26 March 2020

Determining the growth rate of a published chart

A friend drew my attention to this page comparing the Bay Area COVID-19 cases to New York. But it can be hard to judge visually how steep a log chart is.

To calculate the growth rates, for example of New York on the following chart:

  1. Take a screen capture
  2. Paste it into MSPaint, or similar
  3. Drag the selection box from one log-axis marker to the next.
  4. Use MSPaint's ruler indicator (bottom left corner) to see how high the selction is.
    1. E.g. in the following example, measure from the 1,000 grid line to the 10,000 grid line.
    2. The distance is 198 pixels for a 10x multiplication 
    3. Let's call this PixelsPerTenfold
  5. Pick two date lines around the slope being measured
    1. Start dragging the selection box from the point where the data line intersects the first date line.
    2. Drag to a point where the data line intersections the last date line.
    3. Use the ruler in the corner to measure the vertical distance.
    4. Call this PixelsIncreased
    5. In the example below it is 240 pixels.
  6. Apply calculations as follows:
PixelsIncreased  = 240
PixelsPerTenfold  = 198
Days = 3
Daily Multiplier = 10(PixelsIncreased / PixelsPerTenfold / Days) = 2.54
Daily % Increase = 100 * (DailyMultiplier-1) = 154%
Days to Double = loge2 / logeDailyMultiplier = 0.75 days
Days to 10x = loge10 / logeDailyMultiplier = 2.47 days

Exponential growth - Year 10 maths revision

"When will I ever need to use logarithms, or raising something to the the power of 1/n in real life". During Covid-19, it turns out. Since no one likes logarithms, I thought I'd just post a few formulas. The following should work on any system of exponential growth.

(Naturally, for Excel formulas, substitute in the value or cell reference for X, Y, T, etc)

FormulaExcel FormulaExample
Initial CasesX112  Aus # Mar10
Final CasesY2,431  Aus # Mar25
TimeT15  days
Multiplier over durationM = Y/X=Y/X21.71
Daily MultiplierMD = M1/T=(Y/X)^(1/T)1.23
Daily % Increase    =(MD -1) * 100=((Y/X)^(1/T)-1)22.8%  Format as %
Time to doubleT2X = loge2 / logeMD=LN(2)/LN((Y/X)^(1/T))3.38  days
Time to 10xT10X = loge10 / logeMD=(Y/X)^(1/T)/LN(10)11.22  days
Convert from 'time to double' to 'time to 10x'
T10X = T2X * loge10 / loge2=TDOUBLE * LN(10)/LN(2)
Convert from 'time to 10x' to 'time to double'
T2X =T10X * loge2 / loge10=TTEN * LN(2)/LN(10)
Convert from 'time to double' to 'daily multiplier'
MD = elog2 / T2X=EXP( LN(2) / TDOUBLE)
Apply a daily multiplier for N days