Here is a collection of mathematical formulas that professional photographers often find useful. there are formulas ranging from simple bellows extension, to standard deviation used for photographic processing quality control.
Bellows extension is used for exposure compensation when a camera that incorporates a bellows is used and the bellows is extended beyond it's typical range. For example if you are using a view camera to photograph a subject up close you would need to rack the front standard forward and further away from the rear standard. This causes the bellows length to increase between the lens and the film plane. When this happens your exposure will change due to light being scattered diffused and dispersed inside the bellows chamber. This will effectively reduce the intensity of the light striking the film plane and consequently decrease your exposure. By using the following formula we can determins the proper exposure to use in such situations.
(Bellows extension in mm) squared Bellows Extension Factor = ________________________________ (Lens focal length in mm) squared Extension Factor 1x 1.4x 2x 2.8x 4x 5.6x 8x 11x 16x 22x 32x 45x 64x 90x 128x Loss in Stops 0 .5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7
After determining the extension factor we look up the loss of F-stops on the the chart given. We then increase our exposure by the number of stops indicated from the chart. Example: If I am shooting with a 150mm lens and my bellows are racked out 12 inches I must convert the 12 inches to millimeters. This comes to 304.79999 mm or 304.8 when rounded we now punch these numbers into our formula. 304.8sq / 150sq =x or 92903.04 / 22500 = 4.129024. This is closes to 4x on the chart. We must therefor increase our exposure by 1.5 stops to achieve a good negative. If my original exposure were 1/125th at f/16 I would need to change my f-stop to f/8.5.
This formula is basically a fancy way of arriving at an average or mean, let me explain. In photographic process control a lab will set it's own control standards. These standards are the limits in which the chemical process must be maintained. These limits are determined by running a series of processing control strips through a film processor with continuous replenishment of the chemistry. These strips are ran at fifteen minute intervals and are read on a machine called a densitometer. There are a series of patches on these strips these patches are the items that are read on the densitometer. These resultant readings are recorded. For the sake of this discussion we are going to concentrate on just one reading for each strip to avoid confusing the issue. In our example we will use the High Density step on the strip. Now we use the following formula:
The first step is to take all of our numbers that we obtained from the strips and determine the mean (or average). Then we subtract that average from each individual number and then square the result This is done for each number. This is the (x-xbar)squared function. Now we add up all the results, this is the capital sigma function in the equation. Divide this result by n or the total number of samples in the data set. and then finally figure the square root of that result. I highly recommend using a good scientific calculator an becoming familiar with the statisitical functions, this is a much quicker, less painful, and more accurate method.
The result of all of this work is the control point of our process. The lab will run a new control strip at the beginning of each processing shift and will read and record that reading on a control chart. The result we just obtained is the center line on that chart or the mean for which we strive. We can then set our upper and lower control limits from this mean. When the chemistry starts to approach this limit we then adjust the process to bring it back into control.
For further information on Photogrphic QC and process control please visit the Processing section of this site and click on the link "The how-to of procesing" then go to the link "C-41 z manual" to download this manual from Kodak. In the last section of this manual (section 5) you will find more information on this whole process.
Standard deviation is not meant for the home darkroom enthusiast. It is used mainly in commercial and custom labs to control their processing. This information is provided here with no guarantees as to it's accuracy. Any loss or damages incurred from the use of this material is the sole responsibilty of the user. the webmaster for this site and all parties involved with this site can in no way be held liable for it use or misuse.