# How to Find the Critical Z Value (With Examples)

To find the critical z value, you work backward from the significance level $\alpha$ using the standard normal distribution. For a two-tailed test at $\alpha = 0.05$, the critical values are $\pm 1.96$. For a one-tailed test at the same level, the critical value is $1.6449$ on the right or $-1.6449$ on the left. This article shows how to find critical z values step by step, with a worked example and the tools that do the arithmetic for you.

## Quick Answer

- A critical z value is the cutoff on the standard normal curve where the rejection region begins [1].
- Two-tailed test at $\alpha = 0.05$: $z = \pm 1.96$ [2].
- One-tailed test at $\alpha = 0.05$: $z = 1.6449$ (right) or $-1.6449$ (left).
- The formula uses the inverse normal CDF: two-tailed $z = \Phi^{-1}(1 - \alpha/2)$, one-tailed $z = \Phi^{-1}(1 - \alpha)$.
- In Excel, `=NORM.S.INV(1-0.05/2)` returns 1.9600.

## Before You Start

You need three things before you look up a z value.

First, the significance level $\alpha$. Common choices are 0.10, 0.05, and 0.01 [1]. This is the probability of rejecting the null hypothesis when it is actually true, so a smaller $\alpha$ means a stricter test and a larger critical value.

Second, the direction of the test. A two-tailed test splits $\alpha$ between both ends of the curve. A one-tailed test puts all of $\alpha$ in one tail, either the right or the left.

Third, the standard normal distribution. The z distribution has a mean of 0 and a standard deviation of 1. Critical values come from this distribution, not from your sample. That is what separates a z-test from a t-test, whose critical values depend on the sample size through the degrees of freedom [2].

If you also need cutoffs for other test statistics, the same logic applies to the chi-square distribution, which you can read about in [this guide to the chi-square table](/blog/data-analysis/chi-square-table-critical-values).

## Step by Step

1. **State your significance level.** Write down $\alpha$. For this walkthrough, use $\alpha = 0.05$.

2. **Decide one-tailed or two-tailed.** This comes from your alternative hypothesis. "Not equal to" is two-tailed. "Greater than" or "less than" is one-tailed.

3. **Compute the cumulative probability.** For a two-tailed test, the cumulative probability at the upper cutoff is $1 - \alpha/2$, leaving $\alpha/2$ in the upper tail. For a one-tailed right test, it is $1 - \alpha$. For a one-tailed left test, it is $\alpha$.

4. **Apply the inverse normal CDF.** The critical value is the z-score whose cumulative probability equals the tail probability you just computed.

$$z_{\text{two-tailed}} = \Phi^{-1}\!\left(1 - \frac{\alpha}{2}\right)$$

$$z_{\text{one-tailed right}} = \Phi^{-1}(1 - \alpha)$$

5. **Attach the sign.** The right tail is positive. The left tail is negative. A two-tailed test uses both signs.

6. **Compare your test statistic to the critical value.** Reject the null hypothesis when the test statistic falls in the rejection region beyond the critical value [1].

Here is the full set of values at the three most common significance levels.

| Test type | $\alpha = 0.10$ | $\alpha = 0.05$ | $\alpha = 0.01$ |
|---|---|---|---|
| Two-tailed | $\pm 1.6449$ | $\pm 1.9600$ | $\pm 2.5758$ |
| One-tailed (right) | $1.2816$ | $1.6449$ | $2.3263$ |
| One-tailed (left) | $-1.2816$ | $-1.6449$ | $-2.3263$ |

A [critical value table](/blog/data-analysis/critical-value-table) lists these same cutoffs if you prefer reading values instead of computing them.

