Transcript Measures of Spread

Measures of Spread
1. Range: the distance from the lowest to the highest score
* Problem of clustering differences
** Problem of outliers
2. Interquartile Range
*
*
*
*
omits the upper and lower 25% of scores
eliminates the effect of extreme scores
trimmed samples
loss of information
Data Set I:
8, 8, 9, 10, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14,14, 15, 15, 16, 17
Range = 9
Interquartile Range = 3
Data Set II:
1, 2, 3, 10, 11, 11, 12, 12, 12, 12, 12, 13, 13, 13, 14,14, 15, 21, 25, 30
Range = 29
Interquartile Range = 3
Average Deviations:
 ( y  y)
Read: The sum of - y minus the mean of y,
divided by n
n
example data set:
y =3
y: 2, 3, 4, 3, 4, 1, 4
 ( y  y)
n
(2  3)
=
7
=

(3  3)
7

(4  3)
7

(3  3)
7
 (4  3)  (1  3) 
7
0
The average deviation will always be zero!
 ( y  y)  0
n
7
(4  3)
7
Variance:
average of the summed, squared-deviations about the
mean
y  y


2
s
Standard
Deviation:
2
y
n
the square root of the average of the
summed squared deviations about the mean
 y  y
2
sy 
n
These are here defined as descriptive statistics.
As inferential statistics
s y2 
2


y

y

n 1
See the difference
 y  y
2
sy 
n 1
Influence of extreme scores on variance.
2
2


y

y
d


2
s 

y
n
Y: 1, 2, 19, 5, 8, 7
Note:
d = difference score
the difference between a given
score and the mean.
n 1
 y  42
y7
d2 
A score of 7 (d squared = 0) contributes no units to the variance.
A score of 5 contributes 4 units to the variance.
A score of 2 contributes 25 units to the variance.
A score of 19 contributes 144 units to the variance.
Extreme scores contribute disproportionately more.
Watch out for OUTLIERS!