Sample mean and linear relation?

[jwxie]

Homework Statement

The book states the following:

The sample mean is defined by: $$\bar{x}=\sum_{i=1}^{n}x_{i}/n$$

The computation of the sample mean can often be simplified by noting that if for constants
a and b, $$y_{i}=ax_{i}+b$$, then the sample mean of the data set y1 , . . . , yn is: $$\bar{y}=\sum_{i=1}^{n}(ax_{i}+b)/n=\sum_{i=1}^{n}ax_{i}/n+\sum_{i=1}^{n}b/n=a\bar{x}+b$$


Given the question:

The winning scores in the U.S. Masters golf tournament in the years from
1982 to 1991 were as follows: 284, 280, 277, 282, 279, 285, 281, 283, 278, 277

The book computes as follows:

Rather than directly adding these values, it is easier to first subtract 280 from
each one to obtain the new values yi = xi − 280, we obtain:

4, 0, −3, 2, −1, 5, 1, 3, −2, −3

ecause the arithmetic average of the transformed data set is
y_bar = 6/10
̄
it follows that
x_bar = y_bar + 280 = 280.6

(1) Why are we choosing 280?? What is the reason for that? I tried other numbers don't they don't give the same sample mean.

(2) I also want to confirm my understanding of the linear relation given the summation. I know the constant a should be the 1/n, as x1 / n + x2 / n + xn / n is the same as (x1+x2+xn)/n. Why do we need the b, the intercept? Is it just stating the obvious ,a general form? Or is it possible that we will encounter a statistical sample mean that cross the y?

But what I just said don't make sense to "yi = xi − 280". the constant a is 1...Can someone correct me? Thanks!

 

[jashua]

Let's choose 20 instead of 280. Then, new data set becomes

264 260 257 262 259 265 261 263 258 257

hence, y_bar = 260.6. It follows that x_bar = 20 + 260.6 = 280.6

y = ax + b is the general form of the linear relation between x and y.

 

[the_amateur]

The book states that if the mean $$\bar{x}=\sum_{i=1}^{n}x_{i}/n$$

Then for another distribution which is related as $$y_{i}=ax_{i}+b$$

the mean is $$\bar{y}=a\bar{x}+b$$

what they are trying to imply here is that by properly choosing 'a' and 'b' the process of finding mean can be simplified.

in the example solution you mentioned

a = 1, b = -280 (the values can be any real number, but it is chosen in such a way to make the problem simpler)

so the equation becomes y$_{i}$ = x$_{i}$ - 280

this effectively converts the problem into a much simpler problem involving small numbers. Once we compute $$\bar{y}$$
we have to get back
$$\bar{x}$$
This is done by solving $$\bar{y}=a\bar{x}+b$$
( $$\bar{y}$$ , a ,b are known)

This gives the answer.