CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 7. Sampling Distributions and Estimations

μx=∑xipi

μx= 0 ( 0. 04 )+ 3 ( 0. 30 )+ 8 ( 0. 29 )+ 15. 5 ( 0. 17 )+ 35. 5 ( 0. 15 )+ 55 ( 0. 05 )

μx= 13. 93

And

σ^2 x=∑(xi−μx)^2 pi

= ( 0 − 13. 93 )^2 ( 0. 04 )+( 3 − 13. 93 )^2 ( 0. 30 )

+( 8 − 13. 93 )^2 ( 0. 29 )+( 15. 5 − 13. 93 )^2 ( 0. 17 )

+( 35. 5 − 13. 93 )^2 ( 0. 15 )+( 55 − 13. 93 )^2 ( 0. 05 )

≈ 208 .3451 andσx≈ 14. 43

The expected number of fast food meals purchased by a student at the local university is 13.93. This number should
not be rounded since the mean does not have to be one of the values in the distribution. You should also notice that
the standard deviation is very close to the expected value. This means that the distribution will be skewed to the right
and have long tails toward the larger numbers.

Notice that ̄x= 13 .93 andσx= 14 .43.

Linear Transformations of X on Mean of x and Standard Deviation of x

If you add the same number to all values of a data set, the shape or standard deviation of the data remains the same
but the value is added to the mean. This is referred to as recentering the data set. Likewise, if you rescale the data –
multiply all data values by the same nonzero number- the basic shape will not change but the mean and the standard
deviation will each be a multiple of this number. The standard deviation must be multiplied by the absolute value of
the number. If you multiply the mean and the standard deviation by a constantdand then add a constantc, then the
mean and the standard deviation of the transformed values are expressed as:

μc+dx=c+dμx

σc+dx=|d|σx