## Z-Score Table

A z-table, also known as the standard normal table, provides the area under the curve to the left of a z-score. This area represents the probability that z-values will fall within a region of the standard normal distribution. Use a z-table to find probabilities corresponding to ranges of z-scores and to find p-values for z-tests.

The z-table is divided into two sections, negative and positive z-scores. Negative z-scores are below the mean, while positive z-scores are above the mean. Row and column headers define the z-score while table cells represent the area.

Learn how to use this z-score table to find probabilities, percentiles, and critical values using the information, examples, and charts below the table.

## Negative Z-Score Table

## Positive Z-Score Table

## How to Use the Z-table

In the z-score table, its components represent the following:

- Row headings define the z-score to the tenth’s place.
- Column headings add the z-scores’ hundredth’s place.
- Cells within the table represent the area under the standard normal curve to the left of the z-score.

For example, to find the area for z = -2.23, look at the row (-2.2) and column (0.03) intersection, as shown in the truncated z-table below.

The area to the left of z = -2.23 is 0.01287. The probability of values falling below this value in a normally distributed population is 0.01287 or 1.287%.

Here’s how it looks graphically. This z-table chart is a probability distribution plot displaying the standard normal distribution. The shaded area and its probability correspond to the z-score.

Probabilities in z-tables are accurate only for populations that follow a normal distribution.

Learn more about Z-scores, the Normal Distribution, and Probability Distributions.

## Examples of Using the Z-table to Solve Problems

You can use z-score tables to find areas below, above, between, and outside z-scores. In some cases, solving these problems requires simple addition, subtraction, and understanding the symmetric nature of the z-distribution. For the following examples, I include probability distribution plots that correspond to the values in the example. They help link the plain looking z-tables to something more intuitive!

Some of the following z-table methods allow you to use a test statistic from a z-test to find the p-value. The procedure for using z-tables to find p-values depends on whether you’re using a one- or two-tailed test. For one-tailed tests, the directionality is also a factor. In the following sections, I’ve indicated when to use each z-table procedure to find the p-value for different test conditions.

### Using the Z-table to Find Areas Below a Z-score–Percentiles

The most straightforward problem is finding an area below a z-score because the z-table shows that without requiring further calculations. You can use this type of result as a percentile. Additionally, use this method to find the p-value for a one-tailed z-test with the critical region in the left tail. In this testing scenario, when you look up your test statistic, the z-table area value is the p-value.

Suppose you have an apple that weighs 110 grams. Assume that apple weights follow a normal distribution with a mean of 100 grams and a standard deviation of 15.

The z-score for our apple is (110-100) / 15 = 0.67.

The truncated z-table below shows the area for our z-score.

By looking at the intersection for 0.6 and 0.07, the z-table indicates that the area under the curve is 0.74857. This value indicates that 74.857% of all apple weights will be lower than our apple weight of 110 grams. Alternatively, you can say that our apple is at the 74.857^{th} percentile.

The z-table chart below displays this solution visually.

### Finding Areas Above a Z-score

To find areas above a z-score using a z-table, understand that the total area under the curve equals 1. Consequently, to find the area above a Z-score, you just need to find the area below the z-score in the z-table and subtract it from 1.

Area above the Z-score = 1 – area below Z-score

Use this method to find the p-value for a one-sided z-test with the critical region in the right tail. In this testing scenario, the result is the p-value.

For the apple example above, the z-table indicated that the area below z = 0.67 was 0.74857. Consequently, the area above that z-score is 1 – 0.74857 = 0.25143. In other words, 25.143% of apples will weigh more than our apple.

### Finding Areas Between Two Z-scores

Some problems ask you to find the area between two z-scores. The wording might ask you to find the probability of values falling within a specified number of standard deviations from the mean. The number of standard deviations becomes your positive and negative z-scores. However, this process also works for ranges between any two z-scores.

To use a z-table to find this information, do the following:

- Find the area below the higher z-score.
- Find the area below the lower z-score
- Take the larger area and subtract the smaller area from it.

Area for Range Between Z-scores = Larger Area – Smaller Area

For example, the empirical rule describes the areas for z-score ranges of ± 1, 2, and 3 standard deviations from the mean. However, what is the area within ± 1.5 standard deviations from the mean?

The truncated z-tables below show the areas for z-scores of 1.5 and -1.5.

The z-tables indicate that the areas for the higher and lower z-values are 0.93319 and 0.06681, respectively.

Therefore, the area between these two z-scores is 0.93319 – 0.06681 = 0.86638.

We can conclude that 86.638% of values will fall within ± 1.5 standard deviations of the mean in a normally distributed population. The z-table chart below illustrates this answer.