## Worked Example

The dataset is a set of quiz scores from a class of 15 students, used here to set up a one-sample z-test against a hypothesized mean.

| student_id | score |
|---|---|
| 1 | 72 |
| 2 | 85 |
| 3 | 90 |
| 4 | 68 |
| 5 | 77 |
| 6 | 81 |
| 7 | 95 |
| 8 | 64 |
| 9 | 88 |
| 10 | 79 |
| 11 | 83 |
| 12 | 70 |
| 13 | 92 |
| 14 | 75 |
| 15 | 86 |

The summary statistics come out as follows.

| Step | Value |
|---|---|
| Sample size $n$ | 15 |
| Sample mean | 80.3333 |
| Sample standard deviation ($n-1$) | 9.2633 |
| Significance level $\alpha$ | 0.05 |
| Two-tailed cumulative probability | $1 - \alpha/2 = 0.9750$ |
| Two-tailed critical z | $\Phi^{-1}(0.9750) = 1.9600$ |
| One-tailed right cumulative probability | $1 - \alpha = 0.9500$ |
| One-tailed right critical z | $\Phi^{-1}(0.9500) = 1.6449$ |
| One-tailed left critical z | $\Phi^{-1}(0.0500) = -1.6449$ |

The code below reproduces every number.

```python
from scipy import stats
alpha = 0.05
z_two = stats.norm.ppf(1 - alpha/2)   # two-tailed
z_one = stats.norm.ppf(1 - alpha)     # one-tailed right
print(f"Two-tailed critical z = {z_two:.4f}; one-tailed (right) critical z = {z_one:.4f}; one-tailed (left) critical z = {-z_one:.4f}")
```

Output:

```
Two-tailed critical z = 1.9600; one-tailed (right) critical z = 1.6449; one-tailed (left) critical z = -1.6449
```

So if your alternative hypothesis is "the mean is not equal to the hypothesized value," you reject the null when your z statistic is above 1.96 or below -1.96. If your alternative is "the mean is greater," you reject when the statistic exceeds 1.6449.

## Other Ways to Do It

**Excel.** `=NORM.S.INV(1-0.05/2)` returns 1.9600. For a one-tailed right test, use `=NORM.S.INV(1-0.05)`, which returns 1.6449. The function takes a cumulative probability between 0 and 1 and returns the corresponding z-score.

**A z table.** Standard normal tables list cumulative probabilities for z-scores to two decimal places [3]. Find the probability closest to your target tail probability, then read the z-score from the row and column headers. Tables give you about two decimal places of precision, which is enough for most classroom work.

**A critical z value calculator.** Any tool that computes the inverse normal CDF does the job. You enter $\alpha$ and the number of tails, and it returns the cutoff. This is the fastest route when you already know your test setup.

**Statistical software.** Python's `scipy.stats.norm.ppf`, R's `qnorm`, and most stats packages expose the same inverse CDF. The argument is always a cumulative probability.

If your test involves two proportions instead of a mean, the critical z value works the same way, and the [two proportion z-test formula](/blog/data-analysis/two-proportion-z-test-formula) shows how the statistic is built.

## Troubleshooting

**You got a negative value for a right-tailed test.** You passed $\alpha$ instead of $1 - \alpha$ to the inverse CDF. The right tail always uses a probability above 0.5.

**Your value is close to but not exactly 1.96.** That is rounding. The true value is 1.959964, which rounds to 1.96. Tables and most software report 1.96 [2].

**You cannot decide between one and two tails.** Look at your alternative hypothesis, not your data. Choosing the tail after seeing the results inflates your false positive rate.

**Your software asks for a probability and you gave it a percentage.** Enter 0.05, not 5.

**You are unsure whether to use z or t.** Use z when you know the population standard deviation. Use t when you only have the sample standard deviation, since t critical values depend on the sample size [2].

## Common Mistakes

- **Splitting alpha for a one-tailed test.** A one-tailed test puts the whole $\alpha$ in one tail. Splitting it gives you the two-tailed cutoff and makes the test harder to pass than intended. Fix: use $1 - \alpha$ for a one-tailed right test.
- **Using the sample standard deviation with a z critical value.** The z-test assumes you know the population standard deviation [2]. If you substituted the sample value, a t critical value is the correct choice.
- **Reading the table's body as the z-score.** In a standard normal table, the body holds probabilities and the margins hold z-scores [3]. Reading them backward gives nonsense. Fix: locate your probability in the body first.
- **Forgetting the sign on the left tail.** The left critical value is negative. Comparing a negative test statistic to a positive cutoff flips your conclusion.
- **Picking alpha after running the test.** The significance level belongs in your analysis plan before you see data [1]. Changing it afterward turns a null result into a "significant" one.
- **Confusing the critical value with the p-value.** The critical value is a cutoff on the z scale. The p-value is a probability. They answer the same question from opposite directions [1].

