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Describing Location in a Distribution Measures of Relative Standing and Density Curves Measures of Relative Standing O Percentiles- the percent of data that lies at or below a particular value O Standardized test score reports O Baby weight/height/head size O Ogive- cumulative relative frequency graph is a percentile graph Example O Suppose Jenny earns an 86 (out of 100) on her next Statistics exam O Is that a good grade? Should she be happy or disappointed with this grade? Example Continued O Here are the scores of all 25 students in Jenny’s class. How did she perform on this test relative to her classmates? O 79, 81, 80, 77, 73, 83, 74, 93, 78, 80, 75, 67, 73, 77, 83, 86, 90, 79, 85, 83, 89, 84, 82, 77, 72 O Find the mean, median, and standard deviation because the data would produce a symmetric histogram. Example Cont. O Mean = 80 O Median = 80 O Standard Deviation = 6.07 O Jenny did better than average! How many standard deviations above the mean is Jenny? Measures of Relative Standing O Standardized value (z – score)- how many standard deviations away from the mean a given observation is O Useful with symmetric distributions O Positive z score means above the mean O Negative z score means below the mean Example O Jenny’s z score Z= 86 −80 6.07 = 0.99 Converting Jenny’s original score to standard deviation units is called standardizing! Z- Scores O Used to compare the relative standing of individuals in different distributions O The next day Jenny got her Chemistry test back and saw she earned an 82. The test scores were symmetric with a mean of 72 and a standard deviation of 4. O Did Jenny do better on the Chem test or the Stats test? Percentiles O Different Stats books will calculate the percentile differently. O There is no exact science O The median is always the 50th percentile O One formula is… Percentile = 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 𝑏𝑒𝑙𝑜𝑤 𝑋+0.5 𝑡𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑣𝑎𝑙𝑢𝑒𝑠 * 100 Chebyshev’s InequalityNot on AP Exam O Proportion of values from a data set that will fall within k standard deviations of the mean will be 1 at least 1 - 2 , where k is a number greater than 𝑘 1 (k is not necessarily an integer). Works for any shape! O At least ¾ or 75% of the data values fall within 2 standard deviations of the mean O 1 – 1/22 = 1 – ¼ = ¾ = 75% 8 O 9 or 88.89% of the data values will fall within 3 standard deviations of the mean O 1 – 1/ 32 = 1 – 1/9 = 8/9 = 88.89% Density Curve O Always on or above the horizontal axis and has an area of exactly 1 underneath it O The area under the curve for any given interval is equal to the proportion of all observations Notation used for density curves O Observed data Idealized Data O (Sample) (population) Parameters O Statistics MEAN O O s standard deviation μ (mu) σ (sigma) Median of a Density Curve O Median is the equal areas point (the point with half the area under the curve to the left and half to the right) O The quartiles divide the area under the curve into quarters Mean of a Density Curve O Balance point of the curve- the curve would be balanced here if it were made of solid material The median and mean are the same for a symmetric density curve. Both lie at the center of the curve. The mean of a skewed curve is pulled away from the median in the direction of the long tail. Density Curves O Rough estimate by eye for mean, median, and quartiles, but not standard deviation O This is an idealized description of the distribution of data O Distinguish between the mean and standard deviations of samples and parameters by using different notations