Transcript 3-4
Here is the data for the class heights in inches: 69 71 70 67 71 61 68 64 63 64 71 63 64 74 63 72 74 64 69 64 140 64 68 65 65 65 67 68 75 78 What can we conclude from… a) Chebyshev’s Thereom? b) The Empirical Rule? Slide 1 Section 3-4 Measures of Relative Standing Slide 2 Key Concept This section introduces measures that can be used to compare values from different data sets, or to compare values within the same data set. The most important of these is the concept of the z score. Slide 3 z Definition Score (or standardized value) the number of standard deviations given value x is above or below the mean that a -Can be used to compare values from different or the same data sets -Found by converting a value to a standardized score -No units -Measure of position: describes location of a value (in terms of standard deviations) relative to the mean EX: z=2.0: A value is 2 standard deviations above the mean z = -3.0: A value is 3 standard deviations below the mean Slide 4 Measures of Position z score Sample x x z= s Population x µ z= Round z to 2 decimal places Slide 5 Interpreting Z Scores Whenever a value is less than the mean, its corresponding z score is negative Ordinary values: Unusual Values: z score between –2 and 2 z score < -2 or z score > 2 Slide 6 Notice: Unusual/usual scores • In 3-3, we used the range rule of thumb to conclude that a value is “unusual” if it is more than ____ standard deviations away from the mean. • It follows that unusual values have z scores less than _____ or greater than _____. Slide 7 Example Over the past 30 years, heights of basketball players at Newport University have a mean of 74.5 in. and a standard deviation of 2.5 in. The latest recruit has a height of 79.0 in. 1) Find the z-score. 2) Is the height of 79.0 in. unusual among the heights of players over the past 30 years? Why or why not? Slide 8 Example Our tallest president was Lyndon Johnson at 75”. The tallest Miami Heat basketball player is Shaquille O’Neil at 85”. These guys are from very different populations; find who is relatively taller among their populations. Standardize their scores. Slide 9 Definition Q1 (First Quartile) separates the bottom 25% of sorted values from the top 75%. Q2 (Second Quartile) same as the median; separates the bottom 50% of sorted values from the top 50%. Q3 (Third Quartile) separates the bottom 75% of sorted values from the top 25%. Slide 10 Quartiles and Percents • Used for comparing values within the same data set • Also measures of position • Useful for comparing values within the same data set or between different sets of data. Slide 11 Quartiles Q1, Q2, Q3 divide ranked scores into four equal parts 25% (minimum) 25% 25% 25% Q1 Q2 Q3 (maximum) (median) Slide 12 Percentiles Just as there are three quartiles separating data into four parts, there are 99 percentiles denoted P1, P2, . . . P99, which partition the data into 100 groups. Process -Sort data (ascending order) -Count total data = n -Count # of values less than the score you are looking for) Slide 13 Finding the Percentile of a Given Score Percentile of value x = number of values less than x • 100 total number of values Slide 14 Converting from the kth Percentile to the Corresponding Data Value Notation L= k 100 •n n k L Pk total number of values in the data set percentile being used locator that gives the position of a value kth percentile Slide 15 Example • Find the value of the 30th percentile = P30. • K = ___ since we are trying to find the value of the ____th percentile. Slide 16 Converting from the kth Percentile to the Corresponding Data Value Slide 17 Some Other Statistics Interquartile Range (or IQR): Q3 - Q1 Semi-interquartile Range: Q3 - Q1 2 Midquartile: Q3 + Q1 2 10 - 90 Percentile Range: P90 - P10 Slide 18