How to Read a Z Score Table

A Z Score Table is a chart that shows the probability of a value occurring within a standard normal distribution. It helps students, researchers, and professionals find the area under the bell curve for a given z score. This table is very important in statistics and probability because it allows you to calculate probabilities, confidence intervals, and p-values in hypothesis testing. With a z score table, you can easily connect raw scores to their probabilities without complex calculations. In this guide, we explain what a z-score is, how to interpret it, and what a z-score table means. We also provide both positive and negative z tables along with step-by-step examples showing how to read probabilities from them.

What is a Z Score?

A z-score is a statistical measure that shows how many standard deviations a data point is from the mean of a distribution. It is calculated using the formula:

$$z = \frac{x – \mu}{\sigma}$$

where:

x is the raw score,
μ is the mean,
σ is the standard deviation.

A positive z-score means the value lies above the mean, while a negative z-score means it lies below the mean. For example, a z-score of 2 means the score is two standard deviations higher than the average. Conversely, a z-score of -1 means it is one standard deviation lower.

The figure below shows the standard normal distribution curve, where the mean is at the center (0), and each step along the horizontal axis represents one standard deviation. Z-scores allow us to locate any value on this curve and determine its probability.

Standard Normal Distribution Curve

The concept of z-scores is tied to the standard normal distribution. A standard normal distribution is a special type of normal distribution with a mean of 0 and a standard deviation of 1. When raw data is converted into z-scores, it becomes much easier to work with probabilities and compare values across different datasets. This transformation makes statistical analysis more consistent because every value is measured on the same standardized scale.

Z-scores are widely used to calculate probabilities, p-values, and critical values in statistics. They form the foundation for hypothesis testing, confidence intervals, and many statistical tests. By looking up a z-score in a z score table, you can quickly determine the probability associated with that value.

For faster results, you can also try our free Z Score Calculator, which instantly computes z-scores without the need to insert values in the z score formula.

How to Interpret Z Scores

Here is how to interpret z-scores:

A z-score less than 0 represents a value below the mean.
A z-score greater than 0 represents a value above the mean.
A z-score equal to 0 represents a value exactly at the mean.
A z-score of +1 means the value is 1 standard deviation above the mean; +2 means 2 standard deviations above the mean, and so on.
A z-score of -1 means the value is 1 standard deviation below the mean; -2 means 2 standard deviations below the mean, and so on.
In a normal distribution:
- About 68% of values fall between -1 and +1.
- About 95% of values fall between -2 and +2.
- About 99% of values fall between -3 and +3.

What is a Z Score Table?

A Z Score Table is a chart that shows the probability, or area under the curve, for different z-scores in the standard normal distribution. It is sometimes called a Z Table, a Standard Normal Table, or a Standard Normal Distribution Table. The table lists z-scores along the rows and columns, and at the intersection, you find the corresponding probability. In simple terms, it helps you translate a z-score into the probability of a value occurring in a normal distribution. This makes it possible to move from raw scores to meaningful probabilities without doing long calculations.

A Z table is widely used in hypothesis testing, confidence intervals, and general probability analysis. For example, in hypotheses tests using a z test, a Z table is used to find the critical values. It’s also used to determine the p-values for z tests or even to estimate to estimate the range where most data points lie. Because of these uses, the z score table is one of the most important tools in statistics.

Types of Standard Normal Tables

When working with the normal distribution, it is important to understand that z scores can be either positive or negative. Because of this, standard normal tables are usually divided into two main types. Each table makes it easy to look up probabilities depending on whether the value lies to the right or left of the mean.

Positive Z Score Table

The Positive Z Score Table shows probabilities for z-scores greater than zero. These values lie on the right side of the mean in the standard normal distribution. Thus, each entry in the table gives the cumulative probability from the far left of the curve up to the given positive z-score. For example, looking up Z = 1.65 would tell you the probability of a value being less than or equal to that point.

Positive Standard Normal Table

Negative Z Score Table

The Negative Z Score Table lists probabilities for z-scores less than zero. These values lie on the left side of the mean. Because the normal distribution is symmetric, the negative table mirrors the positive one. For instance, the probability of Z = -1.65 will be the same as the area to the left of -1.65, which equals the probability to the right of +1.65.

Negative Standard Normal Table

Reading a z score table may look confusing at first. However, the process is simple once you understand the row and column system. The table is set up with rows and columns that represent parts of the z-score.

