Z-Score (Z-Stat) Calculator
Determine how many standard deviations a value is from the mean.
Understanding the Z-Stat Calculator
In statistics, a Z-score (also known as a standard score or z-stat) describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If a Z-score is 0, it indicates that the data point's score is identical to the mean score. A Z-score of 1.0 would indicate a value that is one standard deviation from the mean.
The Z-Score Formula
The calculation is straightforward but essential for hypothesis testing and comparing different datasets. The formula used by this calculator is:
- x: The individual value or raw score you are testing.
- μ (Mu): The average or mean of the entire population.
- σ (Sigma): The standard deviation, representing the spread of the data.
Why Use a Z-Stat?
Z-scores allow statisticians to compare scores from different distributions. For example, if you want to compare a student's performance on an SAT test versus an ACT test, you cannot compare the raw scores directly because the scales are different. By converting both scores to Z-scores, you can see which student performed better relative to their respective population.
Imagine a class where the average test score (μ) is 70 and the standard deviation (σ) is 10. If a student scores 85 (x):
Z = (85 – 70) / 10
Z = 15 / 10
Z = 1.50
This means the student scored 1.5 standard deviations above the average.
Interpreting Results
A positive Z-score indicates the raw score is higher than the mean average. A negative Z-score reveals the raw score is below the mean average. In a normal distribution, approximately 68% of scores fall between -1 and 1, 95% fall between -2 and 2, and 99.7% fall between -3 and 3.