DIVERGENCE FROM NORMALITY

Dr. Thaiba Reinai (PT) Assistant Professor NIU Divergence from normality

◦ The two types are: ◦ 1. Skewness ◦ 2. Kurtosis. Videos (Watch this)

◦ https://www.youtube.com/watch?v=HnMGKsupF8Q : Normal Distributions, Standard Deviations, Modality, Skewness and Kurtosis: Understanding concepts SKEWNESS:

◦ A distribution is normal when the Mean, Median and Mode coincide together and there is a perfect balance between the right and left halves of the figure. ◦ But when the Mean, Median and Mode fall at different points in the distribution, and the center of gravity is shifted to one side it is said to be skewed. ◦ In a normal distribution the mean equals the Median-Means. ◦ Mean—Median = 0. So the skewness is ‘0’. ◦ Skewness is defined as “a distribution not having equal probabilities above and below the mean.” ◦ So in fact greater the gap between the mean and the median, greater is the skewness. Defining Skewness

Skewnessis the measureof asymmetry of the distribution of areal valuedrandom variable.

It is the standardized 3rdcentral moment of adistribution

Positive Skewness indicates a long right tail

Negative Skewness indicates a long left tail

Zero Skewness indicates a symmetry around the mean

 NORMAL DISTRIBUTION

SKEWNESS NEGATIVE POSITIVE Negatively Skewed

◦ When in a distribution the scores are massed at the high end of the scale i.e. to the right end and are spread out more gradually towards the left side at that time the distribution is said to be Negatively Skewed. ◦ In a negatively skewed distribution the Median is greater than the Mean. ◦ So when the skewness is negative the mean lies to the left of the Median. Positively Skewed

◦ When in a distribution the scores are massed at the low end of the scale i.e. to the left end and are spread out more gradually to the right side at that time the distribution is said to be Positively Skewed.

◦ In a positively skewed distribution the Median is less than the mean. ◦ So when the skewness is positive the mean lies to the right of the Median. ◦ Skewness can be computed in different ways. The following two methods are most widely used: a. Person’s Measure of Skewness:

◦ In this method we can compute skewness from a frequency distribution. ◦ SK= E (Mean-Median)/σ ◦ Where Sk = Skewness ◦ σ = Standard deviation b. Measure of Skewness in terms of percentiles:

◦ In this method we can compute skewness from percentiles.

◦ Sk= P90+P10/2-P50 ◦ where Sk = Skewness

◦ P90 = 90th percentile

◦ P10 = 10th percentile

◦ P50 = 50th percentile, or Median. Example: Skewness

“Positively Skewed Distribution” Suppose that we live in a neighborhood with 100 homes; 99 of them sell for $ 100,000, and one sells for $ 1,000,000.The median and the mode will be $ 100,000, but the mean will be $ 109,000. Hence, the mean has been "pulled" upward (to the right ) by the existence of one home (outlier) in the neighborhood.

For a negatively skewed distribution , the mean is less than the median , which is less than the mode. In this case, there are large, negative outlier which tend to “pull" the mean downward (to the left ).

KURTOSIS:

◦ Kurtosis means the ‘peakedness’ or flatness of a frequency distribution compared to the normal distribution. ◦ The Collins Dictionary of Statistics defines kurtosis as “the sharpness of a peak on a curve of a probability density function”. ◦ The Normal Probability Curve is moderately peaked. ◦ If any frequency curve is more peaked or flatter than the NPC we can say the distribution diverges from normality. ◦ Kurtosis is a measure of such divergence. DEFINING KURTOSIS

KURTOSIS is a a measure of the "peakedness" of the probability distribution of a real-valued random variable. Its the standardizedfourth central moment of adistribution.

Kurtosis for the normal distribution is 3 Positive excess kurtosis indicate flatness (Long, Fat Tails) Negative excess kurtosis indicates peakedness KURTOSIS There are three types of Kurtosis:

◦ 1. Leptokurtic ◦ 2. Mesokurtic ◦ 3. Platykurtic ◦ When the frequency distribution is more peaked at the center then the Normal Curve is called as Leptokurtic. The value of kurtosis of a leptokurtic curve is greater than 0.263. ◦ When the frequency distribution is normally distributed the curve is Mesokurtic. The kurtosis of a Normal curve is 0.263. ◦ When a frequency distribution is flatter than the normal curve it is called as Platykurtic. The value of kurtosis of a platykurtic curve is less than 0.263. THANK YOU