1. The table below shows the number of students who are varsity and junior varsity athletes. Find the probability that a student is a senior given that he or she is a varsity athlete. ClassVarsityJunior Varsity Totals Freshman7269276 Sophomore22262284 Junior36276312 Senior51257308 Totals11610641180 or 18.1% Warm-Up 5/06

3. Rigor: You will learn how to find the measure of center, spread, and position. Relevance: You will be able to find the measure of center, spread, and position of real world data sets.

4. 0-9 Measure of Center, Spread, and Position

5. Measures of Center, Spread and Position Statistics is the science of collecting, organizing, displaying and analyzing data in order to draw conclusions and make predictions. Descriptive Statistics is the branch of statistics that focuses on collecting, summarizing and displaying data.

6. Population is the entire group from which you are collecting data. Variable is a characteristic of the population that can assume different values called data. Univariate Data is data that includes only 1 variable. Sample is a smaller group of the population (subset) when it is not possible to obtain data from every member of the population.

7. Univariate data are often summarized using a single number to represent what is typical of the population. Measures of average are also called Measures of Central Tendency or Measures of Center.

9. Example 1: The number of milligrams of sodium in a 12-ounce can of ten different brands of regular soda are shown. Find the mean, median and mode. 50, 30, 25, 20, 40, 35, 35, 10, 15, 35 10,15, 20, 25, 30, 35, 35, 35, 40, 50 mode 35 occurs most often, so the mode is 35 milligrams.

10. Since two very different data sets can have the same mean, statisticians also use measures of spread or variation to describe how widely the data values vary and how much the values differ from what is typical. Key Concept: Measures of Spread The range is the difference between the greatest and the least values in a data set.

12. Example 2: Two classes took the same midterm exam. The scores of 5 students from each class are shown. Both sets of scores have a mean of 84.2. a. Find the range, variance and standard deviation for Class A. b. Find the range, variance and standard deviation for class B c. Compare your results. Class AClass B 85, 76, 92, 88, 8075, 85, 95, 98, 68 x 8585 – 84.2 = 0.8(0.8) 2 = 0.64 7676 – 84.2 = –8.2(–8.2) 2 = 67.24 9292 – 84.2 = 7.8(7.8) 2 = 60.84 8888 – 84.2 = 3.8(3.8) 2 = 14.44 8080 – 84.2 = –4.2(– 4.2) 2 = 17.64 Sum = 160.8 Range Variance Standard Deviation

13. Example 2: Two classes took the same midterm exam. The scores of 5 students from each class are shown. Both sets of scores have a mean of 84.2. b. Find the range, variance and standard deviation for class B Class AClass B 85, 76, 92, 88, 8075, 85, 95, 98, 68 Range Variance Standard Deviation To find measure of center and spread using a graphing calculator, first clear all list. (2ND + 7 ENTER 2) Press STAT ENTER to enter each data value. Then press STAT and arrow over to the CALC menu and select 1-Var Stats ENTER. maxX – minX

14. Example 2: Two classes took the same midterm exam. The scores of 5 students from each class are shown. Both sets of scores have a mean of 84.2. c. Compare the sample standard deviations of Class A and Class B. Class AClass B 85, 76, 92, 88, 8075, 85, 95, 98, 68 Class A Standard Deviation Class B There is more inconsistency in the sample scores from class B.

15. Quartiles are 3 position measures that divide a data set arranged in ascending order into four groups, each containing about 25% of the data. The median marks the 2 nd quartile Q 2 and separates the data into upper and lower halves. The 1 st or lower quartile Q 1 is the median of the lower half. The 3 rd or upper quartile Q 3 is the median of the upper half. The 3 quartiles along with the minimum and maximum values are called the five-number summary of a data set.

16. Example 3: The number of hours that Liana, worked each week for the last 12 weeks were 21,10,18,12,15,13,20,20,19,16,18 and 14. a. Find the minimum, lower quartile, median, upper quartile and the maximum of the data set. Use your calculator. minimum lower quartile median upper quartile maximum minX =10 Q 1 = 13.5 med =17 Q 3 = 19.5 maxX = 21 b. Interpret this five-number summary. Over the last 12 weeks, liana worked a minimum of 10 hours and a maximum of 21 hour. She worked less than 13.5 hours 25% of the time, less than 17 hours 50% of the time, and less than 19.5 hours 75% of the time.

17. The Interquartile Range (IQR) is the difference between Q 3 and Q 1 ; it contains about 50% of the values. An Outlier is an extremely high or extremely low value when compared to rest of the values in the set. To find an outlier look for data values that are beyond the upper or lower quartiles by more than 1.5 times the IQR.

18. Example 4: The number of minutes each of 22 students in a class spent working on the same algebra assignment is shown below. 15,12,25,15,27,10,16,18,30,35,22,25,65,20,18,25,15,13,25,22,15,28 a. Identify any outliers in the data. Use your calculator. b. Find mean, median, mode, range, and standard deviation of the data set both with and without the outlier. Describe the effect on each measure. minX = 10 Q 1 = 15 med = 21 Q 3 = 25 maxX = 65 Data setMeanMedianModeRangeStandard Deviation with outlier  22.5 211555  11.2 without outlier  20.5 201525  6.5

19. Assignment Prob/Stats #4 WS, 1-11 All Prob/Stats Test Friday 5/16