# Skewness and kurtosis problems pdf

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- CHAPTER 5 skewness, kurtosis and moments.docx
- INDUSTRIAL STATISTICS
- Descriptive Statistics and Normality Tests for Statistical Data
- Skewness - Kurtosis

*While an individual is an insolvable puzzle, in an aggregate he becomes a mathematical certainty.*

## CHAPTER 5 skewness, kurtosis and moments.docx

Descriptive statistics are an important part of biomedical research which is used to describe the basic features of the data in the study. They provide simple summaries about the sample and the measures. Measures of the central tendency and dispersion are used to describe the quantitative data.

For the continuous data, test of the normality is an important step for deciding the measures of central tendency and statistical methods for data analysis. When our data follow normal distribution, parametric tests otherwise nonparametric methods are used to compare the groups. There are different methods used to test the normality of data, including numerical and visual methods, and each method has its own advantages and disadvantages.

In the present study, we have discussed the summary measures and methods used to test the normality of the data. A data set is a collection of the data of individual cases or subjects. Usually, it is meaningless to present such data individually because that will not produce any important conclusions. In place of individual case presentation, we present summary statistics of our data set with or without analytical form which can be easily absorbable for the audience.

Statistics which is a science of collection, analysis, presentation, and interpretation of the data, have two main branches, are descriptive statistics and inferential statistics. Summary measures or summary statistics or descriptive statistics are used to summarize a set of observations, in order to communicate the largest amount of information as simply as possible.

Descriptive statistics are the kind of information presented in just a few words to describe the basic features of the data in a study such as the mean and standard deviation SD. In inferential statistics, most predictions are for the future and generalizations about a population by studying a smaller sample. These statistical methods have some assumptions including normality of the continuous data. In the present study, we have discussed the summary measures to describe the data and methods used to test the normality of the data.

To understand the descriptive statistics and test of the normality of the data, an example [ Table 1 ] with a data set of 15 patients whose mean arterial pressure MAP was measured are given below.

Further examples related to the measures of central tendency, dispersion, and tests of normality are discussed based on the above data. There are three major types of descriptive statistics: Measures of frequency frequency, percent , measures of central tendency mean, median and mode , and measures of dispersion or variation variance, SD, standard error, quartile, interquartile range, percentile, range, and coefficient of variation [CV] provide simple summaries about the sample and the measures.

A measure of frequency is usually used for the categorical data while others are used for quantitative data. Frequency statistics simply count the number of times that in each variable occurs, such as the number of males and females within the sample or population. Frequency analysis is an important area of statistics that deals with the number of occurrences frequency and percentage.

For example, according to Table 1 , out of the 15 patients, frequency of the males and females were 8 Data are commonly describe the observations in a measure of central tendency, which is also called measures of central location, is used to find out the representative value of a data set.

The mean, median, and mode are three types of measures of central tendency. Measures of central tendency give us one value mean or median for the distribution and this value represents the entire distribution.

To make comparisons between two or more groups, representative values of these distributions are compared. It helps in further statistical analysis because many techniques of statistical analysis such as measures of dispersion, skewness, correlation, t -test, and ANOVA test are calculated using value of measures of central tendency.

That is why measures of central tendency are also called as measures of the first order. A representative value measures of central tendency is considered good when it was calculated using all observations and not affected by extreme values because these values are used to calculate for further measures. Mean is the mathematical average value of a set of data.

Mean can be calculated using summation of the observations divided by number of observations. It is the most popular measure and very easy to calculate. It is a unique value for one group, that is, there is only one answer, which is useful when comparing between the groups. In the computation of mean, all the observations are used.

For example, according to Table 2 , mean MAP of the patients was The median is defined as the middle most observation if data are arranged either in increasing or decreasing order of magnitude. Thus, it is one of the observations, which occupies the central place in the distribution data.

This is also called positional average. Extreme values outliers do not affect the median. It is unique, that is, there is only one median of one data set which is useful when comparing between the groups.

There is one disadvantage of median over mean that it is not as popular as mean. Mode is a value that occurs most frequently in a set of observation, that is, the observation, which has maximum frequency is called mode. In a data set, it is possible to have multiple modes or no mode exists. Due to the possibility of the multiple modes for one data set, it is not used to compare between the groups.

For example, according to Table 2 , maximum repeated value is mmHg 2 times rest are repeated one time only, mode of the data is mmHg. Measures of dispersion is another measure used to show how spread out variation in a data set also called measures of variation. It is quantitatively degree of variation or dispersion of values in a population or in a sample.

These are indices that give us an idea about homogeneity or heterogeneity of the data. Variance, SD, standard error, quartile, interquartile range, percentile, range, and CV.

The SD is a measure of how spread out values is from its mean value. It is called SD because we have taken a standard value mean to measures the dispersion. The variance s 2 is defined as the average of the squared difference from the mean. It is equal to the square of the SD s. For example, in the above, SD is Similarly, variance is Standard error is the approximate difference between sample mean and population mean.

When we draw the many samples from same population with same sample size through random sampling technique, then SD among the sample means is called standard error. If sample SD and sample size are given, we can calculate standard error for this sample, by using the formula. For example, according to Table 2 , standard error is 2. The quartiles are the three points that divide the data set into four equal groups, each group comprising a quarter of the data, for a set of data values which are arranged in either ascending or descending order.

Q1, Q2, and Q3 are represent the first, second, and third quartile's value. For example, in the above example, three quartiles, that is, Q1, Q2, and Q3 are 88, 95, and , respectively. As the first and third quartile in the data is 88 and Interpretation of SD without considering the magnitude of mean of the sample or population may be misleading. To overcome this problem, CV gives an idea. For example, in the above, coefficient of the variation is Difference between largest and smallest observation is called range.

