What is Normal Distribution?

The normal distribution, also known as the Gaussian distribution or bell curve, is a continuous probability distribution that is symmetric around its mean. It is characterized by its bell-shaped curve and is one of the most important probability distributions in statistics.

Normal Distribution Properties

Symmetric around the mean
Mean = median = mode
68% of data falls within 1 standard deviation of the mean
95% of data falls within 2 standard deviations of the mean
99.7% of data falls within 3 standard deviations of the mean

Z-Score Formula

The z-score measures how many standard deviations a value is from the mean:

z = (x - μ) / σ

Where:

z = Z-score
x = Value
μ = Mean
σ = Standard deviation

Probability Calculation

The probability of a value falling within a range can be calculated using the cumulative distribution function (CDF) of the standard normal distribution.

How to Use the Calculator

Enter the mean and standard deviation of your distribution
Choose the calculation type (probability or z-score)
Enter the required values
Click "Calculate" to get the result
View the step-by-step solution

Example Calculations

Z-Score Example:

Mean = 100, SD = 15

Value = 115

z = (115 - 100) / 15 = 1

This value is 1 standard deviation above the mean

Probability Example:

Standard normal distribution

P(Z ≤ 1.96) = 0.975

This means 97.5% of values are ≤ 1.96

Common Z-Score Values

Z-Score	Probability	Percentile
-3	0.0013	0.13%
-2	0.0228	2.28%
-1	0.1587	15.87%
0	0.5000	50.00%
1	0.8413	84.13%
2	0.9772	97.72%
3	0.9987	99.87%

Normal Distribution Calculator