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

Knowing the probabilistic range of security returns based on mean and standard deviation can help in making assumptions about the expected future returns of a security as well as in gauging potential risks. Based on one’s risk tolerance, it can also help in stock screening and selection. The normal distribution is found to have a kurtosis of three. A distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic. When skewness is negative, it means that the data is left skewed. If it is positive, then the data is said to be right skewed, as illustrated below.

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. 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.

When the data is scattered uniformly at the central point, it called as Normal Distribution. 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.

## Objective type Questions

In the negatively skewed distribution the position is reversed, i.e., the excess tail is on the left-hand side. The figure above shows a bell-shaped distribution of the residuals. The X-axis shows the residuals, whereas Y-axis represents the density of the data https://1investing.in/ set. Thus this histogram plot confirms the normality test results from the two tests in this article. 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 .

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 discuss the concept of kurtosis. 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 . However, if the distribution is asymmetrical, the mean will be either above or below the median and the mode. If the outliers lie above the mean, the distribution will be positively skewed .

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. 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.

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. 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.

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. 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.

## study material

In the real word however, the distribution of security returns is not always normal. In fact, there is a tendency for security returns to get asymmetric and exhibit skewness and kurtosis. In such a case, skewness and kurtosis would better represent risk. 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 a measure of the asymmetry of the probability distribution of a random variable about its mean. On the other hand, Kurtosis represents the height and sharpness of the central peak relative to that of a standard bell curve. The figure below shows the results obtained after performing the Skewness and Kurtosis test for normality in STATA.

## Forum Statistics

Time series data requires some diagnostic tests in order to check the properties of the independent variables. 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.

- There are two other comparable characteristics called skewness and kurtosis that help us to understand a distribution.
- If the p-value is lower than the Chi value then the null hypothesis cannot be rejected.
- A distribution with kurtosis greater than three is leptokurtic and a distribution with kurtosis less than three is platykurtic.
- When accompanied by low to moderately positive skewness, such distributions would imply at stable returns and low risk.
- 0; a negative kurtosis, known as Platykurtic will have β2–3 More sharing options…

The delivery time for a product when compared between two delivery outlets. 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.

## (c) Negatively skewed

In the next chapter, we will continue our discussion of statistical measures 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.

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. 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.

In this case, the mean will be greater than the median, which in turn will be greater than the mode. On the other hand, if the outliers lie below the mean, the distribution will be negatively skewed . In this case, the mean will be less than the median, which in turn will be less than the mode.

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