Today’s Lecture

Descriptive Statistics:

Measures of Central Tendency

And

Measures of Variability

Studies show that the more beautiful faces are the more average faces.

So variability is ugly…

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).

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

Formula for the Mean

Formula for the Mean

Formula for the Mean

Formula for the Mean

iX

What if we had…

Answer:

Formula for the Mean

iX

Formula for the Mean

iX

Other Measures of Central Tendency

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

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

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.

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:

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:

Dependent Variable: Number of LegsDependent Variable: Number of LegsWhat’s the mode?What’s the mode?

Mode?F

requ

ency

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.

Misleading Properties of the Mean

• What is the mean income of Survivor participants?

• What would the distribution look like?

Mean = $1,506, 000

Misleading Properties of the Mean

Median = $50,000

Misleading Properties of the Mean

Mode = $40,000

Positive Skew: Mean > Median > ModeNegative Skew: Mean < Median < Mode

Variability(Quantitative Variables)

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.

Demo 4…Again