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Developing The Concept

Data

Subtopic 1: Measures of Central Tendency: 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: paper and pencil for each student Say: Yesterday we looked at what happened to the mean, median and mode when outliers or new data were considered. Today we are going to extend that further. Who can tell me what the median for a set of data is and how to find it? Students will say it 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 for a set of data? (It is the number which 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 will say that the mean is found by adding up all the data items and dividing by the total number of items. Write the following data set on the board. 35, 26, 34, 47, 38, 40, 39, 43, 24, 46, 36, 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 in minutes the amount of time that twelve students spend doing math homework in one day. I'd like you to find the median for the data set. What should you do to help you find the median? Students will say that you should order the data from least to greatest, or something similar. Have them go ahead and find the median for the data. Ask: What is the median? (38.5 seconds) Good, what is the mode for this data? (46 seconds) How did you find that? (46 is the only number repeated.) Say: Now find the mean for this data set. (37.833 seconds) Ask: Of the three measures we just computed, which one do you think represents the data the best? Explain why you feel that it does. Students will probably respond that either the mean or the median best represents the data. Those who support the mean will say it takes into account the lower numbers of 24 and 26, while those who support the median will say it is less affected by the two low numbers. Ask: Do you believe that this data set has any outliers? Some students will feel that the times of 24 and 26 minutes are much lower than the other numbers. However, they are not significantly less than the other numbers. Say: Two more students say that they spend 73 minutes and 81 minutes on math homework. If these two numbers 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. Write them on the board. Say: Let's recalculate the mean, median, and mode to find out what changes. Let's first find the median. Have the students recalculate the median by annexing the two new numbers onto the data set. Ask: What is the new median for the data set? (39.5 seconds) What is the mode for the data? (46 seconds) Ask: How do I calculate the new mean for the data set? Since I know the sum of the first 12 numbers is 454, do I need to start from scratch to find the mean? (No, you can just add 73 and 81 to 454 then divide by 14.) Calculate the new mean. Ask: What did you get for the mean? (43.43 seconds) 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 either the mean or median best represents the new data set. Ask: Do you think the new data set has any outliers? If so, what are they? Many students will feel that 73 minutes and 81 minutes are outliers because they are at least 16 minutes greater than any other times. Ask: Think about the data sets we worked with. Between the mean, median, and mode, which of the three seems to be least affected by outliers? Explain. (The mode is least affected.) Which of the three seems to be most affected by outliers? Explain. Students will say that the mean seems to be most affected. In most cases, the median will be only slightly affected.

      Measures of Central Tendency:
      Mean, Median, and Mode

      Sampling Techniques

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