# Formula for Average Percentage Increase/Decrease from chronological data range.

Let's say I have the following data:

$$1,2,3,1,4,0,6,4,3,2,46,7,5,7$$

I'd like to get a percentage based off some sort of average change of the data (in the order) to get an idea of what % the data appears to be increasing/decreasing on average.

The data above is obviously increasing from start to finish (with a few random changes). But what is the average increase in a percentage format?

My first thought was to take the difference between each consecutive number pair (ie: $2-1 = 1, 3-2 = 1, 1-3 = -2,$ etc), then taking the averages of all these difference values. This might get what I want. But I was wondering if there was some more clever formulas that might do this better.

Cheers!

• Percent change suggests you are looking at multiplicative changes, but no multiplier of $0$ yields $6$. Additive changes are fine of course. Also, I assume you omitted a comma in what reads "$46$". – lulu May 12 '17 at 13:04
• @lulu I've thrown in a large number to demonstrate a 'spike' in values which might skew the data slightly. – David May 12 '17 at 13:05
• Ok. Of course with such a short series it has a very strong effect. – lulu May 12 '17 at 13:08
• Are you looking for some sort of exponential growth fit? For example, you go from 1 to 7 in 13 intervals. So you would have 1*(1+p)^13=7, and then 100*p would be your percentage growth rate per interval. There are more sophisticated approaches if this is what you were aiming at. – Samadin May 12 '17 at 13:16
• @Samadin Not sure if I am looking for exponential or not. I'd simply like a final number which represents the overall average percentage of change from in the data set. Thus giving me an idea of 1) if the data is increasing or decreasing and 2) on average, by how much (% format). It's been a really long time since i've done any sort of sin/cos/tan\ ;) – David May 12 '17 at 13:20

$100*$(No of elements in the highest increasing chain)$/$No of total elements is a good measure of percentage of increasing trend in a particular data.
By the largest increasing chain of the sequence $a_1,a_2,a_3,....a_n$, what I meant is a subsequence with largest cardinality which is increasing, that is a subsequence $a_{k_1},a_{k_2},....a_{k_m}$ such that $k_1<k_2<....<k_m$ and $a_{k_1}<a_{k_2},....<a_{k_m}$