# Normal Distribution: What it Means in Statistics

## What is Normal Distribution?

Normal distribution, also known as Gaussian distribution, is a type of 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. In graph form, normal distribution will appear as a bell curve.

### Characteristics of Normal Distribution

A normal distribution has some unique characteristics. Firstly, it is symmetric. A perfectly normal distribution would be split down the middle, with each half being a mirror image of the other. Secondly, the mean, median, and mode of a normal distribution are equal. This means that the most frequently occurring score (the mode) is also the average (mean), and is the middle value (median). Lastly, the distribution is determined by the mean and the standard deviation. If you know these two values, you can determine the entire distribution.

## Uses of Normal Distribution

Normal distribution is widely used in the field of statistics and is often used in the natural and social sciences for real-valued random variables whose distributions are not known. Its importance is partly due to the central limit theorem, which states that under mild conditions, the mean of a large number of random variables independently drawn from the same distribution is distributed approximately normally, irrespective of the form of the original distribution.

### Applications in Real Life

In real life, there are many sets of data which follow the pattern of a normal distribution. Some examples include:

• Heights of people: Most people are of average height, the frequency decreases as height increases or decreases, forming a normal distribution.
• Blood pressure: For a healthy person, a normal blood pressure reading is approximately 120/80. A reading that is slightly higher or lower is normal, but the frequency decreases the further the reading is from the normal range.
• Marks on a test: Most students will score around the average (mean) mark, with fewer students scoring very high or very low.

## How Normal Distribution Works

The normal distribution is defined by the probability density function. The probability of a random variable falling within a particular range of values is given by the integral of this variable’s density over that range—that is, it is given by the area under the density function but above the horizontal axis and between the smallest and largest values of the range. This integral can be computed for any range of values, from minus infinity to plus infinity.

### Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution. It is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one. The normal random variable of a standard normal distribution is called a standard score or a z-score. Every normal random variable X can be transformed into a z score via the following formula:

Z = (X – μ) / σ

where X is a normal random variable, μ is the mean, and σ is the standard deviation.

## Importance of Normal Distribution

The normal distribution is the most important probability distribution in statistics because it fits many natural phenomena. For example, heights, blood pressure, measurement error, and IQ scores follow the normal distribution. It is also used in the natural and social sciences to make informed guesses about unknown parameters in their statistical populations.