What Is Z Table?
A z-table lists the cumulative area under the standard normal curve to the left of a z-score. This calculator gives the same information instantly and for any z, including the area to the right, the area between 0 and z, the area between two values and the area in both tails.
You can also start from a raw value by entering the mean and standard deviation, so the calculator standardises it for you, or work backwards from a probability to the z-score. The middle 95 % and other standard intervals are shown for reference.
Below you will find the formulas, a variables table, a worked example and answers to common questions such as how to read a z-table by hand.
How to Use the Z Table Calculator
- To go from value to probability: enter x, with μ = 0 and σ = 1 if x is already a z-score. Add a second value to get the area between two points.
- To go from probability to z, switch the direction and enter the area to the left, such as 0.975.
- Read the left, right, between and two-tailed areas, and compare with the table slice.
Z Table Formula
Standardising
Cumulative probability
Areas
Z Table Variables Explained
| Symbol | Meaning | Notes |
|---|---|---|
| z | Standard score | (x − μ) ÷ σ |
| μ, σ | Mean and standard deviation | Used to standardise a raw value |
| Φ(z) | Cumulative probability | P(Z < z) |
| x | Raw value |
Z Table Worked Example
With the example values loaded in the calculator above, the steps are:
- z = (x − μ) ÷ σ = (1.96 − 0) ÷ 1 = 1.96.
- Look up Φ(z): the area to the left is 0.9750021.
- Area to the right = 1 − 0.9750021 = 0.0249979.
Result: z-score = 1.96; Area to the left P(Z < z) = 0.9750021; Area to the right P(Z > z) = 0.0249979.
How to Read a z-Table Result
A z-table gives the area under the standard normal curve to the left of z. The calculator also gives the area to the right, the area between 0 and z, the area between −|z| and |z| and the area in both tails. For z = 1.96, the area between −1.96 and 1.96 is about 0.95.
You can enter a z directly, a raw value with a mean and SD, or a probability to get z. The small table around your z reproduces the part of a printed table you would have used.
How to read a z-table
Split your z-score into its tenths and hundredths. For z = 1.96, find the row labelled 1.9 and the column labelled 0.06; the entry there, 0.9750, is the area to the left. So 97.5% of values lie below z = 1.96. For negative z use the negative half of the table, or use symmetry: Φ(−z) = 1 − Φ(z).
| z | Area to the left | Area between −z and z |
|---|---|---|
| 1.00 | 0.8413 | 0.6827 (about 68%) |
| 1.28 | 0.9000 | 0.8000 |
| 1.645 | 0.9500 | 0.9000 |
| 1.96 | 0.9750 | 0.9500 (about 95%) |
| 2.00 | 0.9772 | 0.9545 |
| 2.576 | 0.9950 | 0.9900 |
| 3.00 | 0.9987 | 0.9973 (about 99.7%) |
Left, right and two-tailed areas
| Question | Use |
|---|---|
| Percentile of a value | Area to the left |
| Probability of exceeding a value | Area to the right |
| Probability within a range | Difference of two left areas |
| Two-tailed p-value | 2 × the smaller tail area |
| Value at a given percentile (inverse) | Probability → z, then x = μ + zσ |
Going backwards: probability to z-score
Many problems give a percentile and ask for the cut-off value, for example the score needed to be in the top 10%. Enter 0.90 as the left area to get z ≈ 1.2816, then convert back with x = μ + zσ. Critical values for confidence intervals work the same way: a 95% two-sided interval leaves 2.5% in each tail, so you look up 0.975.
When the normal model applies
- The variable is approximately normally distributed, or the statistic is a mean or proportion from a large sample.
- Check symmetry and the absence of strong outliers before trusting normal probabilities for raw data.
- For small samples with an unknown σ, use the t distribution instead.
Where Z Table Is Used
Probability from z
Find areas for any z without a printed table.
z from probability
Find the cut-off for a given area.
Raw scores
Enter a mean and SD to work with real units.
Intervals
Find the area between two values.
Common Z Table Mistakes to Avoid
- Check whether your area is the left tail, right tail or middle.
- Use μ = 0 and σ = 1 to enter z directly.
- Symmetry: the area left of −z equals the area right of z.
- The normal model assumes roughly bell-shaped data.
Z Table Terms and Definitions
- Cumulative probability
- The probability that a variable is less than or equal to a given value.
- Percentile
- The value below which a given percentage of observations fall.
Z Table FAQs
What is a z-table?
A table of the standard normal distribution that gives the area to the left of each z-score, which is the cumulative probability.
How do I find the probability from a z-score?
Look up the z-score in the table, or enter it here; the left-area is the probability that Z is less than that value.
How do I find a z-score from a probability?
Use the inverse normal: choose the probability-to-z direction and enter the area to the left.
What is the z-score for 95%?
For a two-sided central 95%, z = ±1.96. For the 95th percentile (area 0.95 on the left), z = 1.645.
How do I find the area between two z-scores?
Subtract the left area of the smaller from the left area of the larger. Enter two values above to do it automatically.
Does this work for any mean and standard deviation?
Yes. Enter μ and σ and the calculator standardises for you.