## 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.

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Saradhi Motamarri, PhD says

Hello Jim, thank you for the nice explanation.

Jim Frost says

You’re very welcome! ðŸ™‚

Funsho Olukade says

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Jafar Mohammed Beker says

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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

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