Chapter_4_Measures_of_Distribution_Shape.pdf

# Chapter_4_Measures_of_Distribution_Shape.pdf - SQQS1033...

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SQQS1033 Data Exploratory and Generalisation lyf Oct’14 Page 1 Chapter 4 Measures of Distribution Shape Distribution of the data is important to help us in determining the behaviour of the data before any further analyses can be executed. The histogram and the box-plot can give you a general idea of the shape, numerical measures of shape give a more precise evaluation: Skewness tells you the amount and direction of skew (departure from horizontal symmetry). Kurtosis tells you how tall and sharp the central peak is, relative to a standard bell curve. Figure 4.1: Strategies to identify the distribution shape Skewness and Kurtosis In many fundamental statistical analyses is to characterise the location and variability . More characterisation of the data includes skewness and kurtosis . Skewness : a measure of symmetry that tells us whether the distribution of data is in the form of to the left, to the right or at the same of the location. Kurtosis : a measure that shows whether the data are peaked or flat relative to a normal distribution. Strategy Mathematical formula Histogram Box plot Probability plot Quantile plot

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SQQS1033 Data Exploratory and Generalisation lyf Oct’14 Page 2 Figure 4.2: Shape of distribution with different value of skewness and kurtosis Why skewness and kurtosis are important? Many classical statistical tests and intervals depend on normality assumptions . Therefore, significant skewness and kurtosis clearly indicate that data are NOT NORMAL. If a data set exhibits significant skewness or kurtosis, the use of classical statistical tests is NOT VALID. Some adjustments need to be done to ensure that the normality assumptions are fulfilled. Skewness Skewness is a measure of the symmetry (skewness = 0) of the shape of a distribution . When majority of the data fall to the left of the mean, the distribution is said to be positively or right skewed meaning that the right tail of the distribution is longer than the left When majority of the data fall to the right of the mean, the distribution is said to be negatively or left skewed meaning that the left tail is longer
SQQS1033 Data Exploratory and Generalisation lyf Oct’14 Page 3 To determine the degree of skewness in a distribution, we may use the mathematical formula as follows. For Ungrouped Data For Grouped Data 3 1 3 ) 1 ( ) ( s n x x s n i i k 3 1 3 ) 1 ( ) ( s n x x f s n i i i k In much simpler form, skewness coefficient can be computed the following formulae which are develop by Karl Pearson. Interpretation of the formula : The coefficient of skewness for a normal distribution data is 0, any approximately symmetrical distribution data should have a coefficient of skewness near to 0 (-0.9999 to -0.0001 or 0.0001 to 0.9999).

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