Describing Data with Averages

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Chapter 4

• Describing Data with Averages

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Central Tendency

• Average
• Definition What Average Joe should score.
• The typical observation.
• Central Tendency
• Where the scores tend to be around (aka, their
  center)?

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Central Tendency
Whats the central tendency?
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Central Tendency

• Measures of Central Tendency
• Why create measures?
• Specific rules eliminate fuzziness of human
  judgment.
• Types
• Mode
• Mean
• Arithmetic Mean

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Measures of Central Tendency

• The Mode

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Measures of Central Tendency

• Mode
• Most frequent value.
• Calculation
• 1) Create frequency distribution.
• 2) Report the most frequent type as the median.
• Not the frequency of that type.

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Measures of Central Tendency
Drunkenness Ratings after a 23oz LIT
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What is the mode?
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Measures of Central Tendency

• Bimodal Distributions
• Two Modes
• Two distinct peaks
• Dont have to be equal
• Indicate potential subsets in the distribution

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Measures of Central Tendency
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Measures of Central Tendency
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Measures of Central Tendency

• The Median

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Measures of Central Tendency

• Median
• The Exact Middle
• Score at which the proportion of .50 of the
  distribution is at or below.
• Cumulative Relative Frequency .50
• Score at which the 50 of the distribution is at
  or below.
• 50th Percentile

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Measures of Central Tendency

• Determining the Median
• 1) Order observations from lowest to highest
• Do not group together
• 2) Find the middle position
• (Total Obs. 1)/2
• (n1)/2
• 3) If the position is a whole number, the median
  is the score at that position

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Measures of Central Tendency

• Determining the Median
• 4) If the middle position is not a whole ,
  median is the midway point between the score
  below and above that position.
• (Score below Score above)/2

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Measures of Central Tendency
Drunkenness Ratings after a 23oz LIT
1) Reorder Data
What is the median?
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Measures of Central Tendency
11, 11, 11, 12, 12, 13, 14, 14, 14, 14, 14, 15,
15, 17, 18, 18, 20, 20, 20, 20
2) Find Middle Position (201)/2 21/2 10.5
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Measures of Central Tendency
Hungover Ratings after a 23oz LIT
1) Reorder Data
What is the median?
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Measures of Central Tendency
11, 11, 11, 12, 13, 14, 14, 14, 14, 15, 15, 17,
20, 20, 20, 20
2) Find Middle Position (151)/2 16/2 8
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Measures of Central Tendency

• The Mean

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Measures of Central Tendency

• The Mean
• Arithmetic Mean
• Add up all the scores and divide by the total
  number of scores
• Median vs Mean
• Median Middle Point
• Mean Balance Point
• Considers distance and position

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Measures of Central Tendency
Important Symbols and Terminology
Population
Sample
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Measures of Central Tendency
Formulas
m S(x) / N
? S(x) / n
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? S(x) / n
S(x)
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? S(x) / n
S(x)
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? S(x) / n
S(x)
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Measures of Central Tendency

• Why so many?

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The World According to the Mode

• Duplicates win
• Fine with a low number of possible values
• Great for seeing subsets (Bimodal)
• Problem
• With more precise measure, duplication may not
  happen
• Example Race times measured to the hundreth of a
  second
• 12.25, 12.57, 12.57, 13.32, 14.14, 14.22, 14.23,
  14.25, 14.27, 14.32, 14.34, 14.39, 14.40

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The World According to the Median
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The World According to the Mean
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The Effect of Outliers
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Median
Middle Position
Mean
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The Effect of Outliers
10
Median
Mean
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The Effect of Outliers
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Median
(1012)/2 11
Mean
balanced
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Why use each?

• Median
• Good for skewed data
• Average Salary
• Why use the mean?
• Mathematical
• Predictable Changes
• Stable-izes with a high n

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The World According to the Mean
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The World According to the Mean
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The Mean

• A Look Under the Hood

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How the Mean Works
A Deviation (x - ?)
What does it balance?
S(x - ?) 0
Balances the deviations
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How the Mean Works!
? 10
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Skewness

• How you can tell by the central tendencies!

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The Effect of Outlier
Mean moves farther toward the direction of
skewness
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Median
(1012)/2 11
Mean
balanced
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Outliers Skewness
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Outliers Skewness
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Outliers Skewness

• Normal
• Mode Median Mean
• Mean - Median 0
• Positively Skewed
• Mode lt Median lt Mean
• Mean - Median
• Negatively Skewed
• Mean lt Median lt Mode
• Mean - Median -

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Qualitative Data

• What can you do?

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Qualitative Data

• Nominal Scale
• Mode
• Ordinal Scale
• Mode
• Median

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Qualitative Data
Nominal Scale
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Qualitative Data
Ordinal Scale