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Standard Deviation Understanding the Mean 2009 3.17c 2 Taken from Virginia Department o f Education “Mean Balance Point” Where is the balance point for this data set? X 3 X X X X X Taken from Virginia Department o f Education “Mean Balance Point” Where is the balance point for this data set? X X X X 4 X X Taken from Virginia Department o f Education “Mean Balance Point” Where is the balance point for this data set? X X X 5 X X X Taken from Virginia Department o f Education “Mean Balance Point” Where is the balance point for this data set? X X X X X X Taken from Virginia Department o f Education “Mean Balance Point” 6 Where is the balance point for this data set? 3 is the Balance Point X 7 X X X X X Taken from Virginia Department o f Education “Mean Balance Point” Where is the balance point for this data set? Sum of the distances below the mean 1+1+1+2 = 5 X 8 MEAN Sum of the distances above the mean 2+3=5 X X X X X Taken from Virginia Department o f Education “Mean Balance Point” Where is the balance point for this data set? Move 2 Steps Move 2 Steps Move 2 Steps Move 2 Steps 4 is the Balance Point 9 Taken from Virginia Department o f Education “Mean Balance Point” We can confirm this by calculating: 2 + 2 + 2 + 3 + 3 + 4 + 5 + 7 + 8 = 36 36 ÷ 9 = 4 The Mean is the Balance Point 10 Where is the balance point for this data set? If we could “zoom in” on the Move 1 Step The Balance Point is between 10 and 11 (closer to 10). space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point. Move 2 Steps Move 1 Step Move 2 Steps 11 Taken from Virginia Department o f Education “Mean Balance Point” • Place the 8 sticky notes as a group so that exactly 3 are ‘16’, and one is ’12’. Place the remaining four numbers so that the balance point is 16. Then find the sum of deviations from the center. • Place the 9 sticky notes as a group so that exactly 1 is ‘11’, two are ‘17’ and two are ’16’. None of the remaining ones have ’16’ Place the remaining four numbers so that the balance point is 16. Then find the sum of the deviations from the center. Taken from Virginia Department o f Education “Mean Balance Point” [Which of the following will have the most variability? A. [Heights of people in this room] B. [Ages of people in this room] C. [The number of countries that people have been to in this room?] Variability: How close the numbers are together. x Standard deviation: (Sx or ) Way to measure the variability. Closer to zero is better! Sum of Distances from Center: -2,-2,-2,-1,-1,0,1,3,4 = 0 Sum of Squares of distances: 4,4,4,1,1,0,1,9,16=40 from center: Average (with one less member) of the squares of the distance from the center: VARIANCE 40/8=5 Square root of the VARIANCE: 2.23 so the STANDARD DEVIATION (Sx) is 2.23 Now find the STANDARD DEVIATION of your Poster 15 Taken from Core Plus Mathematics Grams of Fat Big Mac: BK Whopper: Taco Bell Beef Taco: Subway Sub w/toppings: Dominoes Med. Cheese Pizza: KFC Fried Chicken: Wendy’s Hamburger: Arby’s Roast Beef Sandwich: Hardee’s Roast Beef Sandwich: Pizza Hut Medium Cheese Pizza: 31 46 10 44.5 39 19 20 19 10 39 Calculator Method • • • • • 1) Put the numbers into STAT EDIT 2) Do STAT CALC 1-VAR STATS. The is the “mean” x The Sx is the standard deviation The is a standard deviation we will not use • The n is the amount of data (good way of checking) • The ‘med’ is the median (scroll down) x Which would have a lower standard deviation? (Be prepared to explain): A. [The heights of students in this class] B. [The heights of students in this school] Normal Distribution Bell Curve http://en.wikipedia.org/wiki/File:Standard_deviation_diagram.svg http://en.wikipedia.org/wiki/Skewness Determine if the following examples are Normally Distributed, Years of Teaching Experienc e at Forsyth High School Positively Skewed, or Negatively Skewed. 