Understanding the Mean How can we display data? Since this data is not symmetric, it is harder to determine the mean. Between what two numbers would be your estimate of the mean? Your estimate is closer to which of the two numbers? The mean is 14.3.

Understanding the Mean How did you use a line plot to estimate the mean? “Mean as a Balance Point” What does that make you think of? If we have a balance point, then we have equal amounts on both sides. How does this relate to mean?

Where is the balance point for this data set? X X X XXX X X X X XX X X X X If the data set was symmetric, it would be easy to find the balance point. What about when it isn’t symmetric? As long as we make balanced or equal moves toward the center (an equal number of data point “moves”), we can transform the data set without affecting the mean. Now the line plot is symmetrical. The balance point is 3. Instead of looking at this as transforming the data set, we should consider the movements as the distance traveled to the balance point. Adapted from 2010 VDOE Math Institute Grades 6-8

4 is the Balance Point Move 2 Steps Where is the balance point for this data set? Notice the line plot is symmetrical. Adapted from 2010 VDOE Math Institute Grades 6-8

We can confirm this by calculating: = ÷ 9 = 4 Adapted from 2010 VDOE Math Institute Grades 6-8

The Balance Point is between 10 and 11 (closer to 10). Move 2 Steps Move 1 Step Where is the balance point for this data set? If we could “zoom in” on the space between 10 and 11, we could continue this process to arrive at a decimal value for the balance point. Adapted from 2010 VDOE Math Institute Grades 6-8

Mean: Balance Point Use a line plot to represent the data set. Begin with the extreme data points. Balance the moves, moving one data point from each side an equal number of steps toward the center. Continue until the data is distributed symmetrically or until there are only two values left on the line plot.

Mean Problem We will solve this using the concept of the Mean as the Balance Point. Please do not solve this algebraically! Joe has the following test grades: 85, 80, 83, 91, 97 and 72. In order to make the academic team he needs to have an 85 average. With one test yet to take, he wants to know what score he will need on that to have an 85 average.

Mean = Joe’s grades: 85, 80, 83, 91, 97 and 72. Knowing that Joe needs his average (mean) to be 85%, we will use this as the balance point. (Horizontal line) The scores he currently has will be placed above, below, or on the balance point. 85 If 85 is to be the balance point, then after the 7 th test grade is added, the sum of the distances of the data values above the mean must equal the sum below the mean. To relate this to the line plot, the number of moves to the right of the mean must equal the number of moves to the left. Notice that the length of the lines are an indication of the distance from the mean; the longer the line the farther the value is from the mean.

Mean = Joe’s grades sorted: 72, 80, 83, 85, 91, The sum of the distances below the mean is 20. The sum of the distances above the mean is 18. To make Joe’s grades balanced at 85, does Joe’s next test grade need to be above or below the balance point to have a mean of 85? By how much? So therefore, Joe’s next grade must be at least an 87. Above 20-18= Now, the sum of the distances above the mean is 20. It’s important to note that the sums of the values above the mean, 275, is different than sum of the values below the mean, 235. Sometimes displaying the values in numerical order will help the students visualize the difference

Of course, it can be solved algebraicly. What score will “balance” the number line ? Another way to visualize Joe’s dilemma. Joe’s grades: 85, 80, 83, 91, 97 and 72. A value equal to the mean has a distance of 0.

Just a little reminder about sigma notation This is also called summation notation. This is read “The sum of i from i = 1 to i = 8.” This means to start with i = 1 and add that to i = 2 and add that sum to i = 3, etc until i = 8.

Another look at sigma notation and subscript notation How is this read? Add the first 6 terms together x sub 1 plus x sub 2 plus x sub 3 plus x sub 4 plus x sub 5 plus x sub 6

Mean of a Data Set Containing n Elements = µ µ is pronounced “mew” as in amusing. This means the sum of the n elements in the data set divided by the number of elements in the data set.

Mean of a Data Set Containing n Elements = µ In Algebra I and Algebra II, only the population mean will be used. x = Sample mean (pronounced x bar) used when the data only represents a sampling of the population. µ = Population mean

A student counted the number of players playing basketball in the Central Tendency Tournament each day over a two week period. 10, 30, 50, 60, 70, 30, 80, 90, 20, 30, 40, 40, 60, 20 Using your graphing calculator, find the mean of the data set.

Casio Texas Instruments The mean is x The view from each calculator is given. Since the formula for sample mean and population mean are the same, x is μ. 45 is the Mean for the data set.