Scaling average and standard deviation

Question

I want to "scale" some numbers and associated standard deviation.

I am analyzing the data of content in urine for x amount of intake of a certain chemical:

time(d)    value    sd
1         0.5      0.25
7         0.2      0.12

If I want to scale the data for an intake of magnitude x/3, I can divide the values by 3 (assuming the content in urine is linear in intake), but how would I scale the sd? Would the values also be divided by 3?

Answer 1

Yes. If you scale data $X$ by dividing it with $a$ then clearly, mean and standard deviation scale with $a$: $$ \overline{\Bigg(\frac{X}{a}\Bigg)}=\frac{\overline X}{a}\,,\quad\sqrt{\overline{\Bigg(\frac{X}{a}-\frac{\overline{X}}{a}\Bigg)^2}} =\frac{1}{a}\sqrt{\overline{(X-\overline{X})^2}}\,. $$