Normal Distribution Calculator

How to Calculate Normal Distribution Probability

The normal distribution, also known as the Gaussian distribution, is a probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.

Normal Distribution Formula

The formula for the normal distribution is:

\[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2} \]

Where:

• \( x \) is the raw score
• \( \mu \) is the mean
• \( \sigma \) is the standard deviation
• \( e \) is the base of natural logarithms (approximately 2.71828)
• \( \pi \) is pi (approximately 3.14159)

Calculation Steps

1. Calculate the z-score using the formula: \( z = \frac{x - \mu}{\sigma} \)
2. Use a standard normal distribution table or calculator to find the probability associated with the z-score.

Example

Let's calculate the probability for a raw score of 75, with a mean of 70 and a standard deviation of 5.

1. Calculate the z-score:

   \[ z = \frac{75 - 70}{5} = 1 \]

2. Look up the probability for z = 1 in a standard normal distribution table or use a calculator:

   Probability ≈ 0.8413 or 84.13%

This means there's approximately an 84.13% chance of observing a value less than or equal to 75 in this distribution.

Visual Representation

This diagram illustrates a standard normal distribution. The red dashed line represents a z-score of 1, corresponding to our example calculation.