For example, in the above, minimum and maximum observation in the data is 82 mmHg and mmHg. Hence, the range of the data is 34 mmHg also can write like: 82— [ Table 2 ]. The standard normal distribution is the most important continuous probability distribution has a bell-shaped density curve described by its mean and SD and extreme values in the data set have no significant impact on the mean value.

If a continuous data is follow normal distribution then Various statistical methods used for data analysis make assumptions about normality, including correlation, regression, t -tests, and analysis of variance. Central limit theorem states that when sample size has or more observations, violation of the normality is not a major issue. If a continuous data follow normal distribution, then we present this data in mean value.

If our data are not normally distributed, resultant mean is not a representative value of our data. A wrong selection of the representative value of a data set and further calculated significance level using this representative value might give wrong interpretation.

If applicable, then means are compared using parametric test otherwise medians are used to compare the groups, using nonparametric methods. An assessment of the normality of data is a prerequisite for many statistical tests because normal data is an underlying assumption in parametric testing. There are two main methods of assessing normality: Graphical and numerical including statistical tests. Graphical interpretation has the advantage of allowing good judgment to assess normality in situations when numerical tests might be over or undersensitive.

Although normality assessment using graphical methods need a great deal of the experience to avoid the wrong interpretations. If we do not have a good experience, it is the best to rely on the numerical methods. The two well-known tests of normality, namely, the Kolmogorov—Smirnov test and the Shapiro—Wilk test are most widely used methods to test the normality of the data. For both of the above tests, null hypothesis states that data are taken from normal distributed population.

Skewness is a measure of symmetry, or more precisely, the lack of symmetry of the normal distribution. Kurtosis is a measure of the peakedness of a distribution.

The original kurtosis value is sometimes called kurtosis proper. A distribution, or data set, is symmetric if it looks the same to the left and right of the center point.

Although this is a less reliable method in the small-to-moderate sample size i. To overcome this problem, a z -test is applied for normality test using skewness and kurtosis. A Z score could be obtained by dividing the skewness values or excess kurtosis value by their standard errors. If the graph is approximately bell-shaped and symmetric about the mean, we can assume normally distributed data[ 12 , 13 ] [ Figure 1 ].

In statistics, a Q—Q plot is a scatterplot created by plotting two sets of quantiles observed and expected against one another.

For normally distributed data, observed data are approximate to the expected data, that is, they are statistically equal [ Figure 2 ]. A P—P plot probability—probability plot or percent—percent plot is a graphical technique for assessing how closely two data sets observed and expected agree.

It forms an approximate straight line when data are normally distributed. Departures from this straight line indicate departures from normality [ Figure 3 ]. Box plot is another way to assess the normality of the data.

## INDUSTRIAL STATISTICS

Show all documents Normal variance-mean mixtures I an inequality between skewness and kurtosis necessary conditions under which a given statistical model can be fitted to data. In the realm of Quantitative Finance, where skewness and kurtosis play a key role, one is interested in large classes of non-Gaussian distributions, which are able to supersede the ubiquitous Black-Scholes model. A first choice is the normal variance-mean NVM mixture model, which has even been proposed as theoretical foundation for a semi-parametric approach to financial modelling e. Bingham and Kiesel

Sign in. To go straight to the Python code that shows how to test for normality, scroll down to the section named Example. The data set used in the article can be downloaded from this link. Normality means that your data follows the normal distribution. While building a linear regression model, one assumes that Y depends on a matrix of regression variables X. This makes Y conditionally normal on X. Several statistical techniques and models assume that the underlying data is normally distributed.

## Descriptive Statistics and Normality Tests for Statistical Data

Exploratory Data Analysis 1. EDA Techniques 1. Quantitative Techniques 1. A fundamental task in many statistical analyses is to characterize the location and variability of a data set. A further characterization of the data includes skewness and kurtosis.

### Skewness - Kurtosis

The concept of kurtosis is very useful in decision-making. In this regard, we have 3 categories of distributions:. A leptokurtic distribution is more peaked than the normal distribution. The higher peak results from clustering of data points along the X-axis. The tails are also fatter than those of a normal distribution.

Note: This article was originally published in April and was updated in February The original article indicated that kurtosis was a measure of the flatness of the distribution — or peakedness. This is technically not correct see below. Kurtosis is a measure of the combined weight of the tails relative to the rest of the distribution. This article has been revised to correct that misconception. New information on both skewness and kurtosis has also been added.

The term 'skewness' refers to lack of symmetry or departure from symmetry, e.g., when a distribution is not symmetrical (or is asymmetrical) it is called a skewed.

#### Leptokurtic

In probability theory and statistics , the skew normal distribution is a continuous probability distribution that generalises the normal distribution to allow for non-zero skewness. This distribution was first introduced by O'Hagan and Leonard A stochastic process that underpins the distribution was described by Andel, Netuka and Zvara As has been shown, [7] the mode maximum of the distribution is unique. This yields the estimate. Concern has been expressed about the impact of skew normal methods on the reliability of inferences based upon them. The exponentially modified normal distribution is another 3-parameter distribution that is a generalization of the normal distribution to skewed cases.

The third moment measures skewness , the lack of symmetry, while the fourth moment measures kurtosis , roughly a measure of the fatness in the tails. The actual numerical measures of these characteristics are standardized to eliminate the physical units, by dividing by an appropriate power of the standard deviation. In the unimodal case, if the distribution is positively skewed then the probability density function has a long tail to the right, and if the distribution is negatively skewed then the probability density function has a long tail to the left. A symmetric distribution is unskewed. We proved part a in the section on properties of expected Value.