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: 2, 3, 4, 3, 4, 1, 4 = 3

The average deviation will always be zero!

y

7

)32( 7

)33( 7

)34( 7

)34( 7

)33( 7

)34( 7

)31(

0)(

n

yy

n

yy )( Read: The sum of - y minus the mean of y, divided by n

example data set:

n

yy )(=

= 0

Variance:

Standard Deviation:

These are here defined as descriptive statistics.

average of the summed, squared-deviations about the mean

n

yysy

2

2

the square root of the average of the summed squared deviations about the mean

n

yysy

2

As inferential statistics

1

2

2

n

yysy

1

2

n

yysy

See the difference

Influence of extreme scores on variance.

Note:d = difference score the difference between a given score and the mean.

Y: 1, 2, 19, 5, 8, 7y 4 2

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!

d 2

1

22

2

n

d

n

yysy

7y