Your exam score of 82 sounds fine โ but is it? If the class average is 90, 82 is below average. If the average is 60, 82 is outstanding. The z-score tells you exactly where a value sits inside its own dataset, in units of standard deviation.
What Is a Z-Score?
A z-score (standard score) measures how many standard deviations a value is above or below the mean of its dataset. It lets you compare values from completely different distributions โ a height in centimeters against a test score out of 100.
The Z-Score Formula
z = (x โ ฮผ) รท ฯ
Where x is your value, ฮผ is the population mean, and ฯ is the standard deviation.
Worked Example
Heights of adult men: mean = 175 cm, standard deviation = 8 cm. What is the z-score of someone 187 cm tall?
z = (187 โ 175) รท 8 = 12 รท 8 = 1.5
That person is 1.5 standard deviations above the mean โ taller than about 93% of men.
Z-Score to Percentile Conversion
In a normal distribution, the percentile is fixed for each z-score:
| Z-Score | Percentile |
|---|---|
| โ2.0 | 2.3% |
| โ1.0 | 15.9% |
| 0 | 50% (the mean) |
| +1.0 | 84.1% |
| +1.5 | 93.3% |
| +2.0 | 97.7% |
How Z-Scores Are Used
- Education โ comparing exam results across different tests
- Finance โ detecting unusual price movements (a 3-sigma move)
- Quality control โ flagging defective products outside tolerance
- Health โ growth charts for children (height/weight percentiles)
Rule of Thumb: The 68-95-99.7 Rule
In any normal distribution:
- 68% of values fall within ยฑ1 standard deviation
- 95% fall within ยฑ2 standard deviations
- 99.7% fall within ยฑ3 standard deviations
So a z-score beyond ยฑ2 is unusual, and beyond ยฑ3 is extremely rare โ often worth investigating as an outlier or an error.
๐ก Pro Tip: Use the z-score to spot outliers before running any statistical analysis. Values with |z| > 3 are prime candidates for data-entry errors.
Try Our Tools
- Z-Score Calculator โ value, mean, and standard deviation in
- Standard Deviation Calculator โ compute ฯ from raw data
- Weighted Average Calculator โ understand your mean better
*Find your standard score instantly with our free Z-Score Calculator.*
