Question 117072
First sort the numbers into ascending order (from least to greatest):

75,77,78,78,79,80,80,81,84,85,87,89,90,91


Now we need the mean. So add up all of the numbers and divide the sum by the number of numbers (which in this case is 14).

{{{Mean=(75+77+78+78+79+80+80+81+84+85+87+89+90+91)/14=1154/14=82.4285714285714}}}


So the mean is 82.4285714285714





Now that we have the mean, we can use this formula to find the standard deviation:


Standard Deviation:*[Tex \LARGE  \sigma=\sqrt{ \frac{1}{N}\displaystyle\sum_{i=0}^N (x_i-\bar{x})^2}] where *[Tex \LARGE \bar{x}] is the arithmetic mean, *[Tex \LARGE x_i] is the ith number, and *[Tex \LARGE N] is the number of numbers


So we can replace N with 14


*[Tex \LARGE\sqrt{ \frac{1}{14}\displaystyle\sum_{i=0}^{14} (x_i-\bar{x})^2}]


Replace  *[Tex \LARGE \bar{x}] with 82.4285714285714


*[Tex \LARGE\sqrt{ \frac{1}{14}\displaystyle\sum_{i=0}^{14} (x_i-82.4285714285714)^2}]


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



{{{sqrt((1/14)((75-82.4285714285714)^2+(77-82.4285714285714)^2+(78-82.4285714285714)^2+(78-82.4285714285714)^2+(79-82.4285714285714)^2+(80-82.4285714285714)^2+(80-82.4285714285714)^2+(81-82.4285714285714)^2+(84-82.4285714285714)^2+(85-82.4285714285714)^2+(87-82.4285714285714)^2+(89-82.4285714285714)^2+(90-82.4285714285714)^2+(91-82.4285714285714)^2))}}}


Subtract the terms in the parenthesis


{{{sqrt((1/14)((-7.429)^2+(-5.429)^2+(-4.429)^2+(-4.429)^2+(-3.429)^2+(-2.429)^2+(-2.429)^2+(-1.429)^2+(1.571)^2+(2.571)^2+(4.571)^2+(6.571)^2+(7.571)^2+(8.571)^2))}}}


Square each term


{{{sqrt((1/14)(55.190041+29.474041+19.616041+19.616041+11.758041+5.900041+5.900041+2.042041+2.468041+6.610041+20.894041+43.178041+57.320041+73.462041))}}}


Add up all of the terms


{{{sqrt((1/14)353.428574)}}}


Multiply


{{{sqrt(25.2448981428571)}}}


Take the square root


{{{5.02443013115489}}}


So the standard deviation is {{{5.02443013115489}}}


Here is an <a href=http://www.csgnetwork.com/stddeviationcalc.html>online standard deviation calculator</a> that will check your work.