What a z score table represents
A z score table (also called a standard normal table or z table) is used with the standard normal random variable \(Z\), where \(Z \sim N(0,1)\). The most common z score table reports the cumulative distribution function \(\Phi(z) = P(Z \le z)\), meaning the area under the standard normal curve to the left of the given \(z\)-value.
Key idea: The table entry for \(z\) is typically \(\Phi(z)\).
Right-tail and interval probabilities are computed from \(\Phi(z)\) using subtraction and symmetry.
How to locate a value in a z score table
- Round \(z\) to two decimal places (most tables are built that way).
- Row gives the first decimal place (for example, \(1.2\)).
- Column gives the second decimal place (for example, \(0.03\)).
- Cell value is \(\Phi(z) = P(Z \le z)\) for that \(z\).
Mini excerpt of a typical z score table
The excerpt below is consistent with a cumulative-left z score table \(\Phi(z)\). For example, the entry at row \(1.2\) and column \(0.03\) corresponds to \(z = 1.23\).
| z | 0.00 | 0.01 | 0.02 | 0.03 | 0.04 |
|---|---|---|---|---|---|
| 1.20 | 0.8849 | 0.8869 | 0.8888 | 0.8907 | 0.8925 |
| 0.50 | 0.6915 | 0.6949 | 0.6985 | 0.7019 | 0.7054 |
Visualization: area to the left of a z-value
Core formulas used with a z score table
Standardizing: if \(X \sim N(\mu,\sigma)\), then \(Z = \dfrac{X-\mu}{\sigma}\) and \(Z \sim N(0,1)\).
Symmetry: \(\Phi(-z) = 1 - \Phi(z)\).
Right tail: \(P(Z \ge z) = 1 - \Phi(z)\).
Between two values: \(P(a \le Z \le b) = \Phi(b) - \Phi(a)\).
Worked examples using the z score table
Example 1: Find \(P(Z \le 1.23)\)
Locate \(z = 1.23\) in the z score table: row \(1.2\) and column \(0.03\). The table entry is \(\Phi(1.23) \approx 0.8907\).
\[ P(Z \le 1.23) = \Phi(1.23) \approx 0.8907 \]
Example 2: Find \(P(-0.50 \le Z \le 1.23)\)
First find each cumulative probability from the z score table (using the left-tail convention).
\[ \Phi(1.23) \approx 0.8907 \]
\[ \Phi(-0.50) = 1 - \Phi(0.50) \]
From the table, \(\Phi(0.50) \approx 0.6915\), so \(\Phi(-0.50) \approx 1 - 0.6915 = 0.3085\).
\[ P(-0.50 \le Z \le 1.23) = \Phi(1.23) - \Phi(-0.50) \approx 0.8907 - 0.3085 = 0.5822 \]
Example 3: Convert a percentile to a z-value
Suppose the 95th percentile of the standard normal distribution is needed. This means finding \(z\) such that \(\Phi(z) = 0.9500\).
In a cumulative-left z score table, the closest value to \(0.9500\) occurs near \(z = 1.64\) or \(z = 1.65\). Interpolation gives the widely used critical value \(z \approx 1.645\).
\[ \Phi(z) = 0.95 \quad \Rightarrow \quad z \approx 1.645 \]
Applying the z score table to a nonstandard normal variable
If a measurement \(X\) is normally distributed with mean \(\mu = 70\) and standard deviation \(\sigma = 8\), the probability \(P(X \le 80)\) can be found by standardizing and then using the z score table.
\[ z = \frac{80 - 70}{8} = \frac{10}{8} = 1.25 \]
Then \(P(X \le 80) = P(Z \le 1.25) = \Phi(1.25)\). From the z score table, \(\Phi(1.25) \approx 0.8944\) (value depends on the specific table rounding).
Common pitfalls to check
- Table type: Some tables report right-tail area or area between 0 and \(z\). Confirm whether the z score table is \(\Phi(z) = P(Z \le z)\) before computing tails.
- Negative z-values: Use symmetry \(\Phi(-z) = 1 - \Phi(z)\) when the table lists only positive \(z\).
- Rounding: Standardize carefully and round \(z\) consistently with the table’s precision (commonly two decimals).