1. Central Tendency & Scale Types

2. Outline Central Tendency Scale Types
Mean
Median
Mode
Scale Types
Nominal
Ordinal
Interval
Ratio
Different statistics for different variables

3. Central Tendency
Statistics – simplify a large set of data to a single (meaningful) number
Central Tendency
One useful kind of summary information
Intuitively: typical, average, normal value
Three statistics for central tendency
Mean
Median
Mode

4. Mean Sum of scores divided by number of scores Sample mean:
IQ: X = [94, 108, 145, 121, 88, 133]
SX = = 689
M = 689/n = 689/6 =
Equal apportionment
If everyone had mean score, total would be the same
Balance point, seesaw analogy (Fig 3.3)
Equal upward and downward distances 88 94
108
121
133
145 M

5. Population Mean Finite population Infinite population
X = [1,1,2,2,2,2,3,3,3,3,3,4,4,4,5,5,6]
x = 1
f(x) = 2
SX = 1*2
x = 2
f(x) = 4
SX = 2*4
x = 3
f(x) = 5
SX = 3*5
Probability
p(x) = fraction of population with value x

6. Mean of Infinite Population
Half of all leprechauns have 1 pot of gold. The other half have 2. Mean?
92% of dogs have 4 legs. 5% have 3 legs. 2% have 2 legs. 1% have 5 legs.
100 dogs.
92 with 4 legs,
5 with 3 legs, 2 with 2 legs,
1 with 5 legs.
4+… = 392
392/100 = 3.92

7. Median Middle value Not average of minimum and maximum
Higher than half the scores, lower than other half
Not average of minimum and maximum
Sorting approach
X = [4,7,5,8,6,2,1,4,3,5,6,8,7,4,3,6,9]
X = [1,2,3,3,4,4,4,5,5,6,6,6,7,7,8,8,9]
Same as 50th percentile

8. Frequency (.5 M households)
Mean vs. Median
Both based on a notion of balance
Mean sensitive to each datum's distance from middle
Median better for irregular distributions
Skew
Outliers
Median
Mean
Mean
Household Income
Frequency (.5 M households)
Median ($46k)
Mean ($63k)
Mean excluding outlier
Mean

9. Mode
Most common value
Peak in the distribution for continuous variables
Simple and insensitive
Most useful when mean, median not definable
College majors, sex, favorite color

10. Scale types We usually use numbers to represent values of variables
Numbers are just a model or analogy for real world
Some properties relevant, some superfluous
Sex
Males = 1; females = 2
Females not twice males
Analogy still limited for more “numerical” variables
Height, reaction time
Can’t multiply together
Numbers have many properties
Which are relevant for a given variable?
Determines what kinds of statistics make sense
Scale of a variable
Summarizes what numerical properties are meaningful
4 types of scales: Nominal, Ordinal, Interval, Ratio

11. Nominal Scale Values are just labels
Sex: {male, female}
Color: {red, green, blue, …}
No structure or relationships between values
Essentially non-numeric
Can use numbers for “coding” but just as placeholders
Red = 1; green = 2; blue = 3
Only mathematical notion is equality (=)
Two scores are equal, or they’re not
Few meaningful statistics
Frequencies: Number of scores of a given value
Mode: Value with greatest frequency

12. Ordinal Scale Values are ordered, but differences aren’t meaningful
Preferences, contest placings, years of education
1st - 2nd  2nd - 3rd
Mathematical notion of greater-than (>, =)
Additional meaningful statistics
Median, quantiles
Range, interquartile range

13. Interval Scale Differences between scores are meaningful
Today 4° warmer than yesterday
Ratios of scores not meaningful
2° not twice as hot as 1°
No real zero point
E.g. Fahrenheit vs. Celcius; IQ
Mathematical notion of subtraction (–, >, =)
Additional meaningful statistics
Mean
Variance, standard deviation

14. Ratio Scale Zero is meaningful Ratios between scores make sense
Weight, time, etc.
Ratios between scores make sense
Twice as heavy, twice as long
Mathematical notion of division (/, –, >, =)
No notable new statistics

15. Meaningful Operations
Summary of Scale Types
Scale
Meaningful Operations
Mode
Median
Mean
Nominal =
Ordinal
> =
Interval
– > =
Ratio
/ – > =