Logarithmic question

[wseyller]

First off, I work in the Laser printer industry. I deal with a standardized testing procedure that compares remanufactured laser cartridges to one another and Oem cartridges as well. One portion of the test deals with density readings and to calculate averages and other various details.

I use a device called a densitometer that gives density readings showing how dark the print is. I take five of these readings in each series and must average them. I also take 4 background readings to see how much toner has attracted to a blank area. Density average must be reported rounded to 2 decimal places and background to 3 decimal places. The standard calls to convert each number to a reflection percentage, average them, then convert back using logarithm. What I would like to know is how much difference does it make in any case to simply average the readings, then going through the trouble of converting using logs.


EXAMPLE

Density Calculations


1. Take density readings with densitometer.



2. Convert each readings to show the reflection percentage.

This formula will be done for each reading.

Reflection = (1 / 10) ^ Density Reading or
Reflection = 0.1 ^ Density Reading

Example
0.0407 = (1 /10) ^ 1.39
0.0398 = (1 /10) ^ 1.40
0.0389 = (1 /10) ^ 1.41
0.0380 = (1 /10) ^ 1.42
0.0371 = (1 /10) ^ 1.43

In percentage form you get the results of %4.07, %3.98, %3.89, %3.80, %3.71
Reflection(Percentage) = Reflection(Decimal) x 100



3. Average the reflection density values.

(%4.07 + %3.98 + %3.89 + %3.80 + %3.71) / 5 = %3.89 Average Reflection Density

This calculation can be performed using the decimal or the percentage form.



4. Convert the average reflection density to the inverse to acheive the average density.

This calculation requires that the Average Reflection Density to be in decimal form.
Reflection(Decimal) = Reflection(Percentage) / 100

Average Density = (base 10)Log (1 / Average Reflection Density)

1.41 = (base 10)Log (1 / 0.0389)



The Average Density is 1.41

 

[CSMR]

Not much difference, because the data are close together, and in the range of the data log is aproximately linear.

 

[wseyller]

Sometimes the values may be a little different.

Some Density measurements may come up to be lets say: 1.35, 1.40, 1.46, 1.50, 1.59

If averaging using logs you get 1.45, but you get 1.46 if you just average the readings.

So are you saying that if the deviation of the numbers are greater the difference of the method of using logs or not will be greater.

 

[CSMR]

You are comparing average(density) with (log base .1)(average(.1^density)).
Since .1^ is convex, average(.1^density))>.1^average(density), and since log base .1 is decreasing,
average(density)>(log base .1)(average(.1^density)). As you found.
Since densities are small and e^ is almost linear in that range the difference will be small - smaller than 0.01 I should think - perhaps rounding gave a bigger difference than the actual one.

Edit: corrected.