Key Takeaways
- The normal distribution is symmetric and bell-shaped, centered around the mean (μ)
- 68-95-99.7 Rule: 68% of data falls within 1σ, 95% within 2σ, 99.7% within 3σ
- Z-score tells you how many standard deviations a value is from the mean
- For a standard normal distribution: mean = 0 and standard deviation = 1
- The total area under the normal curve equals 1 (or 100%)
What Is Normal Distribution?
The normal distribution (also called Gaussian distribution or bell curve) is the most important probability distribution in statistics. It describes how data is distributed around a central value (the mean) with a specific spread (standard deviation). Many natural phenomena follow this pattern: heights, test scores, measurement errors, and more.
The distribution is characterized by two parameters: the mean (μ) which determines the center of the distribution, and the standard deviation (σ) which determines the spread or width of the curve.
Z-Score Formula Explained
z = (x - μ) / σ
The z-score tells you how many standard deviations away from the mean a particular value lies. A z-score of 0 means the value equals the mean, while a z-score of 2 means the value is 2 standard deviations above the mean.
Common Z-Score Reference Values
| Z-Score | Left Tail P(Z ≤ z) | Right Tail P(Z ≥ z) | Common Use |
|---|---|---|---|
| -2.576 | 0.0050 | 0.9950 | 99% CI lower |
| -1.96 | 0.0250 | 0.9750 | 95% CI lower |
| -1.645 | 0.0500 | 0.9500 | 90% CI lower |
| 0 | 0.5000 | 0.5000 | Mean (50th percentile) |
| 1.645 | 0.9500 | 0.0500 | 90% CI upper |
| 1.96 | 0.9750 | 0.0250 | 95% CI upper |
| 2.576 | 0.9950 | 0.0050 | 99% CI upper |
The 68-95-99.7 Rule (Empirical Rule)
The empirical rule is a quick way to estimate probabilities for normal distributions:
- 68% of data falls within 1 standard deviation of the mean (μ ± 1σ)
- 95% of data falls within 2 standard deviations of the mean (μ ± 2σ)
- 99.7% of data falls within 3 standard deviations of the mean (μ ± 3σ)
Pro Tip: Using the Empirical Rule
If you know a dataset is approximately normal, you can quickly estimate probabilities without a calculator. For example, if test scores have a mean of 75 and standard deviation of 10, about 95% of students scored between 55 and 95 (75 ± 20).
Practical Applications
Quality Control
Manufacturing uses normal distribution to set tolerance limits. If a product dimension should be 10mm with σ = 0.1mm, items outside μ ± 3σ (9.7mm to 10.3mm) might be rejected - this captures only 0.3% of defects.
Test Scores & Grading
Standardized tests like SAT and IQ are designed to follow normal distributions. A z-score lets you compare scores across different tests or years.
Finance & Risk
Stock returns are often modeled (approximately) as normal distributions. Value at Risk (VaR) calculations use z-scores to estimate potential losses at specific confidence levels.
Medical Research
Clinical trials use normal distribution assumptions to determine if treatments are effective. Statistical significance often uses z-scores to calculate p-values.
Frequently Asked Questions
A z-score (or standard score) tells you how many standard deviations a value is from the mean. A z-score of 0 means the value equals the mean. Positive z-scores indicate values above the mean, while negative z-scores indicate values below. For example, a z-score of 1.5 means the value is 1.5 standard deviations above the mean.
Population standard deviation (σ) uses N in the denominator and represents the true spread of an entire population. Sample standard deviation (s) uses n-1 in the denominator (Bessel's correction) and estimates the population spread from a sample. For large samples, the difference is minimal.
Many natural measurements (height, weight, test scores) approximate normal distributions. The Central Limit Theorem states that sample means approach normality as sample size increases (typically n ≥ 30), regardless of the original distribution. Use normality tests (Shapiro-Wilk, Q-Q plots) to verify.
To find P(a ≤ X ≤ b), calculate the z-scores for both values, then find P(Z ≤ z_b) - P(Z ≤ z_a). This calculator handles this automatically when you select "Probability Between Two X Values" mode.
A percentile indicates the percentage of values that fall below a given value. The 90th percentile means 90% of values are below this point. In a standard normal distribution, the 50th percentile equals the mean (z = 0), the 84th percentile is approximately z = 1, and the 97.5th percentile is z = 1.96.