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Measures of Center and Spread Mean Formula: Median Process: Mean Median Where are the measures of center for different shaped distributions? Symmetric Right Skewed Left Skewed MEDIAN Measure of Center: Measure of Spread: Used for: Range: Quartiles: IQR: What is the IQR used for? o Shows… o Helps… IQR Test for possible outliers: Why is it just for POSSIBLE outliers? Example: Supermarket Spending- data below is the amount (rounded to the nearest $) spent in the supermarket. Do a test for possible outliers Supermarket Spending 0 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 3 9 1 5 0 5 2 5 1 5 0 5 1 9 3 6 0 5 4 7 7 8 8 9 0 1 2 3 4 6 6 8 8 8 8 6 9 9 3 4 5 7 9 3 9 0 3 6 6 3 Percentiles: Example: Use the data above to find the percentile for the observation $24. Special Percentiles: Q1 = Median = Q3 = Distribution (using Median): MEAN Measure of Center: Measure of Spread: Used for: What IS Standard Deviation? How do we calculate it? Variance Symbol: Formula: Std. Deviation Symbol: Formula: Note: n – 1 = xi - x= Questions: (p. 52-53 in book) 1. Why do we square the deviations? 2. Why do we divide by n – 1? 3. Why use the std. deviation and not the variance? Notes on Std. Dev: Only use when… S = 0 means… Unless there is no spread,… As observations become more spread out… Easy way to calculate Std. Deviation: AP Statistics Section 1.2 – Measures of Central Tendency 1. Consider the following set of homerun distances (in feet) to center field in 13 ballparks: {387, 400, 400, 410, 410, 410, 414, 415, 420, 420, 421, 457, 461}. Find the mean, median, and mode. Is the data skewed? 2. Suppose that the numbers of unnecessary procedures recommended by five doctors in a 1-month period are given by the set {2, 2, 8, 20, 33}. If it is discovered that the fifth doctor really recommended an additional 25 procedures, how will the median and mean be affected? 3. Suppose the salaries (in dollars) of six employees are 3000, 7000, 15,000, 22,000, 23,000 and 38,000, respectively. a. What are the median and mean salaries? b. What will the new median and mean salaries be if everyone receives a $3000 raise? c. What if everyone receives a 10% raise? 4. Suppose that the faculty salaries at a college have a median of $32,500 and a mean of $38,700. What does this indicate about the shape of the distribution of the salaries? AP Statistics Section 1.2 – Measures of Spread 1. During the years 1929-39 of the Great Depression, the weekly average hours worked in manufacturing jobs were 45, 43, 41, 39, 39, 35, 37, 40, 39, 36, and 37. What is the std. deviation? What is the variance? 2. The following is a sample of 50 shoppers at a grocery store and the amount of money (in dollars) that they spent. 2 6 6 8 9 10 11 11 12 12 13 13 14 14 15 15 15 16 17 18 18 18 19 19 20 20 20 21 23 26 27 28 29 30 31 32 33 33 34 36 37 39 40 43 45 52 61 63 64 69 a. Find the mean and standard deviation. b. Find the five-number summary and the IQR. c. Which summary better summarizes the data? Why? (Hint: Make a graph on your calculator) d. Are there any outliers using the 1.5IQR criterion? If so, what are they? e. On the higher end, what would be the smallest value that would be considered an outlier?