Normal Distribution Calculator
Calculate normal distribution probabilities, z-scores, percentiles, and shaded bell-curve areas. Use this student-friendly calculator for left-tail, right-tail, between-values, outside-values, z-score, and reverse normal distribution problems with clear steps and visual explanations.
Background
The normal distribution is a bell-shaped continuous probability distribution described by its mean μ and standard deviation σ. It is used for many real-world measurements, including test scores, heights, errors, and natural variation.
How to use this calculator
- Enter the mean μ and standard deviation σ.
- Choose whether you want a left-tail, right-tail, between-values, outside-values, z-score, or percentile calculation.
- Enter the x-value, z-score, or percentile required by the selected mode.
- Click Calculate to see the probability, shaded curve, formula, statistics, table, and steps.
- Use quick picks to explore common normal distribution homework examples instantly.
How this calculator works
- It converts x-values to z-scores using the mean and standard deviation.
- It uses the normal cumulative distribution function to calculate area under the curve.
- It shades the left tail, right tail, middle region, or outside region based on your selected mode.
- It can reverse-solve an x-value from a z-score or percentile.
- It explains the result in plain language so students understand what the probability means.
Formula & Equations Used
Normal density: f(x) = 1/(σ√(2π)) · e^(-(x - μ)²/(2σ²))
Z-score: z = (x - μ) / σ
Reverse z-score: x = μ + zσ
Left-tail probability: P(X ≤ x) = Φ(z)
Right-tail probability: P(X ≥ x) = 1 - Φ(z)
Between-values probability: P(a ≤ X ≤ b) = Φ(z_b) - Φ(z_a)
Example Problem & Step-by-Step Solution
Example 1 — Find P(X ≤ 85)
- A test has mean μ = 75 and standard deviation σ = 10.
- We want the probability that a score is at most 85.
- Compute the z-score: z = (85 - 75) / 10 = 1.
- Find the cumulative probability: P(Z ≤ 1).
- The calculator shades the area to the left of 85 and reports the probability.
Example 2 — Find P(70 ≤ X ≤ 90)
- Use μ = 75 and σ = 10.
- Convert the lower value: z_a = (70 - 75) / 10 = -0.5.
- Convert the upper value: z_b = (90 - 75) / 10 = 1.5.
- Subtract cumulative probabilities: Φ(1.5) - Φ(-0.5).
- The calculator shades only the middle area between 70 and 90.
Example 3 — Find the 90th percentile
- Enter mean μ = 75 and standard deviation σ = 10.
- Choose percentile mode.
- Enter 0.90 or 90%.
- The calculator finds the z-score with 90% of the area to the left.
- Then it converts the z-score back to an x-value using x = μ + zσ.
Frequently Asked Questions
Q: What is a normal distribution?
A normal distribution is a bell-shaped continuous distribution described by a mean and standard deviation.
Q: What is a z-score?
A z-score tells how many standard deviations an x-value is above or below the mean.
Q: What does left-tail probability mean?
Left-tail probability means the area under the normal curve to the left of a selected x-value or z-score.
Q: Can this calculator find percentiles?
Yes. It can find the x-value that corresponds to a percentile or cumulative probability.
Q: Is this the same as a standard normal calculator?
It can work as a standard normal calculator by setting the mean to 0 and the standard deviation to 1.
All calculators