Z-Scores & Normal Distribution
Learn to standardize data, interpret bell curves, work with percentiles, and apply z-scores in real-world scenarios.
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Introduction
Z-scores (also called standard scores) allow us to compare values from different distributions by converting them to a common scale. They tell us how many standard deviations a value is from the mean.
This concept is fundamental for understanding probability, hypothesis testing, and comparing measurements across different scales. For example, how do you compare SAT scores to ACT scores? Z-scores provide the answer.
What is a Z-Score?
A z-score measures the distance between a data point and the mean in terms of standard deviations. It answers the question: "How unusual is this value?"
Z-Score Interpretation
Calculating Z-Scores
Z-Score Formula
x
Individual value
μ (mu)
Population mean
σ (sigma)
Standard deviation
Example Calculation
Test scores have μ = 75 and σ = 10. Find the z-score for a score of 90.
z = (x - μ) / σ
z = (90 - 75) / 10
z = 15 / 10
z = 1.5
Interpretation: A score of 90 is 1.5 standard deviations above the mean.
The Normal Distribution
The normal distribution (bell curve) is the most important probability distribution in statistics. Many natural phenomena follow this pattern.
Properties of Normal Distribution
- • Symmetric around the mean (μ)
- • Mean = Median = Mode
- • Total area under curve = 1
- • Tails extend to infinity but never touch the axis
- • Completely defined by mean (μ) and standard deviation (σ)
Standard Normal Distribution
The standard normal distribution is a special normal distribution with μ = 0 and σ = 1. When we calculate z-scores, we're converting any normal distribution to the standard normal.
68%
Within z = ±1
95%
Within z = ±2
99.7%
Within z = ±3
Using the Z-Table
The z-table (standard normal table) gives the probability that a standard normal variable is less than or equal to a given z-score.
Common Z-Score Values
| Z-Score | Area Below | Percentile |
|---|---|---|
| -2.00 | 0.0228 | 2.28th |
| -1.00 | 0.1587 | 15.87th |
| 0.00 | 0.5000 | 50th |
| +1.00 | 0.8413 | 84.13th |
| +1.96 | 0.9750 | 97.5th |
| +2.00 | 0.9772 | 97.72th |
Z-Scores and Percentiles
Z-scores can be converted to percentiles, which tell you what percentage of values fall below a given score.
Example: Test Score Percentile
If SAT scores have μ = 1000 and σ = 200, what percentile is a score of 1200?
1. Calculate z-score: z = (1200 - 1000) / 200 = 1.0
2. Look up z = 1.0 in the table: 0.8413
3. Convert to percentile: 84.13th percentile
This means the score of 1200 is higher than about 84% of all test takers.
Applications
📊 Comparing Different Scales
Convert SAT (200-800) and ACT (1-36) scores to z-scores to compare performance on a common scale.
🔬 Identifying Outliers
Values with |z| > 3 are typically considered outliers (occurs less than 0.3% of the time in normal data).
📈 Quality Control
Control charts use z-scores to detect when a manufacturing process has gone out of control.
🧪 Hypothesis Testing
Z-tests use z-scores to determine if sample results are statistically significant.
Summary
Key Takeaways
- 1.Z-scores measure how many standard deviations a value is from the mean.
- 2.The formula is z = (x - μ) / σ.
- 3.The standard normal distribution has μ = 0 and σ = 1.
- 4.Z-tables convert z-scores to probabilities and percentiles.
- 5.Z-scores allow comparison across different measurement scales.