Scaled vs. Original Socre Mean = 77 Median = 77.1

Transcription

1 Have out... - notebook - colors - calculator (phone calc will work fine) Tests back as you come in! vocab example Tests Scaled vs. Original Socre Mean = 77 Median = Δ vocab example 1

2 2.1 Describing Location in a Distribution vocab example Objectives The student will be able to... - Measure position using percentiles - Interpret cumulative relative frequency graphs - Measure position using z-scores - Transform data - Define and describe density curves 2

3 Percentile The pth percentile of a distribution is the value with p percent of the observations less than it Is your percentile for a test the same as the percent correct? Wins in MLB The stemplot below shows the number of wins for each of the 30 Major League Baseball teams in Key 5 9 represents a team with 59 wins Find the percentiles for the following teams: (a) Colorado Rockies, who won 92 games (b) NY Yankees, who won 103 games (c) KC Royals and Cleavland Indians, who both won 65 games 3

4 1. Create a table Cumulative Relative Frequency Graph 2. Find Relative Frequency, Cumulative Frequency, and Cumulative Relative Frequency 3. Create a graph based on Cumulative Relative Frequency and interpret Age of President Age Frequency Relative Cumulative Relative Freq. Freq. Cum. Freq

5 Cumulative Relative Frequency Graph: Age of President (a) What percentile is the age of 55 at? Interpret. (b) What is the 75th percentile for this distribution? (c) Where is the graph steepest? What does this indicate about the distribution? Age of President Cumulative Relative Frequency Graph: (d) Barack Obama was inaugurated at age 47. Was he unusually young? 5

6 Heights Macy, a 3-year-old female is 100 cm tall. Brody, her 12-yearold brother is 158 cm tall. Obviously, Brody is taller than Macy - but who is taller, relatively speaking? That is, relative to other kids of the same ages, who is taller? According to the Centers for Disease Control and Prevention, the heights of three-year-old females, have a mean of 94.5 cm and a standard deviation of 4 cm. The mean height for 12-year-old males is 149 cm with a standard deviation of 8 cm. Any thoughts on how we could compare Macy and Brody on an "even playing field"? z-score (standardized score) If x is an observation from a distribution that has a known mean and standard deviation, the standardized value of x is: z = x - mean standard deviation The z-score tells us how many standard deviations away from the mean your observation is. Units? Sign? 6

7 Home Run Kings The single-season home run record for MLB has been set just three times since Babe Ruth hit 60 home runs in Roger Maris hit 61 in 1961, Mark McGwire hit 70 in 1998 and Barry Bonds hit 73 in In an absolute sense, Barry Bonds had the best performance of these for players, since he hit the most home runs in a single season. However, in a relative sense this may not be true. Baseball historians suggest that hitting a home run has been easier in some eras than others. This is due to many factors, including quality of batters, quality of pitchers, hardness of the baseball, dimensions of ballparks, and possible use of performance-enhancing drugs. To make a fair comparison, we should see how these performances rate relative to other hitters during the same year. Home Run Kings Calculate the standardized score for each player and compare. 7

8 AP 2.1 Notes WEB.notebook Home Run Kings In 2001, Arizona Diamondback Mark Grace's home run total had a standardized score of z = Interpret this value and calculate the number of home runs he hit. Before you leave... How wide do you think that the classroom is, in meters? Write your guess on a post-it. We will use these guesses tomorrow! Teacher Note: Make a dotplot of guesses and dotplot of guess-true width for tomorrow 8

9 Homework 2.1 (Day 1) Assignment Bellwork In 2001, Arizona Diamondback Mark Grace's home run total had a standardized score of z = Interpret this value and calculate the number of home runs he hit. *have out notes, bw, hw, calc, colors, pen(cil)* 9

10 2.1 Describing Location in a Distribution (DAY 2) vocab example Standardized Scores (z-scores) puts all data on an "even playing field" converts data to have a mean of 0 and a standard deviation of 1 10

11 Guesses of Room Width (insert dotplot) Let's discuss SOCS! Guesses of Room Width What do you think will happen to the distribution if we graph the errors? error = guess - actual 11

12 Guesses of Room Width (insert dotplot of errors) Transforming Data Effect of Adding (or Subtracting) a Constant: Adding the same number a to each observation adds a to the measures of center and location (mean, median, quartiles, percentiles) but does not change the shape of the distribution or measures of spread (range, IQR, st. dev.) 12

13 Guesses of Room Width What do you think will happen to the distribution if we graph the errors converted to feet? feet = meters x 3.28 Guesses of Room Width (insert dotplot of errors converted to feet) 13

14 Transforming Data Effect of Multiplying (or Dividing) a Constant: Multiplying (or dividing) the same number b to each observation, multiplies (divides) the measures of center/location and the measures of spread by b but does not change the shape of the distribution NYC Taxi In 2010, Taxi Cabs in New York City charged an initial fee of $2.50 plus $2 per mile. In equation form, fare = (miles). At the end of the month a businessman collects all of his taxicab receipts and analyzed the distribution of fares. The distribution was skewed to the right with a mean of $15.45 and a standard deviation of $ (a) What are the mean and standard deviation of the lengths of his cab rides in miles? (b) If the businessman standardized all of the fares, what would be the shape, center, and spread of the distribution? 14

15 (a) mean: $15.45 = m miles NYC Taxi st. dev.: $10.20 = m *not affected by +/- * $10.20 = 2m 5.1 miles (b) Subtracting & dividing does not change the shape, so the distribution would remain skewed right. The center would decrease by $15.45 and divide by $ If we use mean as a measure of center, it would become $0. The spread is not effected by subtracting, but would be divided by $ If we use st. dev. it would become $1. Density Curve A density curve is a curve that: - is always on or above the horizontal axis, and - has area exactly 1 underneath it A density curve describes the overall pattern of a distribution. The area under the curve and above any interval of values on the horizontal axis is the proportion of all observations that fall in that interval. 15

16 Density Curve Median of Density Curve- the equal-areas point, the point that divides the area under the curve in half Mean of Density Curve- the balance point, at which the curve would balance if made of solid material Tweet It Out or Ticket Out on a Post-it Why do we find standardized scores? What does the mean and standard deviation change to when we calculate z-scores? 16

17 Homework 2.1 (Day 2) Assignment 17