Download Chapter 4

Review
 Mean—arithmetic average, sum of all scores divided by the
number of scores
 Median—balance point of the data, exact middle of the
distribution, 50th percentile
 Mode—highest frequency, can be more than one
 Find the mean, median, mode
Review
Review
 Find the mean, median, mode
 Mean=sum of all scores(∑fX) /number of scores(N)
Review
 Find the mean, median, mode
 Mean=sum of all scores(∑fX) /number of scores(N)
 Median=middle point (N-1/2)th position
Review
 Find the mean, median, mode
 Mean=sum of all scores(∑fX) /number of scores(N)
 Median=middle point (N-1/2)th position
 Mode=greatest f
Major Points
 The general problem
 Range and related statistics
 Deviation scores
 The variance and standard deviation
 Boxplots
 Review questions
The General Problem
 Central tendency only deals with the center
 Dispersion


Variability of the data around something
The spread of the points
 Example: Mice and Music
Mice and Music
 Study by David Merrell
 Raised some mice in quiet environment
 Raised some mice listening to Mozart
 Raised other mice listening to Anthrax
 Dependent variable is the time to run a straight alley maze after 4
weeks.
Results
 Anthrax mice took much longer to run
 Much greater variability in Anthrax group

See following graphs for Anthrax and Mozart
 We often see greater variability with larger mean
 The range


Range and Related Statistics
Distance from lowest to highest score
Too heavily influenced by extremes
 The interquartile range (IQR)



Delete lowest and highest 25% of scores
IQR is range of what remains
May be too little influenced by extremes
Trimmed Samples
 Delete a fixed (usually small) percentage of extreme scores
 Trimmed statistics are statistics computed on trimmed samples.
 Definition


Deviation Scores
distance between a score and a measure of central tendency
usually deviation around the mean
 Importance
Variance
 Definitional formula
 Example

See next slide
Computing the Variance
Computing the Variance
Standard Deviation
 Definitional formula

The square root of the variance
 Computational formula based on algebraic manipulation

Makes it easier to calculate
Computational Formula
T ry o n e
T ry o n e
T ry o n e
T ry o n e
T ry o n e
Estimators
 M ean

Unbiased estimate of population mean ()
Define unbiased

Long range average of statistic is equal to the parameter being estimated.
 Variance

Unbiased estimate of 2
Estimators--cont.

Using

gives biased estimate

Standard deviation
use square root of unbiased estimate.
Merrell’s Music Study SPSS Printout
Treatment Mean
Quiet
307.2319
Mozart
114.5833
Anthrax 1825.8889
Total
755.4601
 The general problem

N
23
24
24
71
WEEK4
Std. Deviation
71.8267
36.1017
103.1392
777.9646
Boxplots
A display that shows dispersion for center and tails of distribution
 Calculational steps (simple solution)




Find median
Find top and bottom 25% points (quartiles)
eliminate top and bottom 2.5% (fences)
Draw boxes to quartiles and whiskers to fences, with remaining points as
outliers
 Boxplots for comparing groups
Combined Merrell Data
Merrell Data by Group
Review Questions
 What do we look for in a measure of dispersion?
 What role do outliers play?
 Why do we say that the variance is a measure of average
variability around the mean?
 Why do we take the square root of the variance to get the
standard deviation?
Review Questions--cont.
 How does a boxplot reveal dispersion?
 What do David Merrell’s data tell us about the effect of music on
m i c e?