MATH154-4 Quantitative Methods Measures of Shapes PDF

MATH154-4 QUANTITATIVE METHODS COURSE OUTCOME 1 MEASURES OF SHAPES Mapua University – Department of Mathematics  OBJECTIVES At the end of the lesson, the students are expected to Compute and interpret the coefficient of skewness and the kurtosis; Differentiate the coeffici...

MATH154-4 QUANTITATIVE METHODS COURSE OUTCOME 1 MEASURES OF SHAPES Mapua University – Department of Mathematics  OBJECTIVES At the end of the lesson, the students are expected to Compute and interpret the coefficient of skewness and the kurtosis; Differentiate the coefficient of skewness and kurtosis; Identify the three types of skewness; and Identify the three types of kurtosis. Mapua University – Department of Mathematics  SKEWNESS DEFINITION Degree of asymmetry of distribution about a mean. It is a measure on how the data departs from being symmetrical Can be interpreted as symmetric, positively skewed or negatively skewed Mapua University – Department of Mathematics  SKEWNESS DISCUSSION ON SKEWNESS A distribution is said to be symmetric if it can be folded along a vertical axis so that the two sides coincide. A distribution that lacks symmetry with respect to a vertical axis is said to be skewed. Mapua University – Department of Mathematics  SKEWNESS TYPES OF SKEWNESS Mapua University – Department of Mathematics  SKEWNESS TYPES OF SKEWNESS The distribution illustrated in Figure (a) is said to be skewed to the right, since it has a long right tail and a much shorter left tail. In Figure (b) we see that the distribution is symmetric, while in Figure (c) it is skewed to the left. Mapua University – Department of Mathematics  SKEWNESS SKEWNESS OF DATA Positive skewness: mode < median < mean Symmetric: mode = median = mean Negative skewness: mode > median > mean Mapua University – Department of Mathematics  SKEWNESS FORMULA - UNGROUPED DATA σ 𝒙−𝒙ഥ 𝟑 𝑺𝒌 = 𝒏𝒔𝟑 Where : Sk – coefficient of skewness x – score/observation 𝑥ҧ − mean s – standard deviation n – number of observations Mapua University – Department of Mathematics  SKEWNESS FORMULA - GROUPED DATA σ𝒇 𝒙 − 𝒙ഥ 𝟑 𝑺𝒌 = 𝒏𝒔𝟑 Where : Sk – coefficient of skewness x – class mark 𝑥ҧ − mean s – standard deviation f - frequency n- total number of frequency Mapua University – Department of Mathematics  SKEWNESS INTERPRETATION OF VALUES Sk < 0, “negatively skewed” or “skewed to the left” Sk = 0, symmetrical Sk > 0, “positively skewed” or “skewed to the right” Mapua University – Department of Mathematics  SKEWNESS INTERPRETATION OF VALUES Sk +1, the distribution can be called highly skewed. 1 1 −1 ≤ 𝑆𝑘 ≤ − 𝑜𝑟 ≤ 𝑆𝑘 ≤ +1, the distribution can be 2 2 called moderately skewed. 1 1 − ≤ 𝑆𝑘 ≤ + , the distribution can be called approximately 2 2 symmetric. Mapua University – Department of Mathematics  SKEWNESS INTERPRETATION OF VALUES Mapua University – Department of Mathematics  SKEWNESS INTERPRETATION OF VALUES Positive skewness: mode < median < mean Symmetric: mode = median = mean Negative skewness: mode > median > mean Mapua University – Department of Mathematics  KURTOSIS DEFINITION A measure of the degree to which a unimodal distribution is peaked The state or quality of flatness or peakedness of the curve describing a frequency distribution about its mode Mapua University – Department of Mathematics  KURTOSIS TYPES OF KURTOSIS Leptokurtic Mesokurtic Platykurtic Mapua University – Department of Mathematics  KURTOSIS TYPES OF KURTOSIS Mapua University – Department of Mathematics  KURTOSIS FORMULA - UNGROUPED DATA σ 𝒙−𝒙 ഥ 𝟒 𝑲= 𝒏𝒔𝟒 Where : K – coefficient of Kurtosis x – score/observation 𝑥ҧ − mean s – standard deviation n – number of observations Mapua University – Department of Mathematics  KURTOSIS FORMULA - GROUPED DATA σ𝒇 𝒙 − 𝒙ഥ 𝟒 𝑲= 𝒏𝒔𝟒 Where : K – coefficient of Kurtosis x – class mark 𝑥ҧ − mean s – standard deviation f - frequency n- total number of frequency Mapua University – Department of Mathematics  KURTOSIS INTERPRETATION OF VALUES K < 3, “platykurtic” K = 3, “mesokurtic” K > 3, “leptokurtic” Mapua University – Department of Mathematics  EXAMPLE The ff data give the number of dresses made by a factory in 10 days: 52, 57,55,63,50,52,60,58, 54, 56. Compute and interpret the skewness and kurtosis. Mapua University – Department of Mathematics  SOLUTION Create a table of solution x ഥ 𝒙−𝒙 ഥ 𝒙−𝒙 𝟐 ഥ 𝒙−𝒙 𝟑 ഥ 𝒙− 𝒙 𝟒 50 - 5.7 32.49 -185.193 1055.6001 52 -3.7 13.69 -50.653 187.4161 Find the 52 -3.7 13.69 -50.653 187.4161 Mean 54 -1.7 2.89 -4.913 8.3521 σ𝑥 𝑥ҧ = 𝑛 55 -0.7 0.49 -0.343 0.2401 557 56 0.3 0.09 0.027 0.0081 = 10 57 1.3 1.69 2.197 2.8561 𝑥ҧ = 55.7 58 2.3 5.29 12.167 27.9841 60 4.3 18.49 79.507 341.8801 63 7.3 53.29 389.017 2839.8241 ෍ = 𝟓𝟓𝟕 ෍ = 𝟏𝟒𝟐. 𝟏𝟎 ෍ = 𝟐𝟒𝟏. 𝟖𝟏𝟑 ෍ = 𝟒𝟔𝟓𝟏. 𝟓𝟕𝟕 Mapua University – Department of Mathematics  SOLUTION Find the standard deviation Solve for Skewness 𝒙 𝟐 σ 𝒙−ഥ σ 𝒙−𝒙 ഥ 𝟑 𝒔= 𝑺𝒌 = 𝒏−𝟏 𝒏𝒔𝟑 𝟏𝟒𝟐.𝟏𝟎 241.813 𝒔= 𝑆𝑘 = 𝟏𝟎−𝟏 10 3.97 3 𝒔 = 𝟑. 𝟗𝟕 𝑆𝑘 = 0.39 The set of dresses made by the factory for 10 days is positively skewed but slightly symmetrical curve. Mapua University – Department of Mathematics  SOLUTION Solving for Kurtosis 𝒙 𝟒 σ 𝒙−ഥ 𝑲= 𝒏𝒔𝟒 4651.577 𝐾= 10 3.97 4 𝐾 = 1.87 The set of dresses made by the factory for 10 days is has a slightly platykurtic curve. Mapua University – Department of Mathematics  SUMMARY Skewness is the degree of asymmetry of distribution about a mean. -It is a measure on how the data departs from being symmetrical -Can be interpreted as symmetric, positively skewed or negatively skewed Positive skewness: mode < median < mean Symmetric: mode = median = mean Negative skewness: mode > median > mean Mapua University – Department of Mathematics  SUMMARY Kurtosis – is a measure of the degree to which a unimodal distribution is peaked The state or quality of flatness or peakedness of the curve describing a frequency distribution about its mode ❖ K < 3, “platykurtic” ❖ K = 3, “mesokurtic” ❖ K > 3, “leptokurtic” ❖ Sk < 0, “negatively skewed” or “skewed to the left” ❖ Sk = 0, symmetrical ❖ Sk > 0, “positively skewed” or “skewed to the right” Mapua University – Department of Mathematics  REFERENCES Montgomery and Runger. Applied Statistics and Probability for Engineers, 5th Ed. © 2011 Introduction to Business Statistics, Sirug, 2011 Introduction to Quantitative Methods in Business,Bharat Kolluri, 2016 http://en.wikipedia.org/wiki/Statistics http://writing.colostate.edu/guides/research/stats/index.cfm http://irving.vassar.edu/faculty/wl/econ209/dessript.pdf http://www.preciousheart.net/chaplaincy/Auditor_Manual/10d escsd.pdf Mapua University – Department of Mathematics