How do I find the mean of the data set #{x_1,x_2,....,x_25}# given that #sum_(i=1)^25x_i^2=2568.25# and the standard deviation is #5.2#?

1 Answer
Geoff K.
Aug 4, 2018

The mean is #mu=8.7#.

Explanation:

Standard deviation #sigma# is the square root of the variance #sigma^2#. The formula for population variance is

#sigma^2=(sum(x_i-mu)^2)/N#

If we distribute the square, we get

#sigma^2=(sum(x_i^2-2x_imu+mu^2))/N#

#color(white)(sigma^2)=(sumx_i^2-2musumx_i+Nmu^2)/N#

#color(white)(sigma^2)=(sumx_i^2-2Nmu^2+Nmu^2)/N#

#color(white)(sigma^2)=(sumx_i^2)/N-mu^2#

This gives us an equation for #sigma^2# in terms of #sumx_i^2,# the size of the set #(N),# and the mean #(mu)#.

  • We know #sigma^2=5.2^2 = 27.04#.
  • We know #sumx_i^2=2568.25#.
  • We know #N=25.#
  • We want to find #mu#.

We can now plug in all the known values to solve for the one unknown.

#sigma^2=(sumx_i^2)/N-mu^2#

Solving for #mu^2#:

#mu^2 = (sumx_i^2)/N-sigma^2#

#color(white)(mu^2) = (2568.25)/25-5.2^2#

#color(white)(mu^2) = 102.73-27.04#

#color(white)(mu^2) = 75.69#

#=>mu=sqrt(mu^2)=sqrt(75.69)=8.7#

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