Z-Score Calculator

How to Use This Z-Score Calculator

This calculator converts any raw value into a standardized z-score and provides associated probability information. Follow these steps:

• Enter your raw value (x): This is the specific data point you want to standardize. It could be a test score, measurement, or any numerical value from your data set.
• Enter the population mean: The average value of the entire population or distribution you are comparing against.
• Enter the standard deviation: The measure of spread in your population. This must be a positive number.
• Click Calculate: The calculator displays the z-score plus additional information including the percentile rank and probabilities.

The results show not just the z-score but also its interpretation. The percentile tells you what percentage of the population falls below your value. P(X less than x) and P(X greater than x) give the cumulative probabilities in each direction.

What is a Z-Score?

A z-score (also called a standard score or z-value) measures exactly how many standard deviations a specific value lies above or below the mean of a distribution. It transforms raw data into a standardized scale where the mean becomes 0 and the standard deviation becomes 1.

Z-scores are one of the most important concepts in statistics because they allow meaningful comparisons between values from different distributions. Consider comparing a score of 75 on one test to a score of 85 on another test. Without knowing the mean and standard deviation of each test, you cannot determine which performance was actually better. Z-scores solve this problem by placing both scores on the same scale.

Z-scores are fundamental in statistics for many applications:

• Comparing scores from different tests: SAT vs ACT scores, different grading scales, or measurements in different units can all be compared using z-scores.
• Identifying outliers: Values with z-scores beyond plus or minus 2 or 3 are often considered outliers worthy of investigation.
• Calculating probabilities: In normal distributions, z-scores directly correspond to cumulative probabilities, enabling prediction and inference.
• Quality control: Manufacturing and business processes use z-scores to monitor whether outputs fall within acceptable ranges.
• Research and hypothesis testing: Z-scores form the basis for z-tests and relate closely to other statistical tests.

Understanding Z-Score Values

The sign and magnitude of a z-score tell you exactly where a value falls in the distribution:

• z = 0: The value equals exactly the mean
• z = +1: The value is one standard deviation above the mean (approximately 84th percentile)
• z = -1: The value is one standard deviation below the mean (approximately 16th percentile)
• z = +2: The value is two standard deviations above the mean (approximately 98th percentile)
• z = -2: The value is two standard deviations below the mean (approximately 2nd percentile)

In a normal distribution, about 68% of values fall within one standard deviation of the mean (z between -1 and +1), about 95% fall within two standard deviations (z between -2 and +2), and about 99.7% fall within three standard deviations (z between -3 and +3). This is known as the empirical rule or 68-95-99.7 rule.

Z-Score Formula

The z-score is calculated using the standardization formula:

z = (x - μ) / σ

Where x is the raw value you are converting, μ (mu) is the population mean, and σ (sigma) is the population standard deviation. The formula subtracts the mean to center the distribution at zero, then divides by the standard deviation to scale it to unit variance.

To convert a z-score back to a raw value, use the reverse formula:

x = μ + (z * σ)

This is useful when you know what percentile or z-score you want and need to find the corresponding raw value.