In 18th century, a scientist found that the astronomical data he was studying, when plotted against the probability of occurrence gave him a bell-shaped curve, a statistician now knows that the weight of the adults in a city when plotted, will yield him a bell-shaped curve, and the physics professor at your college knows that the mean score of samples taken from the whole class would again give him a bell-shaped curve. This bell-shaped curve which can be found in almost every field of study we know is so normal to the people associated that it is named as normal distribution.

Normal Distribution which is also called Gaussian curve (after the name of Carl Friedrich Gauss) is the most commonly used distribution among the world of distributions. It’s a bell-shaped curve which describes the data that cluster around the mean. The probability distribution function, f(x) of normal distribution plots the data on X-axis and the probability of occurrence of the corresponding data on Y-axis.

 

 

Normal Bell-shaped Curve

 

Properties of Normal Distribution:

  • Probability distribution function, f(x) of Normal distribution is defined only by the mean (µ) and the standard deviation (s) of the data representing.
  • Skewness (Sk) of the normal distribution is zero, which tells us that it is symmetric along the Y-axis.
  • Kurtosis (k) of normal distribution is 3, which is used as the standard to measure the ‘peakedness’ of the distribution.

In various modeling applications the data is comfortably assumed to be normally distributed.

In this article, we will check whether the market data (Nifty), return on which is assumed to be normal by market practitioners for various calculations (like VAR) is actually normal or is it yet another lethal assumption which can lead us to yet another crisis (as the simple assumption of increasing home prices in US did, in shaping the current crisis).

Take the past 10-year Nifty data from www.nseindia.com > Indices > Statistics > Historical > Values.

Take the continuously compounded daily returns in the column adjacent to Turnover.

At the bottom of the sheet, you can see the average daily return, standard deviation, skewness and the kurtosis of the daily returns.

This clearly shows the deviation of the data from the normal behavior. This can be further checked by various test statistics like Kolmogorov-Smirnov test, Anderson-Darling, Wilk-Shapiro test, Jarque-Berra test, Rankit plot, Q-Q plot, P-P plots etc.

 

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