MEASURES OF DISPERSION Standard Deviation and Variance

Warm-up  Which of the following values could be valid mean absolute deviations. Explain why the others are invalid? -400, -3.2, 0, 0.2, 100, 1,000  12 people are travelling together to Ocean City. The amount of spending money each bring for their 3 day trip is given. Determine the mean absolute deviation. {150, 150,175, 200, 200, 225, 225, 250, 250, 300, 300, 300}

Cell phones and driving  The researchers asked student volunteers (subjects) to use a machine that simulated various driving situations. At irregular intervals, a target would flash red or green. Subjects were instructed to press a “brake” button as soon as possible when they detected a red light. The machine calculated the mean reaction time to the red flashing targets for each student in milliseconds.  The subjects were given a warm-up period to familiarize themselves with the driving simulator. Then the researchers had each subject use the driving simulation machine while talking on a Cell Phone about politics to someone in another room and then again with music or a book-on-tape playing in the background (Control). The subjects were randomly assigned as to whether they used the Cell Phone or the Control setting for the first trial. data collected by researchers at the University of Utah (Strayer and Johnston, 2001).

Measures of Dispersions  Give two different words that can replace dispersion?  Why can’t the mean of the deviations be used to indicate how spread out a set of data is?  How does the Mean Absolute Deviation resolve this problem?  What are the units for Mean Absolute deviation? (Look at the formula).

Measures of Dispersions If analyzing the number of text message sent per day in this class 1.What is the unit of a deviation? (x i – μ) 2.What is the unit of the absolute deviations? | (x i – μ)| 3.What is the unit of the mean absolute deviation? 4.What is the unit if we square the deviations? (x i – μ) 2

Variance deviation square it sum (add 1 to n) divided by data value mean # of data

Standard Deviation = deviation square it sum (add 1 to n) divided by s q u a r e r o o t

Mean Absolute Deviation Variance Standard Deviation Units same as the data More resistant to outliers σ 2 Units are squared Sensitive to outliers Can result in large value Variance = (Standard Deviation) 2 σ Units same as data Sensitive to outliers More commonly used Standard Deviation =

Standard Deviation Game

Calculate Find the variance and standard deviation. The heights of the dogs (at shoulders) are: 600mm, 470mm, 170mm, 430mm, and 300mm.

Calculate Variance and Standard Deviation xixi (x i – μ) (x i – μ) 2 x1x1 600 x2x2 470 x3x3 170 x4x4 430 x5x5 300 Sum Mean ( μ) Variance (σ 2 ) Standard Deviation (σ)

Calculate At the Westminster Dog Show, the shoulder height in millimeters of the Old English Sheepdogs is given. Calculate the variance and standard deviation of the contestants’ shoulder heights. ( 533 mm, 555 mm, 568 mm, 520 mm, 525 mm, 578 mm)

Interpret Talk to your partner. Why is there such a difference in standard deviation and variance between the two sets of dogs? Shoulder height in millimeters Variety of Dogs: (170mm, 300mm, 430mm, 470mm, 600mm). Sheepdog: (520 mm, 525 mm, 533 mm, 555 mm, 568 mm, 578 mm) How many dogs are within 1 standard deviation above the mean? How many dogs are within 1 standard deviation below the mean?

Thank Goodness for Calculators! TI Calculator enter the data into a list, STAT, EDIT Select STAT, CALC, 1-Var Stats, and then the list you entered the data into. Central Tendency Park Dispersion Park

1. The temperatures for March 13 from 1996 to 2011 are given in the table. a. Determine the mean, standard deviation, variance, and mean absolute deviation. Use your calculator. μ = _____________ σ = __________________ σ 2 = __________________ M.A.D. = _____________ b. If you were to remove any outliers, which one(s) would it be? Explain how you think it will affect your descriptive statistics. Year Temp (F)

Remove the outliers from your data. Recalculate your measurements. Which measure did it affect the most? The least? μ = _____________ σ = __________________ σ 2 = __________________ M.A.D. = _____________ Below is the data for January 1. Determine the mean, standard deviation, variance, and mean absolute deviation. Use your calculator. μ = _____________ σ = __________________ σ 2 = __________________ Compare the two days using the descriptive statistics. Which day has more variation in temperature? Year Temp (F)

2. The Health Nut Cereal Company is looking at upgrading some of its packing machines. These machines are responsible for pouring 18 ounces of cereal into the box. The plant manager collected data on three of the machine. Reported in the table is the standard deviation of the boxes filled each week. He is not going to be able to replace all of the machines. What would be your recommendation? Week 1Week 2Week 3 Machine Machine Machine Standard Deviation of Cereal Box Weights (ounces)

