What is a Normal distribution?

A normal distribution is a probability distribution where most of the values lie close to the mean, and the probabilities form a “bell” shape.

How to use the normal distribution calculator?

• Select the Calculation type from the drop-down list.
• Find probability (calculate probability for a single value (left-tail or right-tail)).
• Find percentile p (X < x) (Find the value(X) for a given cumulative probability).
• Find p (a < X < b) (Probability between two values).
• Standardize to Z-score (convert a value (X) to its z-score).
• Enter the mean (μ): The average of your dataset.
• Enter the standard deviation (σ): How spread out the values are.
• Enter probability, Lower bound, or Upper bound based on your chosen type.
• After entering the values, click “Calculate”, and in seconds, you will see the result.

Examples

Example 1: Find Probability

Mean (μ) = 50, standard deviation (σ) = 5, X value = 55

Example: Standardize: Z = (X - μ)/σ = (55 - 50)/5 = 1; P(X < 55) = P(Z < 1) = 0.8413

• Probability: 0.8413
• Percentage: 84.13%
• Z score: 1

Example 2: Find P (a < X < b)

Mean (μ) = 100, standard deviation (σ) = 15, lower bound = 90, upper bound = 110

Example: Z₁ = (90 - 100)/15 = -0.6667, Z₂ = (110 - 100)/15 = 0.6667; P(90 < X < 110) = 0.7475 - 0.2525 = 0.495

• Probability: 0.495
• Percentage: 49.5%

Example 3: Standardize to Z-score

Mean (μ) = 70, standard deviation (σ) = 10, X = 85

Example: Z = (85 - 70)/10 = 1.5

• Z-score: 1.5

Example 4: Find Percentile (x for given Probability)

Mean (μ) = 50, standard deviation (σ) = 5, probability p = 0.90

Example: Step 1: Find the z value for p → zₚ = Φ⁻¹(0.90) ≈ 1.2816; Step 2: x = μ + zσ = 50 + (1.2816 × 5) = 56.408

• Percentile x: 56.41
• Probability p: 0.90 (90th percentile)
• Z score: ≈ 1.2816

How does the MathCalc Normal distribution calculator work?

The calculator uses the z-score formula (Z = (X - μ)/σ) for standardizing to a z-score. It utilizes the cumulative distribution function (CDF) to determine probabilities and employs subtraction of CDF values to find P(a < x < b).

Why use the MathCalc Normal distribution calculator?

Get a Quick Result

If you are finding probabilities or percentiles by hand, it can be time-consuming, especially when dealing with large values.

Example: Mean (μ) = 100, σ = 15, lower bound = 90, upper bound = 110 → Result = 0.495

Reduce Human Error

Manual math can lead to minor mistakes. This tool reduces errors and ensures correct results.

Example: Mean (μ) = 50, σ = 5, X = 55 → Result = 0.8413

User-Friendly

Works for any mean & standard deviation. Handles probabilities, percentiles, ranges, and z-scores in one place. Perfect for students, teachers, and data analysts.

Tips for Best Results:

• Always double-check your input.
• Always select the correct option according to your question statement.

FAQ

Does this work for both standard and non-standard normal distributions?

Yes! It works for any mean and standard values.

Can it find both tail probabilities?

Yes, it can find both tail probabilities.

What is the benefit of “Standardize to Z-score”?

It tells you how far a value is from the mean in standard deviation units.