Normal Distribution - The Bell Curve
I heard a lot about the normal distribution during my engineering days, but never really understood the practical applications of it. Only after I started working in the field of data science, I got to know how useful the bell curve is. The normal distribution,also known as Gaussian distribution is a continuous probability distribution that is symmetrical about the mean, so the left side of the centre is a mirror image of that on the right side. It is also called bell curve because of its shape.
The area under the bell curve represents probability and the total area under the curve is equal to 1.
Properties of Bell Curve
- The total area under the bell curve is 1.
- The bell curve is symmetric about its mean. i.e. area to the right side of the centre is exactly same as area to the left side of the bell curve.
- The mean,median and mode are all equal.
- It follows empirical rule. Empirical rule tells us about what percentage of data will fall within certain number of standard deviations from the mean.
- About 68.2% of the data lies within 1 standard deviation.
About 95.5% of the data lies within 2 standard deviations.
About 99.7% of the data lies within 3 standard deviations.
Why is it so important?
The normal distribution is the most important probability distribution in statistics because it fits majority of activities in our everyday life. For example, exam scores, heights, weight, income distribution in society etc., follows the normal distribution.
Height and Weight of Population
Height of the population is an example of normal distribution. Majority of the people are of average height. The number of people too short or too tall from average height are very less. Similarly, even weight of the population follows the normal distribution.
Exam Scores
The marks scored in various competitive exams such as CAT, GATE etc,. follow normal distribution. Majority of the students score average marks. Whereas, the number of students scoring too high or too low marks are very less.
Income Distribution in Society
This is one of the classic example of normal distribution. Most of the people in society belong to middle class. People who are extremely rich or extremely poor are less.
Stock Market
Everyday we keep hearing about the rise or fall of the stock market. The price of the shares hike or crash based on various factors of the market. These changes in the log values of rates, stock prices & stock returns often form a bell-shaped curve. If returns are normally distributed, then about 99.7 percent of the returns are expected to fall within the three deviations of the mean value. Such characteristics of the bell curve allow financial analysts and investors to make statistical inferences about the expected return and risk of investments.
Shoe Size
Most of people in the world have average feet size. People with too small or too large feet length are rare. Hence majority the shoes in the world are of average size and also follow normal distribution. So in Cinderella's story, unless she belonged to that rare group of people, her shoe would have fit any other average woman!
Conclusion
One of the main reason why the normal distribution is so important is that it is extremely useful for the statisticians to work with. Almost all statistical tests assume normal distributions. A large number of random variables, in science and economics, are either approximately or exactly explained by the normal distribution. It is used in almost all fields, hence justifying its name - Normal Distribution.
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