A z-score measures how many standard deviations a data value is above or below the mean: z = (x - \mu)/\sigma. Z-scores standardize measurements from different scales, enabling comparison of apples and oranges and looking up probabilities in standard normal tables.
Definition
A z-score measures how many standard deviations a data value is above or below the mean: z = (x - \mu)/\sigma.
💡 Intuition
A universal measuring stick—z = 2 means '2 SDs above average.'
🎯 Core Idea
Z-scores let you compare values from different distributions.
Example
Formula
Notation
z is the standard score; Z \sim N(0, 1) is the standard normal distribution
🌟 Why It Matters
Z-scores standardize measurements from different scales, enabling comparison of apples and oranges and looking up probabilities in standard normal tables.
💭 Hint When Stuck
Try saying it aloud: 'My value is ___ away from the mean, and one SD is ___.' Divide the first blank by the second.
Formal View
Related Concepts
🚧 Common Stuck Point
A z-score of +2 means the value is 2 standard deviations above the mean — it does not mean 2% probability or 2 units away on the original scale.
⚠️ Common Mistakes
- Subtracting the mean from the standard deviation instead of from the raw score: computing \frac{\mu - x}{\sigma} instead of \frac{x - \mu}{\sigma}
- Interpreting a negative z-score as an error — it simply means the value is below the mean
- Forgetting to divide by the standard deviation — just computing x - \mu gives the deviation, not the z-score
Go Deeper
Frequently Asked Questions
What is Z-Score in Math?
A z-score measures how many standard deviations a data value is above or below the mean: z = (x - \mu)/\sigma.
Why is Z-Score important?
Z-scores standardize measurements from different scales, enabling comparison of apples and oranges and looking up probabilities in standard normal tables.
What do students usually get wrong about Z-Score?
A z-score of +2 means the value is 2 standard deviations above the mean — it does not mean 2% probability or 2 units away on the original scale.
What should I learn before Z-Score?
Before studying Z-Score, you should understand: mean, standard deviation.
Prerequisites
Next Steps
Cross-Subject Connections
How Z-Score Connects to Other Ideas
To understand z-score, you should first be comfortable with mean and standard deviation. Once you have a solid grasp of z-score, you can move on to normal distribution.
Visualization
StaticVisual representation of Z-Score