Quick Answer
Use this Normal Distribution Calculator to find the probability that a normally distributed value is less than, greater than, or between selected values. Enter the mean, standard deviation, and target value; the calculator returns probability, percentage, z-score, and calculation steps.
Choose Between P(a < X < b) when you need the area between two values. The second value field appears only for that mode.
How to Use the Normal Distribution Calculator
- Select the probability mode: less than, greater than, or between.
- Enter the distribution mean.
- Enter the standard deviation.
- Enter value a. If using between mode, also enter value b.
- Read the probability and percentage in the result panel, then check the calculation steps if you need to show work.
Z-Score Formula
The z-score tells you how many standard deviations a value is from the mean. A z-score of 0 is exactly average. A z-score of 2 is two standard deviations above the mean.
Common Probability Modes
- Less than P(X < a): use this for percentile-style questions, such as the share of scores below 110.
- Greater than P(X > a): use this for right-tail questions, such as the chance of exceeding a threshold.
- Between P(a < X < b): use this for the bell-curve area between two cutoffs.
Why the Normal Distribution Matters
Why is the normal distribution so important? Because of the Central Limit Theorem. This theorem proves that if you take enough random samples of anything (even data that is wildly skewed or chaotic), the averages of those samples will always form a perfect, symmetrical Bell Curve. This allows statisticians to make incredibly accurate predictions about giant populations without having to survey everyone.
Real-World Applications
- Standardized Testing: SAT and IQ scores are artificially forced into a normal distribution. If you have an IQ of 130 (Mean 100, SD 15), your Z-score is +2, placing you in the 97.7th percentile.
- Manufacturing Quality (Six Sigma): Ensuring that the dimensions of a machined part fall within an acceptable range. 'Six Sigma' implies a defect rate of only 3.4 per million.
- Human Resources: Evaluating employee performance ratings to ensure grading curves are fair and statistically valid across different departments.