The Chebychev's theorem states that for any numerical data set,
1.) at least [tex]\frac{3}{4}[/tex] of the data lie within two standard deviations of the mean;
2.) at least [tex]\frac{8}{9}[/tex] of the data lie within three standard deviations of the mean;
3.) at least [tex]1-\frac{1}{k^2}[/tex] of the data lie within k standard deviations of the mean, where k is any positive whole number that is greater than 1.
Thus, given a
data set with a mean of 150 and a standard Deviation of 15, 75% of the data represent [tex]\frac{3}{4}[/tex] of the data, and according to Chebychev's theorem, at least [tex]\frac{3}{4}[/tex] of the data lie within two standard deviations of the mean.
Thus, 75% of the data will fall within the interval
[tex]150\pm2(15)=150\pm30=(150-30,\ 150+30)=(120,\ 180)[/tex].
Therefore, 75% of the data will
fall within the interval 120 to 180.
