Shifting data and its affect on z-scores

I have a problem with ‘shifting data’ and how if affects certain measures.

Lets take 5 data values: 1, 2, 3, 4, 5.

An example of a measure of centre would be the mean (=3)

An example of a measure of variation would be the range (=4)

A example of a measure of position would be a percentile (50^{th} percentile=3)

__ADDING A CONSTANT TO ALL DATA VALUES__

Lets ‘shift’ the data by adding 10 to each data value.

Our new data values are: 11, 12, 13, 14, 15.

The 'rules of maths' in my notes state that:

Measures of centre should increase by 10.

Measures of variation should remin unchanged.

Measures of position should increase by 10.

Our new mean is 13 (ie: old mean plus the 10)

Our new range is 4 (ie: same as the old range)

Our new 50^{th} percentile value =13 (ie: old 50^{th} percentile plus 10)

__MULTIPLYING ALL DATA VALUES BY A CONSTANT__

What if we had ‘shifted’ the data by multiplying each data value by 10.

Our new data values are: 10, 20, 30, 40, 50.

The 'rules of maths' in my notes state that:

Measures of centre should increase by a multiple 10.

Measures of variation should increase by a multiple of 10.

Measures of position should increase by a multiple of 10.

Our new mean is 30 (ie: old mean by 10)

Our new range is 40 (ie: old range by 10)

Our new 50^{th} percentile value =30 (ie: old 50^{th} percentile by 10)

I UNDERSTAND ALL THIS. MY PROBLEM IS WITH Z-VALUES.

Lets take the z-value of the number 4 in our original data set. The z-value is found by subtracting the mean and then dividing by the standard deviation.

ie: (4 – 3)/ 1.58114

= 0.63245506406

__ADDING__

Now, as before if I ‘shift’ all the data by adding 10.

The new z-value would be:

(14 – 13)/ 1.58114

= 0.63245506406

ie: the same as the old z-value.

But the 'rules of maths' in my notes state that it should be increased by 10?

__MULTIPLYING__

Now, as before if I ‘shift’ all the data by multiplying by 10.

The new z-value would be:

(40 – 30)/ 15.8114

= 0.63245506406

ie: the same as the old z-value.

But the 'rules of maths' in my notes state that it should be increased by a multiple of 10?

AM I DOING SOMETHING WRONG HERE?

Im so confused!

Many thanks guys.

ps: this is my first post, i hope i have explained myself clearly enough!

Re: Shifting data and its affect on z-scores

I suspect you only shift your sample values but not your population values so your population mean stays as before and if we define normal distribution's z as $\displaystyle z=\frac{\bar{x}-\mu }{\sigma }$ then your population mean $\displaystyle \mu$ is as before while $\displaystyle \bar{x}$ is shifted.