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Math Help - Shifting data and its affect on z-scores

  1. #1
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    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 (50th 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 50th percentile value =13 (ie: old 50th 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 50th percentile value =30 (ie: old 50th 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!
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  2. #2
    Senior Member MaxJasper's Avatar
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    Lightbulb 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 z=\frac{\bar{x}-\mu }{\sigma } then your population mean \mu is as before while \bar{x} is shifted.
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