MEASURES OF CENTER AND SPREAD MEASURES OF CENTER 11/20/2014. What is a measure of center? a value at the center or middle of a data set

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Norman Robertson
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1 MEASURES OF CENTER AND SPREAD Mean and Median MEASURES OF CENTER What is a measure of center? a value at the center or middle of a data set Several different ways to determine the center: Mode Median Mean 1

2 Mode Way to describe a representative or typical value The value that occurs with the greatest frequency Bimodal Multimodal No Mode two data values occur with the same greatest frequency more than two data values occur with the same greatest frequency no data value is repeated Mode is the only measure of central tendency that can be used with nominal data Median Describes the center of a numerical data set in terms of how many data points are above and below it. the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude. Often denoted by (pronounced xtilde ) From Median to Mean Median: describes the center of a numerical data set in terms of how many data points are above and below it Mean: the measure of center obtained by adding the values and dividing the total by the number of values Do these definitions contribute to student understanding? 2

numerical data set in terms of how many data points are above and below it. the middle value when the original data values are arranged in order of increasing (or decreasing) magnitude.

3 From Median to Mean Many students believe that the mean is ALWAYS a better measure of center, no matter what. Students do not understand the advantages of each statistic and when to use each. What are the instructional deficiencies that may lead to these misconceptions? Summarized from: Mean and Median: Are the Really So Easy? By Judith Zawojewski and J. Michael Shaughnessy Nine teachers were asked how many years they have been teaching and they answered as follows: 1, 3, 4, 4, 4, 5, 7, 8, 9 Find the MEDIAN of these values. Transfer this information onto the dot plot using the following color code: for values below the median for values above the median for values that are equal to the median Place an arrow at the location of the median. 3

Summarized from: Mean and Median: Are the Really So Easy? By Judith Zawojewski and J.

4 In your groups respond to the following: How would adding the value of 1 to your distribution change your MEDIAN? How would adding the value of 10 to your distribution change your MEDIAN? What can you say about the effect of data values on the MEDIAN? describes the center of a numerical data set in terms of how many data points are above and below it Why is the MEDIAN a measure of center? 4

How would adding the value of 10 to your What can you say about the effect of data values on the

5 Now find the MEAN of these values. Transfer this information onto the dot plot using the following color code: for values below the mean for values above the mean for values that are equal to the mean Place an arrow at the location of the mean the measure of center obtained by adding the values and dividing the total by the number of values 5

below the mean for values above the mean for values that are equal to the mean Place an

6 In what ways is the mean the center of a distribution? Explore this question with your partner using your dot plot. Hint: How can you find the mean without knowing the formula? Why is the mean a measure of center? Two ways of thinking about mean Fair share Balance point give 3 In your groups respond to the following: How would adding the value of 1 to your distribution change your MEAN? How would adding the value of 10 to your distribution change your MEAN? What can you say about the effect of values on the MEAN? 6

Two ways of thinking about mean Fair share Balance point -3 +3 3 6 9 give 3 In your groups respond to the following: How would

7 From Median and Mean: MEAN In your distribution, which is higher, the MEAN or the MEDIAN? Which is a better measure of the typical number of teaching years? When is mean a better measure of center? When is median a better measure of center? From Median and Mean: MEAN Find two numbers that can be added to the data set and not change the mean. Explain how you chose these two numbers. Find three numbers that can be added to the data set and not change the mean. Explain how you chose these three numbers. Comparing Mean and Median Find two numbers that can be added to the data set that change the mean but not the median. Explain how you chose these two numbers. Find two numbers that can be added to the data set that change the median but not the mean. Explain how you chose these two numbers. 7

Find three numbers that can be added to the data set and not change the mean. Explain how you chose these three numbers.

8 The Mean as a Balance Point The mean is influenced by values other than the mean. When one calculates the mean, a value of zero, if it appears, must be taken into account The sum of the deviations from the mean is always zero. Replace each original value with its distance from the mean. (above is positive, below is negative) Sum the values This is why it s a measure of center! 8

sum of the deviations from the mean is always zero.

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,