Z-Score Calculator

A z-score (also called standard score) measures how many standard deviations a value is from the mean. It standardizes any normal distribution to a common scale. **The Formula**: z = (x - μ) / σ Where: - x = the value being standardized - μ = population mean - σ = population standard deviation **Interpretation**: - z = 0: value equals the mean - z = 1: value is 1 standard deviation ABOVE mean - z = -1: value is 1 standard deviation BELOW mean - z = 2: value is 2 standard deviations above - z = -3: value is 3 standard deviations below **The 68-95-99.7 Rule** (for normal distributions): - 68% of data within ±1 σ (z between -1 and +1) - 95% of data within ±2 σ (z between -2 and +2) - 99.7% of data within ±3 σ (z between -3 and +3) **Z-Score to Percentile**: For a normal distribution: | Z-Score | Percentile | |---------|------------| | -3.0 | 0.13% | | -2.0 | 2.28% | | -1.5 | 6.68% | | -1.0 | 15.87% | | -0.5 | 30.85% | | 0.0 | 50.00% | | +0.5 | 69.15% | | +1.0 | 84.13% | | +1.5 | 93.32% | | +2.0 | 97.72% | | +3.0 | 99.87% | **Why Z-Scores Matter**: 1. **Comparison across scales**: A z-score of 1.5 has the same meaning whether you're measuring height, IQ, or test scores. 2. **Outlier detection**: |z| > 3 is often considered an outlier (less than 0.3% probability). 3. **Hypothesis testing**: Z-scores form the foundation of statistical tests. 4. **Normalization**: Required input for many machine learning algorithms. **Common Z-Score Applications**: **SAT/ACT Scoring**: SAT mean = 1050, SD = 200 Score of 1450: z = (1450-1050)/200 = 2.0 → 97.7th percentile **IQ Tests**: IQ mean = 100, SD = 15 IQ of 130: z = (130-100)/15 = 2.0 → 97.7th percentile (gifted) **Stock Returns**: Stock returns have mean μ and SD σ A return 2 SD above average is exceptionally good (rare) **Medical Tests**: Normal lab values with z-scores beyond ±2 are often flagged as abnormal **Critical Z-Values for Hypothesis Testing**: For a two-tailed test at: - α = 0.10: |z| = 1.645 - α = 0.05: |z| = 1.960 (most common) - α = 0.01: |z| = 2.576 - α = 0.001: |z| = 3.291 If your calculated |z| exceeds the critical value, reject the null hypothesis. **Sampling Distributions**: For sample means (Central Limit Theorem): z = (x̄ - μ) / (σ/√n) The standard error (σ/√n) replaces σ when working with sample means. **Difference of Two Means**: z = (x̄₁ - x̄₂) / √(σ₁²/n₁ + σ₂²/n₂) Used for testing whether two populations differ. **Limitations**: 1. **Requires normality**: Z-scores are most meaningful for normal distributions 2. **Population parameters**: Need true μ and σ (often unknown — use t-scores instead) 3. **Sample size**: For small samples, t-distribution is more accurate 4. **Outliers**: Extreme values dominate calculations **Z-Score vs T-Score**: - **Z-score**: Use when σ is known OR n ≥ 30 - **T-score**: Use when σ unknown AND n < 30 With n ≥ 30, t and z give nearly identical results.