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Section 2.4 Working with Summary Statistics Measures of Center Mean, Median, and sometimes Mode Measures of Spread Standard Deviation, and Quartiles (Q1 and Q3) Remember SD gives an “average” deviation from the mean. The quartiles divide the data into 25% portions. Lets say we know the mean value of the homes in a community along with the total number of homes: $213,500; 412 homes We also know the tax rate: 1.5% How can we use the mean to determine the total tax dollars received by the community? The mean is used to represent the value of every home. When we describe the center of the annual income of a group of people, it is typical to use the median instead of mean…Why? There are typically a large group of people clustered around the low end of the scale with a few having very large incomes. This creates a distribution that is….. Skewed right and therefore the mean gives a measure that is higher than expected. The median filters out these extreme values. Create a dot plot of the following data: City Country Temperature (F) Addis Ababa Algiers Bangkok Madrid Nairobi Sao Paulo Warsaw Ethiopia Algeria Thailand Spain Kenya Brazil Poland 32 32 50 14 41 32 -22 Now create a dot plot of the distance the temperature is from freezing (32o). Positive if above freezing, negative if below. Recentering a set of data is when we add or subtract a constant from each data value. This shifts the data on the number scale, but does nothing to change the shape or spread. The mean will be shifted by the constant added or subtracted. Now use the same data and convert it to celsius. Simply multiply the degrees above or below freezing by 1/1.8. What happened to your data set? Shape Mean? Standard deviation? Notice that the mean and SD are multiplied by 1/1.8, but the shape stays the same. This simply shrinks or if by a number greater than 1, stretches the distribution. A summary statistic is resistant to outliers if it is not changed very much when the outlier is removed from the data. A summary statistic is sensitive to outliers if it is changed significantly when the outlier is removed from the data. Remember our discussion of Mean vs Median Refer to page 77: example of television viewers Percentile: If a value is at the kth percentile, then k% of the data is at or lower than this value. Example: You got a 32 on math portion of the ACT. You are told this is the 86th percentile. That means 86% of the test takers scored at a 32 or lower. It also means that 14% scored above a 32. This is a measure of where a data value lies within the data set. Frequency plot where the plotted points show you the accumulated percent of data up to that point. Example: page 78 Page 80 E47, 49, 53 - 56 Re-Centering happens when you…..?? What happens to the shape? What happens to the center? What happens to the spread? Re-Scaling happens when you….?? What happens to the shape? What happens to the center? What happens to the spread?