3. Suzie’s Lemonade Stand is turning into a huge success. She realizes if she is going to set up an additional stand she is going to know how many glasses of lemonade she is selling. Below is her data. MondayTuesdayWednesdayThursdayFridaySaturdaySunday Determine the mean, standard deviation, variance, and mean absolute deviation. Use your calculator. μ = _____________ σ = __________________ σ 2 = __________________ M.A.D. = ________________ Should Suzie open up a new stand for all 7 days? If not what would you recommend? Glasses of Lemonade Sold

4. The Stem-n-Leaf plot represents the number of Girl Scout cookies sold Girl Scout Boxes Sold in Troop 3452 Key 3|2 = 32 boxes The mean is 35.6 boxes and the standard deviation is 20.8 boxes. a. How many girl scouts sold boxes within 1.5 standard deviation? b. How many girl scouts sold more than 1 standard deviation above the mean? c. How many girl scouts sold less than 1.5 standard deviation below the mean?

Independent Work The following are heights (in inches) of the University of Wyoming’s Men’s Basketball Team: (69, 76, 73, 79, 69, 80, 74, 72, 72, 82, 73, 79, 80, 76, 79, 75, 83) 1. Construct a line plot of the data. 2.Calculate the mean, standard deviation, and variance. μ = ______________ σ = __________________ σ 2 = __________________ 3. How many heights are above the mean? How many are below the mean?

Independent Work The following are heights (in inches) of the University of Wyoming’s Men’s Basketball Team: (69, 76, 73, 79, 69, 80, 74, 72, 72, 82, 73, 79, 80, 76, 79, 75, 83) 4.How many heights are within one standard deviation of the mean. Show/Explain how you arrived at your solution. 5. Let’s say Coach Larry Shyatt wants to add one more player to the team. Which of the following players trying out for team would you suggest the coach add to the team? Explain your choice. a. A player one standard deviation above the mean. b. A player one standard deviation below the mean. c. A player two standard deviations above the mean.

Cell phones and driving  The researchers asked student volunteers (subjects) to use a machine that simulated various driving situations. At irregular intervals, a target would flash red or green. Subjects were instructed to press a “brake” button as soon as possible when they detected a red light. The machine calculated the mean reaction time to the red flashing targets for each student in milliseconds.  The subjects were given a warm-up period to familiarize themselves with the driving simulator. Then the researchers had each subject use the driving simulation machine while talking on a Cell Phone about politics to someone in another room and then again with music or a book-on-tape playing in the background (Control). The subjects were randomly assigned as to whether they used the Cell Phone or the Control setting for the first trial.

Let’s look at the Data and Analyze it! On Cell Phone (milliseconds) Controlled (milliseconds) Calculate the mean, standard deviation, and variance for both data sets. Which measure do you feel is most significant, mean or standard deviation? Using your calculated statistics what inferences can you make? Response Time

Let’s look at the Data and Analyze it! On Cell Phone (milliseconds) Controlled (milliseconds) Which measure do you feel is most significant, mean or standard deviation? Explain. 2.Using your calculated statistics what inferences can you make? 3.Why might the standard deviation and variance be larger when using a cell phone versus not using a cell phone. Cell Phone Controlled μ ms531.5 ms σ 44.5 ms37.6 σ2σ ms ms 2

What size Standard Deviation? You’re in charge of a carnival game, where the player will win a prize by being the first to shoot 3 ping pong balls through a hole. 1. As manager of the game, what type of standard deviation for the accuracy of the ping pong pistols would you want to have? A larger standard deviation would indicate less pistol accuracy and less chance of getting the 3 ping pong balls in the hole. Thus, no prize. 2. What would the participants want? The participants would benefit from a small standard deviation, indicating a greater accuracy with the pistol

What size Standard Deviation? 1.You have been hired by a company that has employees with years of service ranging from 0 to 27 years. What type of standard deviation would you expect in the salaries of the employees? 2.You are considering taking an assistant manager job at Pac Sun Clothing store. Only a high school degree is required for the 1 manager and 2 assistant managers. The remaining 18 employees are high school students. What type of standard deviation for hourly pay do you expect and why?

Which plot has the larger standard deviation? Without calculating, which data set has the greatest standard deviation? Which has the smallest? Explain your answers a Key: 4| 1 = 41 b Key: 4| 1 = 41 c Key: 4| 1 = 41

Exit Ticket 1.The hourly wage for employees at a small retail store is given. Find the mean, standard deviation and absolute deviation. ($8.40, $8.60, $8.60, $8.80, $8.80, $8.80, $9.00, $14.50) 2.If the $14.50 was not included in the data, what type of change if any is expected of in the standard deviation? 3.The variance of all the Algebra 1 students’ semester exams is 36 points squared. What is the standard deviation? 4.Why is standard deviation used more frequently then variance?