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Name: ______________________ Period: ____________ 9/7/2018 Warm Up: Use the stemplot from the problem below to calculate a five number summary and identify if there are any outliers. Here a stem plot of travel time to work for 20 randomly selected New Yorkers. Earlier, we found that the median was 22.5 minutes. 1. Based only on the stemplot, would you expect the mean travel time to be less than, about the same, or larger than the median? Why? 2. Use your calculator to find the mean travel time. Was your answer to Question 1 correct? 3. Would the mean or the median be a more appropriate summary of the center of this distribution of drive times? Standard Deviation ______________________________________________________________ Variance ______________________________________________________________ Hair Measuring Activity -- What is Standard Deviation? 1. Have students record the hair length on the board and then in the table below. So, the variable of interest, x, is hair length. xi 2. What is the average hair length in the class, based on this very “unrandom” sample?_________ This is called x . 3. What is the average distance of hair length away from this mean? To calculate this, subtract the mean form each of the hair lengths (xi). Now add these distances and divide by n (the number of hair lengths recorded). Hmmm. 4. What happened??? 5. How can we adjust what we did before? What can we do? 6. Because this standard deviation is based on a sample (we’re pretending for the moment that we actually took a random sample), it is not an accurate estimate of the true population standard deviation. To make this value an unbiased estimate, we divided by n – 1 instead of n. What we’ve done is develop the formula for the standard deviation. You do not have to memorize it, but you do need to be able to reproduce it (given the formula) and to understand it. Follow- up to hair length and standard deviation. 1.) 2.) 3.) 4.) 5.) 6.) 7.) Measure the longest hair on your head in inches. List the hair length of each member of your group. Find the average hair length of your group. Find the difference from the mean for each person in your group. Square this difference for each person in your group. Take the average of these squared differences. Take the square root of the average of the sum of the squared differences. What did you find? Rhymes with “Lanyard Re-creation” What conclusion(s) can you arrive at if the standard deviation for the hair length of an entire class is zero? One student answered that the class is bald. Are they correct? Are they correct for the right reason? Is having a bald class the only way the standard deviation can be zero? Consider the groups in the room and identify the group with the smallest standard deviation. The largest. A new girl was added to the class and her hair length was the equal to the average hair length for the class. What impact will this addition have to the standard deviation for the class? To the mean? Don’t answer this one too quickly… If everyone cut an inch off of their hair, what would happen to the standard deviation for the class? To the mean? If everyone ate a ton of bananas and their hair length doubled, what would happen to the standard deviation for the class? To the mean? In order for the standard deviation to remain the same what type of hair length(s) of student(s) need to be added to the class? Additional Practice: Here are the foot lengths (in centimeters) for a random sample of 7 fourteen-year-olds from the United Kingdom. 25, 22, 20, 25, 24, 24, 28 Find the “average” squared deviation and the standard deviation.