03 Describing Data Graphically and Numerically Part 2

# Is an integer the median is the average of the values

• Notes
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is an integer, the median is the average of the values in position i and i + 1. (i.e., for an even number of observations, the median is the average of the two middle values.)

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Median Example 40 Note that n = 13 Find the i = (1/2) n position: i = (1/2)(13) = 6.5 Since 6.5 is not an integer, round up to 7 The median is the value in the 7th position: M d = 12 Data array: 4, 4, 5, 5, 9, 11, 12, 14, 16, 19, 22, 23, 24
Mean vs Median 41 The median is the measure of location most often reported for annual income and property value data because a few extremely large incomes or property values can inflate the mean.

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Shape of a Distribution 42 Describes how data is distributed Symmetric or skewed The greater the difference between the mean and the median, the more skewed the distribution Mean = Median Mean < Median Median < Mean Right-Skewed Left-Skewed Symmetric (Longer tail extends to left) (Longer tail extends to right)
Shape of a Distribution 43 Mean = Median Mean < Median Median < Mean Right-Skewed Left-Skewed Symmetric (Longer tail extends to left) (Longer tail extends to right) Exam Scores Weight and height of people Data from business and economics Income Housing prices

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Mode 44 A measure of location The value that occurs most often Not affected by extreme values Used for either numerical or categorical data There may be no mode There may be several modes (2 modes = bimodal) 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Mode = 5 No Mode 0 1 2 3 4 5 6
Weighted Mean The mean of data values that have been weighted according to their relative importance Weighted Mean for a population: Weighted Mean for a sample: Example Application: Computing GPA 45 μ W = w i x i w i x W = w i x i w i x i = value of observation i w i = weight for observation i

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Weighted Mean Example Sample of 26 Repair Projects 46 Days to Complete Frequency 5 4 6 12 7 8 8 2 Weighted Mean Days to Complete: x W = w i x i w i = (4 × 5) + (12 × 6) + (8 × 7) + (2 × 8) 4 + 12 + 8 + 2 = 164 26 = 6.31 days
Which measure of location is the best? 47 Mean is generally used, unless extreme values (outliers) exist It is better to use the median than the mean as a measure of central location when a data set contains extreme values. Mode is good for determining most likely to occur
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