# Quartiles for Normal Distribution

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Let X be a random variable that follows the normal distribution with mean \mu and variance \sigma^2. The quartiles of the normal distribution refer to those values of the random variable which divide the area under the probability distribution curve into exactly four equal parts. Each part covers exactly one-fourth of the total area under the normal distribution curve.

We use the quartiles of the standard normal distribution in order to compute the values of the quartiles for any normal distribution. This is done by converting the normal variable into a standard normal variable by converting it to a Z score.

### Quartiles of the Standard Normal Distribution:

A random variable is said to follow the standard normal distribution if it follows the normal distribution with mean \mu = 0 and variance \sigma^2 = 1. The values of the quartiles for the standard normal distribution can be found by looking at the Z table.

First Quartile = -0.675 (One-fourth of the area under the standard normal curve lies below this value)

Second Quartile = 0 (Half of the area under the standard normal curve lies below this value)

Third Quartile = +0.675 (Three-fourths of the area under the standard normal curve lies below this value)

### Quartiles of Normal Distribution:

Let X be a random variable that follows the normal distribution with mean \mu and variance \sigma^2. We first convert the random variable into a Z score by the formula,

Z = \frac{X - \mu}{\sigma}

We then set the value of Z equal to the value of the quartile of the standard normal curve given above and solve for X. This gives us the value of the quartile for the given normal distribution.

The first, second, and third quartiles of a normal distribution are given by the formulae,

\text{First Quartile} = \mu - 0.675\sigma \text{Second Quartile} = \mu \text{Third Quartile} = \mu + 0.675\sigma

#### Example:

Let X be a normal random variable with mean 110 and variance 100. Then the quartiles of the normal distribution are given as,

\text{First Quartile} = \mu - 0.675\sigma = 110 - 0.675\times 10 = 103.25 \text{Second Quartile} = \mu = 110 \text{Third Quartile} = \mu + 0.675\sigma = 116.75

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