section 3.1 measures of center hawkes learning systems math courseware specialists copyright © 2008...
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Section 3.1
Measures of Center
HAWKES LEARNING SYSTEMS
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Copyright © 2008 by Hawkes Learning
Systems/Quant Systems, Inc.
All rights reserved.
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• A measure of central tendency describes a central, or typical, value in a data set.
• The mean, median, and mode are all measures of central tendency.
Numerical Descriptions of Data
3.1 Measures of Center
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• The mean is what we typically call the “average” of a data set.
• To calculate the mean, simply add all the values and divide by the total number in the data set.
• Formula:
Calculating the Mean:
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It is possible for the mean not to be a number in the data set.
Numerical Descriptions of Data
3.1 Measures of Center
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63 68 71 67 63 72 66 67 70
Calculate the sample mean of the following heights in inches:
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Numerical Descriptions of Data
3.1 Measures of Center
Solution:
When calculating the mean, round to one more decimal place than what is given in the data.
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• The median is the middle value in an ordered set.
• To calculate the median, first put the numbers in numerical order. Then,
a. if n is odd, the median is the number in the center.
b. if n is even, the median is the mean of the center two numbers.
Calculating the median:
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Numerical Descriptions of Data
3.1 Measures of Center
It is possible for the median not to be a number in the data set.
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a. 15 16 11 22 19 10 17 22
Calculate the median of the following sets of data:
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Numerical Descriptions of Data
3.1 Measures of Center
Solution:
10 11 15 16 17 19 22 22
b. 2.6 3.3 5.0 1.8 0.7 2.2 4.1 6.1 6.7Solution:
0.7 1.8 2.2 2.6 3.3 4.1 5.0 6.1 6.7
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• The mode is the data value(s) that occur(s) most frequently.
• A data set may have one mode (unimodal), two modes (bimodal), or many modes (multimodal).
• If each data value occurs the same number of times, then there is no mode.
Calculating the mode:
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Numerical Descriptions of Data
3.1 Measures of Center
The mode will always be a number in the data set.
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a. 63 68 71 67 63 72 66 67 70
Calculate the mode of each data set:
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3.1 Measures of Center
Solution:
63 68 71 67 63 72 66 67 70
b. 51 77 54 51 68 70 54 65 51Solution:
51 77 54 51 68 70 54 65 51
c. 1 5 7 3 2 0 4 6Solution:
No mode
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This all depends on the data:
• For qualitative data, the mode should be used.
• For quantitative date, the mean should be used unless the data set contains outliers.
• Quantitative data sets with outliers should use the median.
Which measure of the “average” is the best to use?
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Numerical Descriptions of Data
3.1 Measures of Center
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Choose the best measure of center for the following data sets:
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Numerical Descriptions of Data
3.1 Measures of Center
a. The average t-shirt size (S, M, L, XL) of American women.
Mode
b. The average salary for a professional team of baseball players.
c. The average price of houses in a subdivision of similar houses.
Mean
Median