Normal Distribution Calculator
Calculate z-scores, probabilities, and areas under the normal curve. Visualize where your value falls on the bell curve.
Quick Answer
The z-score formula is z = (x - μ) / σ. It measures how many standard deviations a value is from the mean. Use the standard normal table (or this calculator) to convert z-scores to probabilities.
Enter Parameters
Results
Normal Distribution Curve
Common Z-Scores Reference
| Confidence | Z-Score | Area in Tails |
|---|---|---|
| 90% | ±1.645 | 10% |
| 95% | ±1.960 | 5% |
| 99% | ±2.576 | 1% |
| 99.9% | ±3.291 | 0.1% |
About This Tool
The Normal Distribution Calculator computes z-scores and probabilities for the normal (Gaussian) distribution, the most important probability distribution in statistics. Enter a mean, standard deviation, and either an x-value or z-score, and instantly see the corresponding probability areas with a visual bell curve. This tool supports both directions: converting raw values to z-scores and converting z-scores back to raw values, making it a versatile resource for students, researchers, and professionals.
What Is the Normal Distribution?
The normal distribution, often called the bell curve or Gaussian distribution, is a continuous probability distribution defined by two parameters: the mean (μ) and standard deviation (σ). It is symmetric around the mean, with most values clustering near the center and probabilities tapering off exponentially in both tails. The shape is completely determined by these two parameters — the mean sets the center and the standard deviation controls the spread. The standard normal distribution is the special case where μ = 0 and σ = 1, and any normal distribution can be converted to standard normal using the z-score transformation.
Understanding Z-Scores
A z-score (also called a standard score) tells you how many standard deviations a particular value is away from the mean. The formula z = (x - μ) / σ standardizes any normal distribution to the standard normal. A z-score of 0 means the value equals the mean. A z-score of +2 means the value is two standard deviations above the mean. Z-scores allow you to compare values from different normal distributions on a common scale. For example, you can compare a test score from one class (mean 75, SD 10) with a score from another class (mean 80, SD 15) by converting both to z-scores.
The 68-95-99.7 Rule
One of the most useful properties of the normal distribution is the empirical rule (also called the three-sigma rule). Approximately 68.27% of values fall within one standard deviation of the mean, 95.45% within two standard deviations, and 99.73% within three standard deviations. This means that values beyond three standard deviations from the mean are extremely rare (only about 0.27% of the time). This rule provides quick mental estimates for probabilities without needing a calculator. Quality control processes like Six Sigma are named after this property, aiming for defect rates beyond six standard deviations.
Applications Across Fields
The normal distribution appears throughout nature, science, and business. In education, standardized test scores (SAT, IQ) follow normal distributions. In finance, stock returns are often modeled as normally distributed (though with heavier tails in reality). In manufacturing, product measurements cluster normally around target specifications. In biology, traits like height and blood pressure are approximately normal in large populations. In physics, measurement errors follow normal distributions due to the central limit theorem. Understanding the normal distribution is a prerequisite for nearly every branch of applied statistics.
The Central Limit Theorem
The central limit theorem (CLT) is the reason the normal distribution is so pervasive. It states that the sampling distribution of the mean of any independent random variables approaches a normal distribution as the sample size increases, regardless of the original distribution shape. This means that even if your underlying data is skewed or non-normal, the average of sufficiently large samples will be approximately normally distributed. The CLT justifies the use of z-tests, t-tests, and confidence intervals in practice, and it explains why so many natural phenomena appear to follow bell curves — they are often the result of many small, independent effects summing together.
Left Tail, Right Tail, and Two-Tailed Probabilities
This calculator provides three probability areas. The left tail P(X < x) is the probability that a randomly selected value falls below x — this is the cumulative distribution function (CDF). The right tail P(X > x) is the complement, giving the probability above x. The two-tailed probability P(-|z| < Z < |z|) gives the area between the negative and positive z-score, representing the probability of falling within that range of the mean. These three values cover the most common hypothesis testing scenarios: left-tailed tests, right-tailed tests, and two-tailed tests.
Frequently Asked Questions
What is a z-score and how do I interpret it?
When should I use the normal distribution?
What is the difference between z-score and t-score?
Can I use this for non-standard normal distributions?
What does the area under the curve represent?
How accurate is this calculator?
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