confirm the table type: left-tail \(\Phi(z)\) versus mean-to-z \(A(z)\).
- Handle negatives consistently: use symmetry \(\Phi(-z)=1-\Phi(z)\) when only nonnegative \(z\) values appear.
- Use complements and differences: right tail \(1-\Phi(z)\); between two values \(\Phi(b)-\Phi(a)\).
- Round z-scores appropriately: many tables are built for two decimals; interpolation is an optional refinement.