Math
Normal Distribution Calculator
Find z-scores, PDF, CDF, and interval probabilities instantly
Z-score
1.5000
PDF value
0.129518
CDF value
93.3193%
Probability between bounds
68.2689%
The CDF values above are shown as percentages. For example, a CDF of 0.9332 means about 93.32% of the distribution falls at or below that x value.
Normal Distribution Calculations
This calculator works with any normal distribution defined by a mean and a standard deviation. Give it a single x value to compute the z-score, density, and cumulative probability, or give it a lower and upper bound to compute the probability inside an interval.
Why The Z-Score Matters
The z-score standardizes any raw value into standard deviations from the mean. That makes it easy to compare very different scales and to map your input onto the standard normal curve.
Use Cases
Normal-distribution calculations are common in quality control, testing, admissions thresholds, finance, and any workflow that needs percentiles or tail probabilities under a bell-curve assumption.
Frequently Asked Questions
What is the difference between PDF and CDF?
The PDF is the density at a point on the curve. The CDF is the cumulative probability to the left of a point. For a continuous distribution, the probability of exactly one point is zero, so interval probabilities come from CDF differences.
What does the z-score mean?
The z-score tells you how many standard deviations an x value is above or below the mean. Positive z-scores are above the mean and negative z-scores are below it.
How is interval probability calculated?
For bounds a and b, the probability is P(a <= X <= b) = CDF(b) - CDF(a) after standardizing each bound with the normal transform.
Can I use non-standard normals?
Yes. Enter any mean and positive standard deviation. The tool converts your x values into standard-normal z-scores behind the scenes.