Mathematics
Z-Scores
Quick fact
A z-score of 1.96 is famously used as the threshold for statistical significance at the 95% confidence level in normal distributions.
Why this is interesting
You got a 75 on a history test and a 68 on a math test. Which is actually the better score? The answer isn't as simple as you think.
Read the full explanation
Understanding Z-Scores
Imagine you're comparing two runners from different races. One finishes 10 seconds after the winner, and the other finishes 10 seconds after their winner. Who performed better relative to their field? You'd need to know how spread out the times were. A z-score does exactly that: it tells you how many standard deviations a value is from the mean of its group. To calculate it, you subtract the mean from the value and divide by the standard deviation. The result is a standardized number that means the same thing regardless of the original scale or units. A z-score of 0 is exactly average, a positive z-score is above average, and a negative one is below average. This lets you compare apples to oranges, or history scores to math scores, in a fair way.
A deeper explanation
The z-score formula z = (x - μ) / σ works because it transforms a raw score into a position on a common scale. The mean shifts the center to zero, and the standard deviation stretches or compresses the scale so that one unit equals one standard deviation. This standardization is powerful because it allows us to apply the properties of the normal distribution: if your data is approximately normal, you can convert a z-score directly into a percentile. For example, a z-score of 1 corresponds to the 84th percentile, and a z-score of 2 is about the 97.7th percentile. Z-scores also help identify outliers—scores with z-scores beyond ±2 or ±3 are often considered unusual. And they are the backbone of many statistical techniques, like comparing effectiveness across different experiments, constructing confidence intervals, and testing hypotheses.