## Limitations

The critical z value approach assumes the test statistic follows a standard normal distribution under the null hypothesis. That assumption holds well for large samples by the central limit theorem, but it can fail for small samples from skewed or heavy-tailed populations. In those cases the nominal $\alpha$ is not the true rejection rate, and your conclusions can be off even though the arithmetic is correct.

The method also says nothing about practical importance. A result can clear the critical value and still be too small to matter, especially with a large sample where tiny effects become statistically detectable. The critical value is a decision rule, not a measure of how much the effect matters. For a broader view of how cutoffs work across distributions, see [how to find Q1 and Q3](/blog/data-analysis/how-to-find-q1-and-q3) for the quantile logic behind these values.

## Frequently Asked Questions

### What is the critical z value for a 95% confidence level?

For a two-tailed test at $\alpha = 0.05$, the critical values are $\pm 1.96$ [2]. The 95% refers to the confidence level, which is $1 - \alpha$. The same 1.96 appears in confidence intervals for a mean when the population standard deviation is known.

### How do I find the critical z value for a two-tailed test?

Divide $\alpha$ by 2, subtract that from 1, and take the inverse normal CDF of the result. At $\alpha = 0.05$, that is $\Phi^{-1}(0.9750) = 1.9600$. The negative of that value is the lower cutoff.

### Is the critical z value always 1.96?

No. 1.96 applies only to a two-tailed test at $\alpha = 0.05$. At $\alpha = 0.01$ the two-tailed value is 2.5758, and at $\alpha = 0.10$ it is 1.6449. One-tailed tests at $\alpha = 0.05$ use 1.6449.

### Can I use a critical z value calculator instead of a table?

Yes. A calculator that computes the inverse normal CDF returns the same values as a table, usually with more decimal places. Tables typically report two decimals, which is enough for most work [3].

### What is the difference between a critical value and a p-value?

A critical value is a cutoff on the test statistic scale. A p-value is the probability of observing a statistic at least as extreme as yours, assuming the null hypothesis is true [1]. You reject the null when the statistic exceeds the critical value or, equivalently, when the p-value falls below $\alpha$.

## References

1. [7.1.3.1. Critical values and p values](https://www.itl.nist.gov/div898/handbook/prc/section1/prc131.htm)
2. [Z-test - Wikipedia](https://en.wikipedia.org/wiki/Z-test)
3. [5.3.2: Table of Critical Values of z - Statistics LibreTexts](https://stats.libretexts.org/Courses/Taft_College/PSYC_2200%3A_Elementary_Statistics_for_Behavioral_and_Social_Sciences_(Oja)/01%3A_Description/05%3A_Using_z/5.03%3A_Introduction_to_the_z_table/5.3.02%3A_Table_of_Critical_Values_of_z)

## Further Reading

- [NIST/SEMATECH e-Handbook of Statistical Methods](https://www.itl.nist.gov/div898/handbook/index.htm)
- [Wasserstein RL, Lazar NA (2016). The ASA Statement on p -Values: Context, Process, and Purpose. The American Statistician](https://doi.org/10.1080/00031305.2016.1154108)
- [Krzywinski M, Altman N (2013). Significance, P values and t-tests. Nature Methods](https://doi.org/10.1038/nmeth.2698)

## Related Articles

- [Chi-Square Table: How to Read It and Find Critical Values](/blog/data-analysis/chi-square-table-critical-values)
- [Two Proportion Z-Test: Formula and Worked Example](/blog/data-analysis/two-proportion-z-test-formula)
- [Critical Value Table: How to Read and Use One](/blog/data-analysis/critical-value-table)
- [How to Find Q1 and Q3: Quartiles Explained with Examples](/blog/data-analysis/how-to-find-q1-and-q3)