Common Core Math I Unit 1: One-Variable Statistics Measures of Center and Spread

Describing Data Numerically Measures of Center – mean, median Measures of Spread – range, interquartile range, standard deviation Again, just to remind students where they are in the study of one-variable stats. Today’s focus is on the mean. S-ID.2  Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

Think About the Situation Lets imagine stacked cubes representing each household from Group A . Use the stacks to answer the following questions. How could we rearrange to find the mean? Family 5 Family 6 This is also printed in the handouts for today. Questions appear on this slide and following slides. What is the median of these data? The median is 3.5 people per household. Family 4 Family 1 Family 3 Family 2 Adapted from Data About Us, Connected Mathematics 2, Grade 6

Finding the Mean Make stacks all the same height by moving cubes. Make the stacks all the same height by moving the cubes. Family 1 Family 5 Family 3 Family 2 Family 4 Family 6 Adapted from Data About Us, Connected Mathematics 2, Grade 6

Finding the Mean How many cubes are in each stack? By leveling out the stacks to make them equal height, you have found the average, or mean, number of people in a household. What is the mean number of people per household? How many cubes are in each stack? 4 By leveling out the stacks to make them equal height, you have found the average, or mean, number of people in a household. What is the mean number of people per household? 4 people per household Ruth Ossie Gary Paul Leon Arlene Adapted from Data About Us , Connected Mathematics 2, Grade 6

Thinking About the Situation Answers to questions/additional questions: There are six households in each group. There are 2 + 3 + 3 + 4 + 6 + 6 = 24 people in Group A and 3 + 3 + 4 + 4 + 4 + 6 = 24 people in Group B. The number of people divided by the number of households tells us the number of people per household, which is the mean. This leads into finalizing the formula for mean (next slide)

Mean You can also think of mean as a balancing point, or making sure everyone has the same amount.

The Formula Discuss what symbols mean! x-bar is the symbol for the mean of the is the capital Greek letter sigma, which stands for “sum.” x is an individual data value n is the number of data values. Note: Students do NOT need to memorize this formula. They need to know, use, and be able to describe the process for finding the mean.

Investigation 2: Using the Mean

Investigation 3: Using the Mean Find the following: the total number of students the total number of movies watched the mean number of movies watched A new value is added for Carlos, who was home last month with a broken leg. He watched 31 movies. How does the new value change the distribution on the histogram? Is this new value an outlier? Explain. What is the mean of the data now? Compare the mean from question 1 to the new mean. What do you notice? Explain. Does this mean accurately describe the data? Explain. Answers to questions/additional discussion questions: a) 10 90 9 movies per student a) There would be an additional bar between 30 and 35 with a frequency of 1. This would make the distribution appear to be skewed to the right. Yes, there is a large gap between 31 and the rest of the data. 121/11 = 11 movies per student It is higher than the previous mean by 2. Take students opinions on this. Some may say yes, some no. Note during the class discussion that 5 of the 11 students have watched 11 movies or more. If time: let’s look at the median number of movies. What is the median number of movies watched both before Carlos is added (6.5) and after Carlos is added (7)? Which value seems to describe the data better in each case – the mean or the median? This discussion lays the groundwork for the next part of the lesson on mean vs. median.

Data for eight more students is added. Add these values to the list in your calculator. How do these values change the distribution on the histogram? Are any of these new values outliers? What is the mean of the data now? The distribution is more clustered to the left, so it now appears skewed right. The new values are not outliers – they all fall in the bottom cluster. The mean is 7.9, or approximately 8, movies per student.

How do I know which measure of central tendency to use? http://regentsprep.org/REgents/math/ALGEBRA/AD2/measure.htm

Investigation 4: Mean vs. Median The heights of Washington High School’s basketball players are: 5 ft 9in, 5 ft 4in, 5 ft 7 in, 5ft 6 in, 5 ft 5 in, 5 ft 3 in, and 5 ft 7 in. A student transfers to Washington High and joins the basketball team. Her height is 6 ft 10in. Discuss and solve in your groups! Let’s talk about the mean vs. the median. How do you know which one to use? Consider this situation: Read the problem, then give groups time to answer the questions on the worksheet. Discuss answers. What is the mean height of the team before the new player transfers in? (65.9 in.) What is the median height? (66 in.) What is the mean height after the new player transfers? (67.9 in.) What is the median height? (66.5 in.) What effect does her height have on the team’s measures of center? (The mean increased by 2 in. and the median increased by .5 in.) How many players are taller than the new mean team height? (2) How many players are taller than the new median team height? (4) Which measure of center more accurately describes the team’s typical height? Explain. (The median gives a more accurate description of the team’s typical height. Half of the players are taller than the median (and half shorter) but only two players are taller than the mean. Using the mean would lead someone to conclude that the team is taller than they really are.)

Mound-shaped and symmetrical (Normal) Mean vs. Median Mound-shaped and symmetrical (Normal) Skewed Left Skewed Right Ask them to discuss with a shoulder buddy where the mean and median for each distribution would be located. They should make a quick sketch on their paper and mark where these values would be. What is the location of the mean relative to the median in each type of distribution? Why does this happen? In a symmetrical distribution, the mean and the median are approximately equal. In skewed distributions, the outliers tend to “pull” the mean towards them, in order to maintain the mean as a balance point for the data set.

Ticket out the Door What happens to the mean of a data set when you add one or more data values that are outliers? Explain. What happens to the mean of a data set when you add values that cluster near one end of the original data set? Explain. Explain why you think these changes might occur.