Normal Distribution: In Statistical Studies

Normal Distribution is a very important part of the statistical studies as it fits for many of the natural events. The best examples for this can be heights, blood pressure, measurement errors, and the IQ scores which follow the normal distribution. This is also termed as the Gaussian Distribution or the Bell Curve.

In our earlier topics we have seen how to compute the normal distribution by using the Empirical Rule. In this article we will be studying the normally distributed data examples, parameters, and the properties of the normal distribution.

The normal distribution is the probability function that determines the values of the different variables and how they are distributed. The normal distribution data is always symmetric. Most of the inspected values are at the peak and the probability values are away from the mean which is narrowly distributed in both the directions.

Example Of The Normally Distributed Data: Heights

As per the definition of the Normal Distribution, the height data is normally distributed as shown in the below graph. The distribution data is the real data taken from the survey of 14 years old girls.

The distribution of the height data is the typical pattern for all the normal distributions. Many of the girl’s height is average height i.e. 1.512 meters. You can see that there is a slight difference between the individual height and the mean of the distribution. The standard deviation computed will be 0.0741 meters, which indicates that the individual girls are liable for the descending manner from the mean height.

As seen earlier this distribution is symmetric. The number of shorter girls in the height equals to the number of girls taller in the height than the average height girls.

Parameters Of The Normal Distribution

The parameters of the normal distribution defines the shape of the distribution and the probabilities in any of the probabilities computations. There are two parameters of the normal distribution, viz. Mean and the Standard Deviation. The normal distribution will not be in a single form always, instead, the shape will be changing as per the parameter values, you can see it in the graph below.

• Mean

The central tendency of the distribution is known as the Mean of that distribution. Mean also defines the location of the peak value in the distribution. It can be seen that most of the values are gathered near the mean. If the mean is changed it means that the entire curve will be shifted to the right or the left of the mean on the X-axis.

• Standard Deviation

The measurement of the variability in the distribution is known as the standard deviation. This also defines the width of the normal distribution. The standard deviation indicates how far the values fall from the mean in the normal distribution. The distance between the observations and the average is represented by the standard deviation.

The width of the normal distribution is either tighten or spread by the standard deviation along the X-axis. The large standard deviations produce the distribution that is spread out more.

When there are more narrow distributions the probability is much higher for the values which will not fall from the mean. As the spread of the distribution is increased the chances of observations will be away from the mean also increases accordingly.

• Population Parameters V/S Sample Estimates

The parameters, mean, and standard deviation are applied to the total populations. For the normal distributions the statisticians indicate these parameters by the Greek notations, i.e. μ (pronounced as mu) for the mean and σ (pronounced as sigma) for the standard deviation.

But generally the population parameters are unknown as it is impossible to measure the entire population. But still you can use the samples to calculate the estimated values of these parameters.

Properties Of The Normal Distribution

Despite the different shapes all the forms of the normal distribution, has the characteristics of the properties as below

• Normal distribution is always symmetric. The normal distribution model cannot be altered.
• The values of the mean, median, and mode are all equal.
• Half of the population is less than that of the mean population and that the half of the population is greater than that of the mean.
• The Empirical Rule allows you to compute the proportionate value of the particular distances from the mean.

Conclusion: As we have seen that the Normal Distribution is important in the statistical studies, it is one of the probability distribution which does not fit for all the population computations.