Compare and contrast the following measures of dispersion (range, standard deviation) in terms of what each of these represent and in terms of the relative adva

Question

Compare and contrast the following measures of dispersion (range, standard deviation) in terms of what each of these represent and in terms of the relative advantages/disadvantage of using one over the other.

Answer 1

Range = largest statistic - smallest statistic.

 

Standard Deviation = square root ( variance) / (n-1)

    where n is the number of statistics and the variance 

    is equal to the total of the square of the varirances

    where each variance is the statistic minus the mean/average

 

They both give an idea of how wide-spread the data is.

 

The problem with the range it is very sensitive to errant

statistics and outliers.

 

The standard deviation allows you to divide the statistics

into levels or layers, like in a grading scale.

Students with scores within one-half standard deviation from

the mean-average score a C. Students scoring within 1/2 and 

1.5 standard deviations above the mean score a B, while students

scoring 1/2 to -1.5 standard deviations below the mean-average

score a D. Students scoring 1.5 standard deviations above the mean

receive a grade or A, while those scoring below -1.5 standard 

deviations below the mean-average receive a grade of F.

The teacher of course has the power to SCALE the grades, or

sometimes called CURVE them by expanding the range for

a grade of C to -1.5 to 1.5 standard deviations above the mean,

1.5 to 2.5 standard deviations above the mean for a grade of B,

and -1.5 to -2.5 standard deviations below the mean for 

a grade of D. 

 

The problem with the standard deviation, though more precise,

is that it depends on the mean-average, in order to have any

meanigful value.

 

For example two classes taking the same exam and averages

of 65.5 and 77.7 respectively. Is the second class smarter?

Absolutely not. The first class has a standard deviation of 5.7

while the second has a standard deviation of 10. So the grades

are more distributed in the second class than the first.