Skewness and kurtosis

Skewness & Kurtosis
types, rules

1. Skewness and Kurtosis SUBJECT : BIOSTATISTIS H.Kalimani II M.Sc Zoology Hajee Karutha Rowther Howdia College Uthamapalayam

2. Skewness In probability theory and statistics, skewness is a measure of asymmetry of the probability distribution of a real-valued random variable about its mean ( mean & median fall at different points in the distribution ). The skewness values can be zero, negative or unidentified. Skewness discovered by Karl Pearson (1894).

3. If one tail is longer than another, the distribution is skewed. These distributions are sometimes called asymmetrical distribution as they don’t show any kind of symmetry. Where one tail is long but other tail is fat, skewness does not obey a simple rule . Within each graph, the values on the right side of the distribution taper differently from the values on the left side. These tapering sides are called tails , and they provide a visual means to determine which of the two kinds of skewness a distribution has: 1) Positive skewness, 2) Negative skewness

4. Negative S kewness The left tail is longer; the mass of the distribution is concentrated on the right of the figure. Long tail in the negative direction on the number line. The mean is also the left of the peak . The distribution is left-skewed , negatively-skewed distributions. Positive S kewness The right tail is longer; the mass of the distribution is concentrated on the left of the figure. Long tail in the positive direction on the number line. The mean is also right of the peak . The distribution is right-skewed , positively-skewed distribution.

5. Rules for Skewness If the skewness is between -0.5 to +0.5  Approximately symmetrical . If the skewness is between -1 to +1  Moderately skewed . If the skewness is less than -1 or greater than +1  Highly skewed .

6. Skewed Distribution There are three types: Symmetrical distribution A.M= Median = Mode Positively skewed distribution A.M > Median > Mode Negatively skewed distribution A.M < Median < Mode

7. Measures of Skewness

8. Karl Pearson’s coefficient of skewness When mode is defined When mode is ill defined Skp = Mean – Mode/ ϭ Skp = 3(mean – median) / ϭ SKp = Karl Pearson’s Coefficient of skewness Ϭ = Standard deviation

9. Bowleys coefficient of skewness More than 0 = Positively skewed Less than 0 = Negatively skewed Skewness is equal to 0 = Symmetrical Quartile based Sk B = Q3 + Q1 – 2 Median / Q3 – Q1 SKB = Bowley’s Coefficient of skewness Q1 = Quartile first Q2 = Quartile second Q3 = Quartile third

10. Kelly’s coefficient of skewness Sk k = P 90 – 2P 50 + P 10 / P 90 – P 10 SK k = Kelly’s Coefficient of skewness P 90 = Percentile 90 P 50 = Percentile 50 P 10 = Percentile 10 Percentile based

11. Application In a data set Concentration is high or low. Distribution is normal or not.

12. Kurtosis In a probability theory and statistics, kurtosis is a measure of the “ tailedness ” of the probability distribution of a real-valued random variable. Kurtosis refers to peakedness or flatness or curvedness of a distribution. The larger the kurtosis, the more peaked will be distribution. Kurtosis is always positive number and its normal distribution is 3.

13. In some distribution the values of mean, median and mode are the same . ( all 3 frequency curves are same skewness, same average & same dispersion ) But if a curve is drawn from the distribution then the height of curve is either more or less than the normal probability curve , since such type of deviation is related with the crest of the curve, it is called kurtosis. The standard measure of a distribution’s kurtosis, originating with Karl Pearson , is a scaled version of the fourth moment of the distribution . This number is related to the tails of the distribution , not its peak.

14. Types of Kurtosis Three types of kurtosis that can be exhibited by an distribution. Mesokurtic - same as the normal distribution with zero. Platykurtic - less than normal distribution, short-tailed distribution (thin tail), negative kurtosis . Leptokurtic – more than normal distribution, heavy-tailed distribution (fatter tail), positive kurtosis.

15. Calculation of Kurtosis Kurtosis is measured by ( β 2 ) If the value of β 2 > 3 , the curve is more peaked than the normal i.e. Leptokurtic If the value of β 2 < 3 , the curve is less peaked than the normal i.e. Platykurtic If the value of β 2 = 3 , the curve is having normal peak , i.e. Mesokurtic Excess kurtosis = kurtosis – 3. Excess kurtosis ranges from -2 to infinity. Kurtosi ranges from 1 to infinity.

16. Application and Formula Tail of distribution Kurtosis helps in determining whether resource used within an ecological guild is truly neutral or which it differs among species. The measure of kurtosis is β 2 = µ 4 / µ 2 2

17. Reference https://corporatefinanceinstitute.com/resources/knowledge/other/skewness/ https://codeburst.io/2-important-statistics-terms-you-need-to-know-in-data-science-skewness-and-kurtosis-388fef94eeaa https://corporatefinanceinstitute.com/resources/knowledge/other/kurtosis/ https://www.statisticshowto.com/probability-and-statistics/statistics-definitions/kurtosis-leptokurtic-platykurtic / http:// ecoursesonline.iasri.res.in/mod/resource/view.php?id=4742

18. QUESTIONS? THANKYOU!