6 5 4 3 Frequency 1 Number of Shoes Owned per Person 0 0-5 1 6-10 6 11-15 10 16-20 11 21-25 9 >26 8 2 0-4 5-9 10-14 15-19 20-25 26-30 30 + IQ's of Randomly Selected People <50 51-60 61-70 71-80 81-90 91-100 101-110 111-120 121-130 131-140 141-150 >150 20 15 10 5 0 Place the following under negatively skewed, normally distributed, or positively skewed, or random? A) The amount of chips in a bag B) The sum of the digits of random 4-digit numbers? C) The number of D1’s that students in this class have gotten? D) The weekly allowance of students E) Age of people on a cruise this week F) The shoe sizes of females in this class Which is more likely to make a better bell curve, measuring the heights of people in this room or measuring the heights of people in this school? http://www.shodor.org/interactivate/activities/NormalDistribution/ SAT Scores 200 300 400 5000 500 600 700 800 The SAT’s are Normally Distributed with a mean of 500 and a standard deviation of 100. A) Give a Title and fill in the bottom row B) What percentage of students score above a 600 on the SAT? 15.8 C) What percentage of students score between 300 and 500? 47.7 D) If Jane got a 700 on the SAT, what percentile would she be? 100-2.2 E) Mt. Tabor has 1600 students, how many students are expected 97.8 to get at least a 700? .022*1600 = 35 students IQ’s of Humans 50 66.7 83.3 5000 100 116.7 133.3 150 The IQ’s are Normally Distributed with a mean of 100 and a standard deviation of 16.667. 2.2 A) What percentage of people have an IQ below 66.7? B) A genius is someone with IQ of at least 150? What percentage? .1 15.8 C) If Tom’s IQ is 83.3, what percentile would he be? D) Spring School has 1000 students, how many students are expected .022 * 1000 = 22 to have at least a 133.3 IQ? E) What number represents a Z-score of 1.5? 100+1.5(16.667) = 125 The following is the amount of black M&M’s in a bag: 12, 13,14, 15, 15, 16, 17, 20, 21,22,23,24,25 Find the mean and standard deviation A. [18.23, 4.46] B. [18.23, 4.28] The following is the amount of black M&M’s in a bag: 12, 13,14, 15, 15, 16, 17, 20, 21,22,23,24,25 What percentage is above 22.6 black M&M’s? 15.8 0.3 Memory Game Dog, cat, monkey, pig, turtle, apple, melon, banana, orange, grape, desk, window, gradebook, pen, graph paper, Stove, oven, pan, sink, spatula, Shoes, tie, bracelet, necklace, boot A) Find the mean and standard deviation with your calculator B) Is it positively skewed, negatively skewed, or normally skewed? • The more people with same data means lower standard deviation. • A lower standard deviation means less variability. • Z score is how many standard deviations you are from the mean. The higher of the absolute value of the z-score indicates the less likelihood of the event happening. Ex: Z score of 2 is more remarkable than a z score of 1 Ex 2: A mean of 7, stdeviation of 3. A z-score of -1.5 would be 7+(-1.5)*3 = 2.5 Debate: • Side 1) You are trying to convince your teacher to always curve test grades to a standard deviation • Side 2) You are trying to convince your teacher to never curve test grades to a standard deviation Summarize the Mathematics A) Describe in words how to find the standard deviation. B) What happens to the standard deviation as you increase the sample size? C) Which measures of variation (range, interquartile range, standard deviation) are resistant to outliers. Explain D) If a deviation of a data point from the mean is positive, what do you know about its value? What if the deviation is zero? E) What do you know about the sum of all the deviations of the mean? F) Suppose you have two sets of data with an equal sample size and mean. The first data set has a larger deviation than the second one. What can you conclude? Adult female dalmatians weigh an average of 50 pounds with a standard deviation of 3.3 pounds. Adult female boxers weigh an average of 57.5 pounds with a standard deviation of 1.7 pounds. The dalmatian weighs 45 pounds and the boxer weighs 52 pounds. Which dog is more underweight? Explain…. http://www.rossmanchance.com/applets/NormalCalcs/NormalCalculations.html One way to measure light bulbs is to measure the life span. A soft white bulb has a mean life of 700 hours and a standard deviation of 35 hours. A standard light bulb has a mean life of 675 hours and a standard deviation of 50 hours. In an experiment, both light bulbs lasted 750 hours. Which light bulb’s span was better? http://www.rossmanchance.com/applets/NormalCalcs/NormalCalculations.html Think back to the two overweight people shown on the first slide. How could we now determine which one is more overweight?