### Finding Areas that Fall Outside Two Z-scores

Problems might ask you to use a z-table to find the area for distribution regions that are more extreme than the given z-scores. In this context, more extreme refers to values that are further away from the mean than the two z-scores.

Use this method to find the p-value for a two-sided z-test with critical regions in both tails. Z tests use Z-scores to determine statistical significance. Learn more about Z-Tests: Uses, Formula & Examples.

To calculate this type of information using a z-table, do the following:

- Find the area above the higher z-score using the method described earlier.
- Find the area below the lower z-score.
- Add the two areas for the total.

For example, a problem might ask you to find the area of z-scores that are more extreme than ± 2.5 standard deviations from the mean.

There’s a shortcut for this method when you have the ± values of the same z-score, as we do in this example. The shortcut involves the fact that the standard normal distribution is symmetrical. Simply double the area of the negative z-value to find the total area in both tails.

Use this shortcut method when a problem provides a test statistic from a two-sided z-test and asks you to use the z-table to find the p-value. Find the area for the negative z-value and double it to obtain the p-value.

For our example, the area less than z = -2.5 equals the region greater than z = 2.5, thanks to the symmetry.

In this case, the z-table indicates that the area less than z = -2.5 is 0.00621. Consequently, the area greater than z = 2.5 is also 0.00621.

Summing these two areas finds the total area for these regions: 0.00621 + 0.00621 = 0.01242.

We can conclude that 1.242% of observations will fall more than ± 2.5 standard deviations away from the mean in a normally distributed population.

If this z-value is a test statistic, the p-value for a two-sided z-test is 0.01242.

The z-table chart below illustrates this result.

### Using Z-Tables to Find Critical Z-values

Z-tables can help you find the critical z-values for a z-test. To find these values, you need to know the significance level and whether you’re performing a one- or two-tailed test.

In a hypothesis test, the results are statistically significant when the test statistic exceeds a critical value. Z-tests use z-values for its test statistic. If you’re performing a t-test, use the T-Distribution Table of Critical Values instead. For a chi-square test, use the chi-square table. For F-tests, use the F-table.

For this procedure, you will first find an area in the negative z-table and then identify the corresponding z-value. That’s the opposite of the other examples where we looked for the z-score first and then found the area. Use the negative table regardless of the number of tails and the directionality (for one-tailed z-tests). It’ll save you from some math!

Look in the negative z-table to find the area that corresponds to your significance level (α) and the number of tails:

- One-tailed test: Look for α in the negative z-table.
- Two-tailed test: Look for α / 2 in the negative z-table.

Suppose we need to find the critical values that correspond to the significance level of 0.05. We’ll find the critical values for both a one- and two-tailed z-test.

Learn more about significance levels, test statistics, critical values, and one- vs. two-tailed hypothesis tests.

### One-tailed Critical Z-value Example

For a one-tailed z-test, look in the negative z-table for the area that equals the alpha of 0.05.

In the truncated negative z-table, I’ve highlighted a cell close to our target alpha of 0.05. The area is 0.04947. This area is at the row and column intersection for the z-value of -1.65. That’s our critical value!

Because our alpha falls between the z-values of -1.64 and -1.65, you can interpolate to find a more precise critical value between them. Instead, I chose the marginally more conservative critical z-value of -1.65.

If the one-tailed z-test is for the left tail, use -1.65 for the critical value. However, if the test is for the right tail, take advantage of the distribution’s symmetry and use +1.65.

The z-table charts below illustrate the critical regions for the two one-tailed z-tests.

### Two-tailed Critical Z-value Example

For a two-tailed z-test, you need to divide your alpha in half because the test splits the area between the upper and lower tails. For a significance level of 0.05, look for the area of 0.05 / 2 = 0.025 in the negative z-table.

In the truncated negative z-table, I’ve highlighted the cell that matches our target area of 0.02500. This area is at the row and column intersection for the z-value of -1.96. That’s our critical value! Because it’s a two-tailed test, we need to use -1.96 and +1.96 for the two critical values.

The z-table chart below illustrates these critical regions.

Saradhi Motamarri, PhD says

Hello Jim, thank you for the nice explanation.

Jim Frost says

You’re very welcome! ðŸ™‚

Funsho Olukade says

Prof Jim is always on point with his explanations of statistical concepts. Well done Prof.

Jafar Mohammed Beker says

Thanks for your contributed by nice explanations

Francis says

one of the most precise and concise tutorials that introduce the concepts and applications in a coherent and easily understandable way! Thank you so much

Jim Frost says

Thanks so much, Francis. I really appreciate your kind words!

Anubrata Das says

Hello Jim, thanks a lot for this nice explanation