Question 86018
First sort the numbers into ascending order (from least to greatest):

 22.5, 22.6, 23.0, 23.1,23.8, 23.8, 24.0, 24.2


a)
To find the mean, add up all of the numbers and divide the sum by the number of numbers (which in this case is 8).

{{{Mean=( 22.5+ 22.6+ 23.0+ 23.1+23.8+ 23.8+ 24.0+ 24.2)/8=23.375}}}


b)
To find the median, count off 4 spaces until you hit the middle number. Since the middle is in between 2 numbers, add them up and divide that sum by 2.

{{{Median=( 23.1+23.8)/2=23.45}}}



c)
To find the mode, simply look for any repeating numbers. If there is one number that occurs more frequently than any other number, then that number is the mode. Since  23.8 repeats itself 2 times, the mode is  23.8



d)
To find the range, subtract the largest number (in this case  24.2) and the smallest number (in this case  22.5) like this:



{{{Range= 24.2- 22.5=1.7}}}



e)
Use this formula to find the standard deviation:


*[Tex \LARGE Standard Deviation: \sigma=\sqrt{ \frac{1}{N}\displaystyle\sum_{i=0}^N (x_i-\bar{x})^2}]

where *[Tex \LARGE \bar{x}] is the average, *[Tex \LARGE x_i] is the ith number, and *[Tex \LARGE N] is the number of numbers


So we can replace N with 8


*[Tex \LARGE\sqrt{ \frac{1}{8}\displaystyle\sum_{i=0}^8 (x_i-\bar{x})^2}]


Replace  *[Tex \LARGE \bar{x}] with 23.375


*[Tex \LARGE\sqrt{ \frac{1}{8}\displaystyle\sum_{i=0}^8 (x_i-23.375)^2}]


Expand the summation (replace each {{{x[i]}}} with the respective number)



{{{sqrt((1/8)(( 22.5-23.375)^2+( 22.6-23.375)^2+( 23.0-23.375)^2+( 23.1-23.375)^2+(23.8-23.375)^2+( 23.8-23.375)^2+( 24.0-23.375)^2+( 24.2-23.375)^2))}}}


Subtract the terms in the parenthesis


{{{sqrt((1/8)((-0.875)^2+(-0.775)^2+(-0.375)^2+(-0.275)^2+(0.425)^2+(0.425)^2+(0.625)^2+(0.825)^2))}}}


Square each term


{{{sqrt((1/8)(0.765625+0.600625+0.140625+0.075625+0.180625+0.180625+0.390625+0.680625))}}}


Add up all of the terms


{{{sqrt((1/8)3.015)}}}


Multiply


{{{sqrt(0.376875)}}}


Take the square root


{{{0.613901457890434}}}


So the standard deviation is {{{0.613901457890434}}}