AEM 201 1st Edition Lecture 7PREVIOUS LECTUREI. Numerical Methods-Measures of Location-Quantitative DataII. Measures of Variability or Dispersion-Quantitative DataCURRENT LECTUREI. Measures Of Variability Or Dispersion-Quantitative DataII. Using Measures Of Relative Location To Identify OutliersMEASURES OF VARIABILITY OR DISPERSION-QUANTITATIVE DATA- Mean Absolute Deviation (MAD): measure of relative dispersion for a data setbased on the average distance that the observations in a data set lie from their meano Computed o Mean absolute deviation gives you an idea of how spread out the data are- Range cannot be negative. The lowest it can get is 0 (all values in the data set are the same)- Interquartile range cannot be negative. The lowest it can be is zero- Mean absolute deviation cannot be negative. The lowest it can be is zero- The larger the mean absolute deviation is, the more spread out the datais- You can break mean absolute deviation into stepso Put the data in an arrayo Find the sample meano Subtract the sample mean from each data valueo Take the absolute value of each answers acquired from the last stepo Add all of the resulting valueso Divide by the number of observations in the data (result is the mean absolute deviation- Variance: measure of relative dispersion based on the squared distance that the observations in a data set lie from their meanGradeBuddyo Computed as:- You have to subtract n by 1 for a sample otherwise it will not be equal to s squared- Variance is the same is the same as mean absolute deviation except we square the values to get rid of the negatives rather than the absolute value- The only reason variance should ever be zero is if every value is equal to the mean (all numbers are the same)- You can also find variance by following a step-by-step processo Put data in an arrayo Find the meano Subtract the mean from each data valueo Square each of the values acquired from aboveo Divide by the number of observations minus 1o Variance is usually always bigger than mean absolute deviation except if the values are between -1 and 1- Standard Deviation: measures relative dispersion that is equal to the square root of the varianceo Computed as:- The smallest the standard deviation can be is zero- Literally is the square root of the variance- Very similar to the mean absolute deviance but is not the same (they have different properties)- Coefficient of Variation: measure of relative dispersion that standardized in relation to its meano Computed as:- This problem cancels out units so you can compare data that doesn’t have thesame units- Standard deviation is _____% of the mean (interpretation)GradeBuddy- Every measure of location must be more than or equal to 0. If it equals 0 it is constantUSING MEASURES OF RELATIVE LOCATION TO IDENTIFYOUTLIERS- We can’t just say a value is an outlier- Outlier: an observation associated with an unusually extreme (small or large) value of a variable- Z-Score: number of standard deviations xi lies from the mean. Often referred to as the standard valueo Computed as:- The z-score tells us how many standard deviations away from the mean- Larger in magnitude, the less likely the value it is to occur- Z-scores have special properties:o Chebyshev’s theorem: at least 1-(1/z squared) of the observations willbe within z standard deviations of the mean. Z must be greater than orequal to 1 This will work for any z above 1 and up- You cannot find an exception to Chebyshev’s theorem. There is a calculus proof- The z values of the range must be symmetric (the same standard deviations from the mean)- The Empirical Rule: for data with a bell shaped (normal) distribution (for now, the only way we can know if a distribution is normal is if someone tells us)o Approximately 68% (68.26%) of all observations in a data set are within z=1 standard deviation of the meano Approximately 95% (95.44%) of all observations in a data set are within z=2 standard deviations of the meano Over 99% (99.72%) of all observations in a data set are within z=3 standard deviations of the meano No at least with the empirical rule. These numbers are