What Is the Normal Distribution?
The normal distribution (also called the Gaussian distribution or bell curve) is the most important probability distribution in statistics. It describes data that clusters around a central average value with no bias to the left or right. When you measure heights of adults, test scores in a large class, blood pressure readings, or manufacturing tolerances, the data almost always forms this characteristic bell shape.
μ), which determines the center of the curve, and the standard deviation (σ), which determines the spread or width. A small σ produces a tall, narrow bell; a large σ produces a short, wide bell. The mathematical formula is f(x) = (1/(σ√(2π))) · e^(−(x−μ)²/(2σ²)), but you rarely need to use this directly — instead, you use z-scores and tables.The curve is perfectly symmetric about the mean, which means the mean, median, and mode are all the same value. The total area under the curve equals 1 (representing 100% probability), and the curve extends infinitely in both directions, getting closer and closer to zero but never actually touching the x-axis.
Properties of the Bell Curve
The normal distribution has several key properties that make it incredibly useful.
μ ± σ). Approximately 95% falls within 2 standard deviations (μ ± 2σ). Approximately 99.7% falls within 3 standard deviations (μ ± 3σ). This means virtually all data in a normal distribution lies within 3 standard deviations of the mean.Symmetry
The left half is a mirror image of the right half. This means P(X < μ) = P(X > μ) = 0.5. It also means that the probability of being a certain distance below the mean equals the probability of being that same distance above.
Area = Probability
The area under the curve between two values equals the probability that a randomly selected data point falls in that range. For example, the area between μ − σ and μ + σ is 0.68, meaning there's a 68% chance a random value falls in that interval.
Why is it everywhere?
The Central Limit Theorem explains why: when you average many independent random variables, the result tends toward a normal distribution regardless of the original distribution. This is why sample means, measurement errors, and natural variations so often follow the bell curve.
The Z-Score Formula
z = (x − μ) / σ, where x is the data value, μ is the mean, and σ is the standard deviation. A positive z-score means the value is above the mean; a negative z-score means it is below the mean; a z-score of 0 means the value equals the mean.Test scores have a mean of 75 and standard deviation of 10. A student scored 90.
Their z-score is z = (90 − 75)/10 = 1.5.
1.5 standard deviations above averageHeights of women have a mean of 64 inches and SD of 2.8 inches. A woman is 59 inches tall.
Her z-score is z = (59 − 64)/2.8 = −1.79.
1.79 standard deviations below the meanWhy z-scores matter: Z-scores convert any normal distribution (with any mean and SD) to the standard normal distribution, which has μ = 0 and σ = 1. This standardization lets you compare values from completely different datasets — for example, comparing a student's SAT score to their ACT score, even though the tests use different scales.
Using the Z-Table
How to read it
Find the row for the first two digits of your z-score (e.g., 1.5) and the column for the hundredths digit (e.g., 0.03). The cell gives the cumulative probability. For z = 1.53, the table gives approximately 0.9370, meaning 93.7% of values fall below this z-score.
Common lookups
| z-score | P(Z ≤ z) | Percentage |
|---|---|---|
z = 0 | 0.5000 | 50% |
z = 1.00 | 0.8413 | 84.13% |
z = 1.96 | 0.9750 | 97.5% |
z = −1.00 | 0.1587 | 15.87% |
z = 2.00 | 0.9772 | 97.72% |
P(Z ≤ z), the probability of being above z is P(Z > z) = 1 − P(Z ≤ z). For example, P(Z > 1.53) = 1 − 0.9370 = 0.0630 or 6.3%.P(a < Z < b) = P(Z < b) − P(Z < a). For example, P(−1 < Z < 1) = 0.8413 − 0.1587 = 0.6826, which confirms the 68% rule.Finding Probabilities — Worked Examples
IQ scores are normally distributed with μ = 100 and σ = 15. What percentage of people have an IQ above 130?
Calculate: z = (130 − 100)/15 = 2.00.
From the z-table, P(Z ≤ 2.00) = 0.9772.
So P(IQ > 130) = 1 − 0.9772 = 0.0228.
about 2.3%A factory produces bolts with mean length 10 cm and SD 0.2 cm. What proportion of bolts are between 9.7 cm and 10.3 cm?
Calculate both z-scores: z₁ = (9.7 − 10)/0.2 = −1.50 and z₂ = (10.3 − 10)/0.2 = 1.50.
From the table: P(Z ≤ 1.50) = 0.9332 and P(Z ≤ −1.50) = 0.0668.
So P(9.7 < X < 10.3) = 0.9332 − 0.0668 = 0.8664.
about 86.6%Exam scores have μ = 72 and σ = 8. What score puts a student in the top 10%?
We need P(Z > z) = 0.10, so P(Z ≤ z) = 0.90.
From the table, z ≈ 1.28.
Convert back: x = μ + zσ = 72 + 1.28(8) = 82.24.
about 82 or above puts you in the top 10%Real-World Applications
The normal distribution and z-scores appear in virtually every field.
Education
Standardized tests (SAT, GRE, IQ tests) are designed to produce normal distributions. Z-scores let you compare performance across different tests and years. A z-score of 2.0 on any normally distributed test always means "top 2.3%" regardless of the test's specific scale.
Quality control
Manufacturing uses the normal distribution to set acceptable tolerance ranges. The "six sigma" methodology aims for processes where defects fall beyond 6 standard deviations from the mean — statistically, just 3.4 defects per million.
Medicine
Blood pressure, cholesterol levels, birth weights, and many other biological measurements follow normal distributions. Doctors use z-scores to determine whether a patient's measurement is within the normal range or is unusually high or low.
Finance
Stock returns are often modeled (approximately) as normal distributions. Z-scores help assess how unusual a day's return is compared to historical averages.
Not all data is normally distributed. Income, city populations, and earthquake magnitudes follow skewed distributions. Always verify normality before applying these methods — a histogram or normal probability plot can help.
If you want to calculate z-scores, find probabilities, or check your statistics work, scan the problem with Solver AI for a complete step-by-step solution.