Z-Score Table

A Z-score table helps you find the probability of a value in a standard normal distribution. It shows how much area lies to the left of a particular z-value. The distribution is bell-shaped with a mean 0 and standard deviation 1.

• The rows and columns of the table define the z-score, and the table cells represent the area.
• For example, the z-score 1.50 corresponds to the area 0.9332, which is the probability that a random variable from a standard normal distribution will fall below 1.50.

Note: The negative z-scores are below the mean, while the positive z-scores are above the mean.

The z-score table is divided into two sections:

1. Positive Z-Score Table: A data point is above the median if its Z-score is positive (greater than 0), with a higher value denoting a larger divergence from the mean.

2. Negative Z-Score Table: A negative Z-score indicates that the data points are nearer the mean.

How to Use a Z-Score Table

Step 1: Calculate the Z-score: Use the formula to find how many standard deviations X is from the mean.

Step 2: Open the Z-score table: Z-values appear up to two decimals (0.00, 0.01, 0.02, ...).

Step 3: Locate the Z-score: Find the row for the first decimal and the column for the second decimal.

The table value gives P(Z ≤ z).

Example: A school has a normally distributed test score with a mean (μ) of 75 and a standard deviation (σ) of 10. A student wants to know the probability of scoring less than 80 on a test.

Solution:

Calculate the Z-score:

Z = 80 −75/10
⇒ Z = 0.5

Look at the Z-scores in the Z-score table to find the corresponding cumulative probability. Let’s say 0.6915.

Thus, the probability of a student scoring less than 80 would be 0.6915 or 69.15%.

How to Interpret Z-Score

Positive z-score -> value is above the mean.

Example: Z = 2 -> 2 standard deviations above the mean.

Negative z-score -> value is below the mean.

Example: Z = −1.5 -> 1.5 standard deviations below the mean.

Applications of Z Score

Z-scores are widely used in many areas, such as:

• Comparing data in statistics and detecting outliers
• Financial analysis (e.g., Altman Z-score for bankruptcy prediction)
• Medical and growth assessments using reference charts
• Performance comparison in sports and academics
• Hypothesis testing, confidence intervals and general data analysis

Solved Examples

Example 1: If the Z-score is 1.5. Find the probability that a randomly selected data point falls below this Z-score.

Solution:

To determine the probability that a randomly selected data point falls below the Z score, we can do the following.

Using a Z-score table or calculator, look for a Z-score of 1.5 and get a corresponding probability of about 0.9332. This means there is a 93.32% probability that the data point falls below a Z-score of 1.5 in the standard normal distribution.

Example 2: Find the probability that the Z score is greater than -1.2

Solution:

To determine the probability that the Z-score is greater than -1.2.

Using the Z-score table, find the cumulative probability associated with -1.2, which would be 0.1151. Subtract this value from 1 to find the probability of greater than -1.2:

1 − 0.1151 = 0.8849

Thus, the probability that the Z-score is greater than -1.2 is approximately 0.8849 or 88.49%.

Practice Questions

1. A class of 100 students took a math test. The mean score is 75, with a standard deviation of 10. What is the Z-score of a student who scored 85 on the test?
2. In applied physics, students measure the time it takes for a ball to fall from a certain height. It is 3 seconds with a standard deviation of 0.5 seconds. If a student measures a fall time of 2.2 seconds, what is the Z-score for this measurement?
3. A company is conducting an employee compensation audit. The average salary is 50,000 with a standard deviation of 8,000. What is the Z-score of an employee with a salary of 56,000?
4. A doctor is measuring the height of a child to compare it with a group of children of the same age. The height of this group is 120 cm and the standard deviation is 5 cm. If the child is 130 cm tall, what is the Z-score for this measurement?
5. In a study of test anxiety among students, the average test anxiety score was 60, with a standard deviation of 10. If a student scores a test anxiety score of 75, what is the Z-score of this score?

Related Articles

• Normal Distribution
• Probability Distribution
• Probability Density Function
• Poisson Distribution