1. Unit 6A
Characterizing Data
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2. Definition
The distribution of a variable (or data set) describes the values taken on by the variable and the frequency (or relative frequency) of these values.
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3. Measures of Center in a Distribution
The mean is what we most commonly call the average value. It is defined as follows:
The median is the middle value in the sorted data set (or halfway between the two middle values if the number of values is even).
The mode is the most common value (or group of values) in a distribution.
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4. Mean vs. Average
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5. Finding the Median for an Odd Number of Values
Example: Find the median of the data set below.
(data set)
(sorted list)
(odd number of values)
median is 6.44
exact middle
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6. Finding the Median for an Even Number of Values
Example: Find the median of the data set below.
(data set)
(sorted list)
(even number of values) 2
median is 5.02
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7. Finding the Mode Example: Find the mode of each data set below.
Mode is 5
Bimodal (2 and 6)
No Mode a b c
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8. Effects of Outliers
An outlier is a data value that is much higher or much lower than almost all other values.
Consider the following data set of contract offers:
$0 $0 $0 $0 $2,500,000
The mean contract offer is
As displayed, outliers can pull the mean upward (or downward). The median and mode of the data are not affected.
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9. Shapes of Distributions
Two single-peaked (unimodal) distributions
A double-peaked (bimodal) distribution
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10. Symmetry
A distribution is symmetric if its left half is a mirror image of its right half.
Help students make connections between the overall distribution and the mean, median, mode and outliers of a population or data set. You may want to use concrete examples such as physical heights tend to be symmetrically distributed whereas the annual salaries of a medium-sized company may be right-skewed due to high-paying salaries of management.
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11. Skewness
A distribution is left-skewed if its values are more spread out on the left side.
A distribution is right-skewed if its values are more spread out on the right side.
Help students make connections between the overall distribution and the mean, median, mode and outliers of a population or data set. You may want to use concrete examples such as physical heights tend to be symmetrically distributed whereas the annual salaries of a medium-sized company may be right-skewed due to high-paying salaries of management.
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12. Variation
Variation describes how widely data values are spread out about the center of a distribution.
From left to right, these three distributions have increasing variation.
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