A z-score tells you how many standard deviations a value is from the mean, calculated as z = \frac{x - \mu}{\sigma}. Z-scores allow comparing values from different distributions.
Definition
A z-score tells you how many standard deviations a value is from the mean, calculated as z = \frac{x - \mu}{\sigma}. Positive z-scores are above the mean; negative z-scores are below. Z-scores allow comparison of values from different distributions.
💡 Intuition
Z-scores put everything on the same scale. A z-score of +2 means 'two standard deviations above average' - unusually high. A z-score of -1 means 'one SD below average' - somewhat low but normal.
🎯 Core Idea
A z-score converts any value to a standardized number of standard deviations from the mean, allowing fair comparison across different scales and units.
Example
Formula
Notation
z is the z-score, x is the raw value, \mu is the population mean, and \sigma is the population standard deviation. Z \sim N(0,1) is the standard normal variable.
🌟 Why It Matters
Z-scores allow comparing values from different distributions. Is being 6'2" more unusual than earning \$100k? Z-scores can answer this.
💭 Hint When Stuck
First, identify the value x, the mean \mu, and the standard deviation \sigma. Then compute z = (x - \mu) / \sigma. Finally, interpret the result: a z-score of +2 means the value is 2 standard deviations above average, which is unusual in a normal distribution.
Formal View
Related Concepts
See Also
🚧 Common Stuck Point
Students interpret z-scores as percentages. A z-score of 2 does not mean 2% — you need a normal distribution table to convert it to a percentile.
⚠️ Common Mistakes
- Forgetting negative z-scores exist
- Misinterpreting magnitude
- Using with non-normal data carelessly
Go Deeper
Frequently Asked Questions
What is Z-Score (Standard Score) in Statistics?
A z-score tells you how many standard deviations a value is from the mean, calculated as z = \frac{x - \mu}{\sigma}. Positive z-scores are above the mean; negative z-scores are below. Z-scores allow comparison of values from different distributions.
What is the Z-Score (Standard Score) formula?
When do you use Z-Score (Standard Score)?
First, identify the value x, the mean \mu, and the standard deviation \sigma. Then compute z = (x - \mu) / \sigma. Finally, interpret the result: a z-score of +2 means the value is 2 standard deviations above average, which is unusual in a normal distribution.
Prerequisites
Next Steps
How Z-Score (Standard Score) Connects to Other Ideas
To understand z-score (standard score), you should first be comfortable with standard deviation intro and mean fair share. Once you have a solid grasp of z-score (standard score), you can move on to stat normal distribution and percentiles.