What is Z-Score Table? Formula, Example, and Chart

What is Z-Score Table?

The number of standard deviations from the mean is defined as the Z-score. A data point represents the number of standard deviations below or above the mean. A raw score in the form of a Z-score is also known as a standard score, and it can be plotted on a normal distribution curve. The Z-score ranges from -3 to +3 standard deviations. 

A Z-score may be used to calculate the difference or distance between a value and the mean value. When a variable is “standardized,” its mean becomes zero and its standard deviation becomes one. 

What is Z-Score Formula?

It is a method of comparing test findings to those of a “normal” population. 

If X is a random variable with a mean (μ) and standard deviation (σ), its Z-score may be determined by subtracting the mean from X and dividing the result by the standard deviation. 

\(z= \frac{(x- \mu )}{ \sigma }\)

μ is the mean value 

x is the test value 

σ is the standard deviation 

Z-Score Interpretation

Here’s how to read z-scores: 

• A z-score less than 0 denotes an element that is less than the mean. 
• A z-score larger than 0 denotes an element that is greater than the mean. 
• A z-score of 0 indicates that the element is mean. 
• A z-score of 1 indicates an element that is one standard deviation above the mean; a z-score of 2 represents an element that is two standard deviations above the mean; and so on. 
• A z-score of -1 denotes an element that is one standard deviation below the mean; a z-score of -2 denotes an element that is two standard deviations below the mean; and so on. 
• If the set has a high number of items, about 68% of the elements have a z-score between -1 and 1; almost 95% have a z-score between -2 and 2; and approximately 99% have a z-score between -3 and 3. 

Standard Normal Probabilities

Example for Z-Score

Let’s understand this better with an example

Example: The mean of the test scores of students in a class test is 60, with a standard deviation of 12. What is the most likely percentage of students who scored higher than 85? 

Solution: The provided data’s z score is, 

z= (85-60)/12=2.08 

According to the z-score table, the proportion of the data included inside this score is 0.9812. 

This indicates that 98.12% of the students have test scores below 85, while the percentage of students who have test scores over 85 is (100-98.12) % = 1.88%. 

Final Verdict

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FAQ’s

What are the Types of Z Score Tables? 

Positive Z Score Table: This indicates that the observed value is greater than the mean of all values. 

Negative Z Score Table: This indicates that the observed value is less than the mean of all values.