Z-Score Calculator

What is a Z-Score?

A z-score (also called a standard score) indicates how many standard deviations a data point is from the mean of a dataset. It allows you to compare values from different datasets by standardizing them to a common scale.

Z-Score Formula

The formula for calculating a z-score is:

Interpreting Z-Scores

Z-scores tell you how unusual or common a value is within a dataset:

• A z-score of 0 means the data point equals the mean
• A positive z-score indicates the data point is above the mean
• A negative z-score indicates the data point is below the mean
• A z-score of +1 or -1 means the data point is 1 standard deviation above or below the mean
• A z-score of +2 or -2 means the data point is 2 standard deviations above or below the mean

The Standard Normal Distribution

In a standard normal distribution (bell curve):

• About 68% of values fall within 1 standard deviation of the mean (z-scores between -1 and +1)
• About 95% of values fall within 2 standard deviations of the mean (z-scores between -2 and +2)
• About 99.7% of values fall within 3 standard deviations of the mean (z-scores between -3 and +3)

Applications of Z-Scores

• Identifying outliers in a dataset
• Comparing scores from different distributions
• Creating standardized test scores
• Calculating probabilities using the standard normal distribution
• Quality control in manufacturing

Example

Suppose a student scores 85 on a test where the mean score is 75 and the standard deviation is 5.

The z-score would be: z = (85 - 75) / 5 = 2

This means the student's score is 2 standard deviations above the mean, which is better than approximately 97.7% of the class (assuming a normal distribution).