I think this is an easy one but I'm unfortunately a bit stumped. In short, why are standard deviations of actual values different than the standard deviations of those values expressed as a percentage of a total?
Say I sell 30 total widgets on day 1 of three different types (A-20, B-7, C-3). Day 2 sells 30 again, and the breakdown is (A-24, B-2, C-4).
For widget A, the 2 day data set calculates a mean of 22.0 and standard deviation of 2.8. For widget B, the 2 day data set calculates a mean of 4.5 and standard deviation of 3.5. For widget C, the 2 day data set calculates a mean of 3.5 and standard deviation of 0.7.
A new day of widgets sold comes in. Day 3 also sells a total of 30 widgets , and the breakdown is (A-27, B-2, C-1). When I calculate Day 3's sales standard deviations from the mean I get (27-22.0)/2.8 = 1.77. I apply the same math for B and C to calculate their respective day 3 sales in terms of standard deviations from the mean.
Now when I convert widget A's sales to percentages of the total widgets sold for each day (day 1 = 20/30 = 66.67%, day 2 = 24/80 = 80%, and day 3 = 27/30 = 90%), and calculate Day 3's mean of 73.3% and sd of 9.43%, I'm able to calculate Day's 3 standard deviations from the mean (90%-73.3%) / 9.43% = 1.77.
Good stuff - it ties out! The below is the actual part I'm stumped on.
Say we change day 2 to reflect 25 total widgets sold, with A selling 24 widgets. The SD from the mean stands at 1.77 but this certainly changes the percentages. The previous 66.7%, 80%, and 90% breakdown with mean of 73.3% now changes to 66.7%, 96% (=24/25), and 90% with mean of 81.33%. The previous SD of 9.43% changes to 20.74%. New SD from mean for percentages is at 0.42 = (90%-81.3%)/20.74%.
Why isn't it still 1.77?