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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