🔔 Normal Distribution Calculator
Find z-scores, cumulative probability, upper-tail probability, and density for a normal distribution.
📊 Statistics📐 Formula included🧠 Practice questions✓ Free
Please enter valid values.
Formula and Explanation
Find z-scores, cumulative probability, upper-tail probability, and density for a normal distribution.
| Concept | Formula or rule |
|---|---|
| Normal Distribution Calculator | z = (x − μ) / σ |
How to Use This Calculator
- Enter the sample, counts, probabilities, or summary statistics requested above.
- Keep paired lists in the same order and separate values with commas.
- Select the correct test direction or sample/population option when available.
- Press Calculate and interpret the result together with the supporting statistics.
Worked Example
Example: For x=115, mean=100, and SD=15, z=1.
✓ P(X≤115) ≈ 0.8413
Where This Is Used
School & coursework
Check homework, understand formulas, and verify step-by-step statistics exercises.
Research
Explore estimates, uncertainty, distributions, and relationships before using full statistical software.
Business & data analysis
Summarize variation, compare groups, and communicate data-driven findings clearly.
Common Mistakes to Avoid
⚠️
Using variance as σ
The denominator is standard deviation, not variance.
The denominator is standard deviation, not variance.
⚠️
Assuming every dataset is normal
Check context and distribution shape.
Check context and distribution shape.
⚠️
Confusing density with probability
The curve height is not itself a probability.
The curve height is not itself a probability.
Practice Questions
1. What is z for x=70, mean=50, SD=10?
Answer: 2
2. What is z at the mean?
Answer: 0
3. Approximately what cumulative probability corresponds to z=0?
Answer: 0.5
Frequently Asked Questions
What is a z-score?▾
The number of standard deviations a value lies from the mean.
What percentage lies within 1 SD?▾
About 68% for a normal distribution.
Can a z-score be negative?▾
Yes, for values below the mean.