Stats Question Finding Range, Variance, and Standard Deviation

Question

Find the range, variance, and standard deviation. Express answers using appropriate units, such as "minutes."

 

Presidential Longevity: Refer to Data Set 12 in Appendix B and use the numbers of years that U.S. presidents have lived after their first inauguration. 

 

Presidents: Washington J. Adams Jefferson Madison Monroe J. Q. Adams Jackson Van Buren Harrison Polk Taylor Pierce Buchanan Lincoln Grant Hayes Garfield Cleveland Harrison McKinley T. Roosevelt Taft Wilson Harding Coolidge Hoover F. Roosevelt Truman Eisenhower J. Kennedy Johnson Nixon Carter Reagan G. H. W. Bush Clinton G. W. Bush Obama

 

Years: 10 29 26 28 15 23 17 25 0 4 1 16 12 4 17 16 0 24 12 4 18 21 11 2 9 36 12 28 16 3 9 25 23

Answer 1

Range is the difference between the highest and the lowest

Range = 36 - 0 = 36 years

 

variance is given by the formula:

Variance = s^2 = [sum(x^2) - (sum x)^2/N]/(N-1)        where sum(x^2) equals the sum of all the numbers squared

                                                                                  sum(x) equals the sum of all the numbers

                                                                                   N equals the number of numbers

Variance = [10478) - (496)^2/33]/32

Variance = 94.5

 

standard deviation is the square root of variance

sd = 94.5^0.5 = 9.71 years

Answer 2

The range is simply the maximum value minus the minimum value.  In this case, 36 - 0 = 36.

 

The variance follows the formula: Σ(x - μ)²/(n-1).  Basically, take each data point, subtract the mean from it and square the result.  Add up the squared values and then divide by the number of data points divided by 1.  (There is statistics software and apps that allow you to do this quickly, but we'll do it the long way just in case.)

 

The mean of the data was 496/33 ≈ 15.03

 

(10 - 15.03)² = 25.3009

(29 - 15.03)² = 195.1609

(26 - 15.03)² = 120.3409

...

(23 - 15.03)² = 63.5209

 

The sum of the squared values is 3022.9697.

 

Therefore the variance is 3022.9697/32 ≈ 94.4678.  (Note there were 33 values, so we divided by 33-1 or 32)

 

The standard deviation is the square root of the variance.  √94.4678 ≈ 9.72