# Are percentages scalable?

I have data made up of percentages but no idea how they were calculated. I am using this data to demonstrate Benford’s Law and so far it does follow this law!

But one of the properties of sets that follow Benford’s Law is that they are scale-invariable, so multiplying all the elements by a constant will keep the frequency distribution of the leading digits in the data the same. So I want to test scale invariability in my data by multiplying all the percentages by a constant but I am worried this is wrong on my part because percentages may not be scalable.

By scalable, my understanding is that it can be converted to different ‘units of measurement’/scales. For example, feet and inches are scalable to meters.

Could someone clear this up for me, please?

• What do you precisely mean by "invariability of data"? And what do you precisely mean by scalable percentages? Also, please tell more about the data you have. Some numerical example would be useful. Jan 23 '19 at 14:01
• Think of percentages as being "parts per hundred". If you multiply all your percentages by $2$ you are converting "parts per hundred" to "parts per two hundred". If you multiply all your percentages by $10$ you are converting "parts per hundred" to "parts per thousand". This is just the same as multiplying measurements by $12$ to convert units from feet to inches. Jan 23 '19 at 15:18
• @gandalf61 If you multiply percentages by $2$ don't you convert to "part per fifty?" Jan 23 '19 at 15:58
• @saulspatz No - to convert to parts per fifty you would divide the percentage value by $2$. For example $20\%$ = $20$ parts per hundred = $40$ parts per two hundred = $10$ parts per fifty. Jan 23 '19 at 16:44
• @gandalf61 Yes, I must still be asleep. Jan 23 '19 at 16:47