# Explain Kurtosis relative to a normal distribution with the help of diagrams

- October 13, 2022
- Forex Trading

Contents

A leptokurtic distribution is one that has kurtosis greater than a mesokurtic distribution. Besides normal distributions, binomial distributions for whichp is close to 1/2 are considered to be mesokurtic. If the p-value is lower than the Chi value then the null hypothesis cannot be rejected. After performing the above procedure, ‘sktest – Skewness and kurtosis test for normality’ box will appear . Select the main variable to test for normality (here it is ‘resid’). Explain Kurtosis relative to a normal distribution with the help of diagrams.

The X-axis shows the residuals, whereas Y-axis represents the density of the data set. Thus this histogram plot confirms the normality test results from the two tests in this article. The preceding articles showed how to conduct https://1investing.in/ time series analysis in STATA on a range of univariate and multivariate models including ARIMA, VAR and VECM . Time series data requires some diagnostic tests in order to check the properties of the independent variables.

If it is positive, then the data is said to be right skewed, as illustrated below. While the graphical representation provides a very quick and easily understandable comparison of the skewness or bias on the data distribution, the skewness measure helps in quantifying the same. This will be particularly important for decision making while comparing distributions which appear similar, but have smaller differences in skew that may not show up well on the graph. Data distributions based on life times of certain products, like a bulb or other electrical devices, are right skewed. The smallest lifetime may be zero, whereas the long lasting products will provide the positive skewness.

Here median, mode and mean are at the same point and the skewness is zero. Kurtosis is typically measured with respect to the normal distribution. A distribution that has tails shaped in roughly the same way as any normal distribution, not just the standard normal distribution, is said to be mesokurtic. The kurtosis of a mesokurtic distribution is neither high nor low, rather it is considered to be a baseline for the two other classifications. Distributions of data and probability distributions are not all the same shape.

We would talk about the usage of these two parameters now. As explained above, these definitely help us to know about the shapes of the distribution; more importantly whether we are working with normal distribution or not. In the next chapter, we will continue our discussion of statistical measures discuss the concept of kurtosis. of risk by talking about covariance and correlation. Based on the above table, let us now calculate the possible range of log returns within which Nifty could trade over the next one month. We can find how much the frequency curve is flatter than the normal curve using measure of kurtosis.

## Business Mathematics & Statistics – Notes | Study Business Mathematics and Statistics – B Com

It should be noted that in moderately symmetrical distribution the interval between the mean and the median is approximately one-third of the interval between the mean and the mode. It is this relationship which provides a means of measuring the degree of skewness. One of the main uses of Kurtosis is to use it as underlying factor for testing Normality, since many of the statistical techniques depend on the normality of distribution. In statistics kurtosis refers to the degree of flatness or peakedness in the region about the mode of a frequency curve.

When accompanied by low to moderately positive skewness, such distributions would imply at stable returns and low risk. This sort of distribution is something that would suit a conservative investor. For a distribution that is perfectly symmetrical, the mean will be equal to the median and the mode .

0; a negative kurtosis, known as Platykurtic will have β2–3 More sharing options… An application oriented question on the topic along with responses can be seen below. The best answer was provided by Vishwadeepak Choudhary on 27th August 2018.

In order to predict the residuals from the regression model, use the below command. Now that we have a way to calculate kurtosis, we can compare the values obtained rather than shapes. The next article discusses the tests for heteroscedasticity. Heteroscedasticity is a violation of an important ordinary least squares assumption that all residuals belong to apopulationthat has a constant variance . In order to perform this test, use the command ‘jb resid’ in the command prompt. Next, use the below command in order to generate the residuals in the data set.

## (b) Positively Skewed

There are two other comparable characteristics called skewness and kurtosis that help us to understand a distribution. In addition to this the discrete probability distribution from a single flip of a coin is platykurtic. One of the most well known leptokurtic distributions is Student’s t distribution. Leptokurtic distributions are sometimes identified by peaks that are thin and tall. The tails of these distributions, to both the right and the left, are thick and heavy. Leptokurtic distributions are named by the prefix “lepto” meaning “skinny.”

