Last updated on August 5th, 2025
In statistics, the normal distribution is a key concept that describes how data values are distributed around the mean. It is often called the bell curve due to its shape. In this topic, we will learn the formula for normal distribution and its significance.
The normal distribution is a probability distribution that is symmetric around the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean. Let’s learn the formula to calculate the normal distribution.
The normal distribution, also known as the Gaussian distribution, is described by its mean (μ) and standard deviation (σ).
The formula for the probability density function of a normal distribution is: [ f(x) = frac{1}{sigma sqrt{2pi}} e{-frac{(x-mu)2}{2sigma^2}} ] where:
( f(x) ) is the probability density function
( mu ) is the mean of the distribution
( sigma ) is the standard deviation
( x ) is the variable
The normal distribution is fundamental in statistics and real life, as it helps in understanding data variations and making predictions.
Here are some important aspects of the normal distribution:
Students often find math formulas daunting, but with some tips and tricks, they can master the normal distribution formula.
The normal distribution plays a major role in various real-life applications:
Students often make errors when working with the normal distribution formula. Here are some common mistakes and tips to avoid them.
What is the probability density of a normal distribution with mean 0 and standard deviation 1 at x = 1?
The probability density is approximately 0.242
Using the formula: [ f(x) = frac{1}{1 cdot sqrt{2pi}} e{-frac{(1-0)2}{2 × 12}} ] [ f(x) = frac{1}{sqrt{2pi}} e{-0.5} approx 0.242 ]
Calculate the probability density for a normal distribution with mean 5 and standard deviation 2 at x = 7.
The probability density is approximately 0.120
Using the formula: [ f(x) = frac{1}{2 cdot sqrt{2pi}} e{-frac{(7-5)2}{2 × 22}} ] [ f(x) = frac{1}{2sqrt{2pi}} e{-0.5} approx 0.120 ]
Find the probability density for a normal distribution with mean 10 and standard deviation 3 at x = 10.
The probability density is approximately 0.133
Using the formula: [ f(x) = frac{1}{3 cdot sqrt{2pi}} e{-frac{(10-10)^2}{2 × 3^2}} ] [ f(x) = frac{1}{3sqrt{2pi}} e{0} approx 0.133 ]
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