Rows show the z-score up to one decimal place (the integer and first decimal).
Columns show the second decimal place (hundredths).
The intersection of the row and column gives the cumulative probability (area under the curve to the left of that z-score).

To find the probability of a value being less than or equal to any z score, follow these steps:

Locate the row containing the first digit of the z score and the first decimal
Move across to the column to find the second decimal
Look for the intersection of these two values. This will give you the probability corresponding to that z score

Worked Examples Using Z Score Table

Example 1. Find Probability From Positive Standard Normal Table

Find the probability of a value being less than or equal to a z-score of 1.65.

To find this probability, go the the positive Z score table and follow these steps:

Step 1: Go to the row and look for 1.6.
Step 2: Move to the column and look for 0.05 (Since 1.6 + 0.05 = 1.65).
Step 3: Look for the instersection of 1.6 and 0.05. This gives you 0.9505.

Thus, P(Z < 1.65) = 0.9505. This implies that the probability of a value being less than or equal to 1.65 is 95.05%

Example 2. Find Probability from Negative Standard Normal Table

Find the probability of a value being less than or equal to a z-score of -1.23.

To find this probability, you’ll need to use the negative Z score table, and follow these steps:

Step 1: Go to the row and look for -1.2.
Step 2: Move to the column 0.03 (Since -1.2 + (-0.03) = -1.23).
Step 3: Look for the intersection of -1.2 and 0.03. This will give you 0.1093.

Thus, P(Z < -1.23) = 0.1093. This means that the probability of a value being less than or equal to -1.23 is 10.93%.

Example 3. Find Z score from Probability

A researcher wants to construct a 95% confidence interval for a population mean using the standard normal distribution. What z-scores mark the boundaries of the middle 95% of the distribution? Use the z score table to find your answer.

Solution

A 95% confidence interval means that 95% of the area is in the center, leaving 5% split equally between the two tails.
- Left tail area = 0.025
- Right tail area = 0.025
To find the cutoff points, we look for the z-scores that correspond to these probabilities.
- For the lower bound, we want P(Z ≤ z) = 0.025.
- For the upper bound, we want P(Z ≤ z) = 0.975.
From the z score table:
- P(Z ≤ -1.96) = 0.0250
- P(Z ≤ 1.96) = 0.9750
Therefore, the z-scores that capture the middle 95% of the distribution are: z = -1.96 and z = +1.96

Thus, the middle 95% lies between -1.96 and 1.96.

Z Score Table vs. Z Score Calculator

A Z Score Table is a traditional way of finding probabilities from z-scores. It is especially helpful in exams, textbooks, or classroom learning where manual calculation is required. However, looking up values in the table can sometimes be slow, especially when you are working with many different z-scores. In such cases, using a calculator is much faster because it gives instant results without flipping through rows and columns.

That is why we created an easy-to-use Z Score Probability Calculator for quick lookups. A calculator is best when doing research, assignments, or data analysis using software. On the other hand, knowing how to read a z score table is still useful in tests, lectures, and when no digital tools are allowed. Both methods are valuable, and together they make learning statistics more flexible and practical.

Frequently Asked Questions

What is a Z Score Table?

A Z Score Table, also known as a Standard Normal Table or Normal Distribution Z Table, is a chart that lists the cumulative probabilities for different z-scores in the standard normal distribution.

How do you read a Standard Normal Table?

To read the table, look at the row for the first decimal place of the z-score and the column for the second decimal place. The value at the intersection shows the probability to the left of that z-score under the normal curve.

What is the difference between a Positive and Negative Z Table?

A positive z table lists probabilities for z-scores greater than the mean (right-hand side of the bell curve). On the other hand, a negative z table lists probabilities for z-scores less than the mean (left-hand side of the bell curve).

How is the Standard Normal Distribution Table used in hypothesis testing?

In hypothesis testing, the table helps convert z-scores into p-values and find critical values. This determines whether to reject or accept the null hypothesis.

What are cumulative and complementary Normal Distribution Tables?

A cumulative z table shows the probability from the far left up to a z-score. However, a complementary table shows the tail probability (area to the right). Thus, both describe the same distribution but present probabilities differently.

How do you interpret z-scores using the Standard Normal Table?

A z-score shows how many standard deviations a value is from the mean. For instance, a z-score of +1.65 corresponds to a probability of 0.9505, which means the value is higher than about 95% of all values in the distribution.