We study skewness to have an idea about the shape of the curve drawn from the given data. When the data set is not a symmetrical distribution, it is called a skewed distribution and such a distribution could either be positively skewed or negatively skewed. A histogram plot also indicates the normality of residuals. A bell-shaped curve shows the normal distribution of the series. In order to generate the histogram plot, follow the below procedure.

Based on the above, what do you think will be the range of returns for Nifty over the next one month, which is roughly equivalent to 21 trading sessions? Well, to find that out, we need to convert daily mean and standard deviation to monthly figures. The degree of kurtosis of distribution is measured relative to the peakedness of normal curve. In other words, measures of kurtosis tell us the extent of which a distribution is more peaked or flat-topped than the normal curve.

## Skewness and Kurtosis

There are several normality tests such as the Skewness Kurtosis test, the Jarque Bera test, the Shapiro Wilk test, the Kolmogorov-Smirnov test, and the Chen-Shapiro test. This article shows two tests; Skewness Kurtosis and Jarque Bera tests because they are simple and popular. In economics, the skewness measure is often used to study income distributions that are skewed to the right or to the left. We are a team of dedicated analysts that have competent experience in data modelling, statistical tests, hypothesis testing, predictive analysis and interpretation. Given the skewness and Kurtosis we could predict the shape of a probability distribution. One of the most important thing that one would like to infer from a descriptive statistics output for any data is how much does the data distribution comply or deviate from a normal distribution.

- In the negatively skewed distribution the position is reversed, i.e., the excess tail is on the left-hand side.
- A bell-shaped curve shows the normal distribution of the series.
- If it is positive, then the data is said to be right skewed, as illustrated below.
- The preceding articles showed how to conduct time series analysis in STATA on a range of univariate and multivariate models including ARIMA, VAR and VECM .
- The chart below depicts how a normal distribution looks like – resembling a bell-shaped curve that is spaced evenly on either side of the mean (µ), which is represented by the vertical line.

This article explains how to perform a normality test in STATA. Further it is also interesting to know that when we check the data points using the Box plot if the mean of the dataset is greater that the median then its negative skewed and when the mean is less than median then its positive skewed. Data may be distributed either spread out more on left or on the right or uniformly spread. For a normal distribution, the data will be spread uniformly about a central point, and not skewed. When the data is scattered uniformly at the central point, it called as Normal Distribution.

In the above table, notice that Tata Motors had the highest standard deviation as well as the highest excess kurtosis. This means that since the start 2021 till the time of writing, compared to the other two stocks, Tata Motors not only had higher dispersion around the mean return but also had longer tails. Furthermore, a moderate level of positive skewness suggests that the returns of Tata Motors are right-skewed. Skewness is the measure of asymmetry in a statistical distribution or a comparative measure of the two tails. Right skewed distributions will have a positive skew while left skewed distributions will have a negative skew.

## Viva Questions

To answer these kinds of questions we need not just a qualitative description of kurtosis, but a quantitative measure. The formula used is μ4/σ4 where μ4 is Pearson’s fourth moment about the mean and sigma is the standard deviation. Finally, click on ‘OK’ to generate the histogram plot showing the normality distribution of the residuals . The time series result will identify the residuals from the regression analysis. The normality test helps to determine how likely it is for a random variable underlying the data set to be normally distributed.

Skewness and Kurtosis are measures that quantify such deviation, often referred to as measures for ‘shape’ related parameters. These measures will be particularly useful while comparing 2 distributions, and decide on the extent of normality – For eg. The delivery time for a product when compared between two delivery outlets.

In short, a positively skewed distribution will have a tail that stretches to the right, while a negatively skewed distribution will have a tail that stretches to the left. The image below shows distributions that exhibit positive skewness, zero skewness, and negative skewness. Earlier in this chapter, we spoke about skewness and kurtosis, which are the third and the fourth central moment, respectively, in statistics . When observations in the data set are normally distributed about the mean, one can use standard deviation as an effective measure of risk. That said, keep in mind that standard deviation assumes a distribution that is normal.

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