click to edit master title style today’s lecture descriptive statistics: measures of central...
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Today’s Lecture
Descriptive Statistics:
Measures of Central Tendency
And
Measures of Variability
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Studies show that the more beautiful faces are the more average faces.
So variability is ugly…
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The “Average” Score andMeasures of Central Tendency
• A measure of central tendency ‘locates’ the distribution or histogram on the X axis.
• Remember: We are taking the
average value of a variable, so the variable must be quantitative.
• Measure #1 (of 3): The “Mean” is the sum of the scores divided by number of scores in the set (n).
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Summation Notation
(Capital Sigma)
XXXX n
n
ii
... 211
YYYY n
n
ii
... 211
https://www.khanacademy.org/math/trigonometry/seq_induction/geometric-sequence-series/v/sigma-notation-sum
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Formula for the Mean
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Formula for the Mean
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Formula for the Mean
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Formula for the Mean
iX
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What if we had…
Answer:
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Formula for the Mean
iX
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Formula for the Mean
iX
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Other Measures of Central Tendency
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Summarizing a Distribution:“Measures of Central Tendency”
• The Median = score that divides the set of scores into two halves, "smaller" & "larger".
4, 6, 9, 3, 5, 11, 2
Median:
234569
11
"Median Position": .42
)17(
2
)1(
n
1234567
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Summarizing a Distribution:“Measures of Central Tendency”
• The Median = score that divides the set of scores into two halves, "smaller" & "larger".
4, 6, 9, 3, 5, 11, 2
Median:
234569
11
"Median Position": .42
)17(
2
)1(
n
1234567
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Summarizing a Distribution:“Measures of Central Tendency”
The Median = score that divides the set of scores into two halves, "smaller" & "larger".
4, 6, 9, 3, 11, 2
Median:
23469
11
"Median Position": .5.32
)16(
2
)1(
n
123456
So, take the average of the third and fourth scores in the set.
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Summarizing a Distribution:“Measures of Central Tendency”
One number to representthe entire population
The Mode = the mostfrequent score
4, 6, 4, 3, 11, 4, 3
Mode:
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Summarizing a Distribution:“Measures of Central Tendency”
One number to representthe entire population
The Mode = the mostfrequent score
4, 6, 4, 3, 11, 4, 3
Mode:
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Dependent Variable: Number of LegsDependent Variable: Number of LegsWhat’s the mode?What’s the mode?
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Mode?F
requ
ency
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Why 3 Different Measures of Central Tendency?
• Each has it’s advantages and disadvantages.
• The mean or median values might be rare or not even exist (so beautiful faces can be rare).
• The mode might be an ‘extreme’ on the edge of the distribution.
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Misleading Properties of the Mean
• What is the mean income of Survivor participants?
• What would the distribution look like?
Mean = $1,506, 000
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Misleading Properties of the Mean
Median = $50,000
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Misleading Properties of the Mean
Mode = $40,000
Positive Skew: Mean > Median > ModeNegative Skew: Mean < Median < Mode
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Variability(Quantitative Variables)
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Variety: the Spice of Life?Variety: the Spice of Life?
Variability in your population Variability in your population Unreliability in your sample.Unreliability in your sample.
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Demo 4…Again