The Z-Score Formula
Standardization
Your value
Mean
Standard Deviation
The z-score transforms any value into 'number of standard deviations from the mean.' Subtract the mean (center it at 0), then divide by SD (scale it to unit variance).
Before and After
Standardization shifts the distribution so the mean is 0 and rescales so the SD is 1. The shape stays the same - only the axis labels change.
Calculate a Z-Score
Population: μ = 100, σ = 15
Your score: x = 130
An IQ of 130 is 2 standard deviations above average. This is easier to interpret than '30 points above average' because we know what ±2 SD means in a normal distribution.
Interpreting Z-Scores
What Z-Scores Tell You
Z-scores have intuitive interpretations: within ±1 is typical, ±2 is unusual, ±3 is rare. For normal distributions, we know exactly what percentile each z-score corresponds to.
The Empirical Rule
For normal distributions: 68% of values fall within z = ±1, 95% within z = ±2, and 99.7% within z = ±3. This is the famous 68-95-99.7 rule.
Z-Scores to Percentiles
| Z-Score | Percentile | Interpretation |
|---|---|---|
| -3.0 | 0.1% | Extremely low |
| -2.0 | 2.3% | Very low |
| -1.0 | 15.9% | Below average |
| 0 | 50% | Average |
| +1.0 | 84.1% | Above average |
| +2.0 | 97.7% | Very high |
| +3.0 | 99.9% | Extremely high |
Each z-score corresponds to a percentile. z = 0 is the 50th percentile (median). z = +1.645 is the 95th percentile. These conversions are found in z-tables or calculators.
Comparing Different Scales
Apples vs Oranges
Student B: 680 on SAT Math (out of 800)
Who did better? We can't compare directly!
Raw scores on different scales can't be compared directly. An 85 out of 100 might be average while 680 out of 800 might be exceptional. Z-scores put everything on the same scale.
Worked Comparison
| Class Test | SAT Math | |
|---|---|---|
| Score | 85 | 680 |
| Mean (μ) | 75 | 500 |
| SD (σ) | 10 | 100 |
| Z-Score | (85-75)/10 = +1.0 | (680-500)/100 = +1.8 |
The class test score of 85 is 1 SD above average (z=1.0). The SAT of 680 is 1.8 SDs above average (z=1.8). The SAT performance is relatively stronger.
Where Z-Scores Are Used
Z-scores appear everywhere: standardized tests use them, doctors interpret lab values with them, financial analysts detect outliers with them, and engineers monitor processes with them.
Detecting Outliers
The |z| > 3 Rule
If you find one, it's either:
• A genuine extreme value
• A data entry error
• Evidence the data isn't normal
Values with |z| > 3 are more than 3 standard deviations from the mean - extremely rare in normal data. They warrant investigation as potential outliers or errors.
Finding Outliers
$150,000 is 6.7 standard deviations above the mean - almost certainly an outlier. Could be an executive, data error, or sign that salary data isn't normal (it's often right-skewed).
Real-World Examples
Apply your knowledge to practical scenarios
Pediatric Growth Charts
A pediatrician measures a 2-year-old's height at 88 cm. The mean for 2-year-olds is 86 cm with SD = 3 cm. Is this child's height normal?
Key Insights
- Medical charts typically show z-score bands (-2, -1, 0, +1, +2)
- A single measurement matters less than the trend over time
- Different populations have different norms - context matters
Comparing Athletes Across Eras
Who had the better batting season: Ted Williams (.406 in 1941) or Tony Gwynn (.394 in 1994)?
Key Insights
- Z-scores enable fair comparison across different contexts
- Being 4+ SDs above mean is extraordinary in any era
- Statistics must account for changing baselines
Z-Scores Quiz
Test your understanding • Passing score: 70%