## Lesson: Mean, Median, and Mode Developing the Concept

Now that you've introduced your students to what happens when outliers or new data are added or deleted from a data set, it is time for them to explore this further on their own. Continue to stress the use of proper vocabulary during the lesson.

Materials: pencil and paper for each student

• Say: We have looked at what happened to the mean, median, and mode when outliers or new data were added or removed. Today we are going to extend that. Who can tell me what the median of a set of data is and how to find it?
Students should say that the median is the middle number or the average of the two middle numbers. To find it, one needs to put the data in either ascending or descending order.
• Say: Good! Now what is the mode of a set of data? (It is the number that is most frequently repeated in the data set.) Good! Now what is the mean, and how do I find it for a data set?
Students should say that the mean is found by adding all the data values and then dividing the sum by the number of data points in the set. Write the following data set on the board: 35, 26, 34, 47, 38, 40, 39, 43, 24, 46, 36, and 46. (Instead of using the data above, you could have your students collect their own data on a topic of their choosing.)
• Say: These numbers represent the scores of 12 students on a math test. I'd like you to find the median of the data set. What do you need to do to help you find the median?
Students should say that they need to order the data from least to greatest, or something similar. Have them go on and find the median for the data.
• Ask: What is the median? (38.5) Good! What is the mode for this data? (46) How did you find the mode? (46 is the only number repeated.)
• Say: Now find the mean for this data set. (37.83)
• Ask: Of the three measures we just computed, which one do you think best represents the data? Explain why you think so.
Students will probably respond that either the mean or the median best represents the data. Those who support the mean might say it takes into account the lower numbers 24 and 26, while those who support the median will say that it is less affected by those two low numbers.
• Ask: Does this data set have any outliers?
Some students might say that the scores 24 and 26 are much lower than the other numbers. However, they are not significantly less than the other numbers.
• Say: Two more students scored 73 and 81 on the math test. If these scores are added to our data set above, what do you think will happen to the mean, median, and mode?
Students will make a number of conjectures; list them on the board.
• Say: Let's recalculate the mean, median, and mode to find out what happens when we add 73 and 81 to the data set. Let's first find the median.
Have the students recalculate the median after adding the two scores to the data set.
• Ask: What is the new median of the data set? (39.5) What is the mode of the data? (46)
• Ask: How do I calculate the new mean for the data set? Since I know the sum of the first 12 scores is 454, do I need to start over to find the mean? (No, you can just add 73 and 81 to 454 and then divide the sum by 14.) Calculate the new mean.
• Ask: What value did you get for the mean? (43.43)
• Ask: What is the mode of the data? (46)
• Ask: What values changed? (The median and mean changed.) Which of the three values do you think best represents this new data set? Explain your reasoning.
Again, students will probably say that either the mean or median best represents the data set.
• Ask: Do you think the new data set has any outliers? If so, what are they?
Many students will feel that 73 and 81 are outliers, because they are at least 16 points greater than any other scores.
• Ask: Think about the data sets. Among the mean, median, and mode, which of the three seems to be least affected by outliers and which seems to be most affected? Explain.
Students will say that the mode is least affected and that the mean seems to be most affected. In most cases, the median will be only slightly affected.

Wrap-Up and Assessment Hints
One way to deepen students' understanding of mathematical concepts is to ask them open-ended questions instead of closed questions. For example, we usually give students a set of data and ask them to find the mean, median, and mode of the data. Instead, we should open up that question by asking students to find 10 numbers that have a mean of 35, a mode of 33, and a median of 36. The answer is not unique and requires students to have a firm grasp of the three concepts. Have students explain how they got their answer. A sample answer follows.

Since the mean is 35 and there are 10 numbers, the sum of all the data points must be 10 x 35, or 350.

Since the mode is 33, I decided to have three 33s in my data set and make sure no other number was repeated more than twice.

33, 33, 33

Since the median is 36, I know the average of the fifth and sixth numbers is 36, so I made them both 36.

33, 33, 33, 36, 36

Four of the five remaining scores have to be greater than 36, and one has to be less than 36. I added the numbers 37, 37, 38, and 39 to the set.

33, 33, 33, 36, 36, 37, 37, 38, 39

Since the sum of these 9 numbers is 322, the remaining number must be 28 if the mean is to be 35.

28, 33, 33, 33, 36, 36, 37, 37, 38, 39