Service Forester's Handbook

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====== Statistics for Prescription ======
Statistics are numerical data about a subject or a name applied to
analysis of the reliability of statistical information by sampling.
\\
The person who analyzes the forest land statistics commonly needs 
only 6 items - Mean, Standard Deviation, Standard Error, Limit of Error,
Coefficient of Variation, and Number of Plots to Sample a Stand to a Desired Accuracy.
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====Mean- M or Average==== 
A total of all sample counts divided by the number of samples.\\
M = (sum of sample counts)/(number of samples)
\\
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====Standard Deviation- SD==== 
A statistician's measure of the spread between the individual
sample from the mean. The SD is just a tool in calculating error, CV and N.\\
SD = √[(∑(Dev)<sup>2</sup>)/(n-1)]
\\
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====Standard Error- SE==== 
A statistician's stepping stone toward determination of LE.\\
SE = (SD)/√N)
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====Limit of Error- LE==== 
We cannot calculate the actual volume and cannot know the exact error.
We calculate a LE or a selected probability. We might say "using gambling odds of 67 out of 100,
(2 out of 3), we have a mean volume of 20 cords per acre, plus or minus 2 cords." The mean is 20,
the Standard Error is ±2, and the Limit of Error is .10 or 10%. Increasing the gambling odds to 
95 out of 100 doubles the LE. (20%). For odds of 99 out of 100 the LE triples (30%). \\
LE = (SE)/(M)
\\
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====Coefficient of Variation- CV==== 
This describes the relative uniformity of he stand. The CV is 
our most important statistical term. For planted stands the CV may vary .10 to .30 (10%-30%). 
Natural stands may vary from about .30 to over 1.00. Different stands with the same CV require 
the same number of samples to produce means of equal accuracy, regardless of the areas of these
stands. The CV comes from the SD and the calculated average or mean.\\
CV = SD/M
\\
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====Plots Needed to Sample a Stand to a Desired Accuracy==== 
The number of samples (N) has a basic
relationship with CV, LE, and the probability.\\
N = (t)<sup>2</sup>(CV)<sup>2</sup>/(LE)<sup>2</sup>\\
"t" is the probability or gambling odds. Use "l" or "t" for one SD where values involved in the 
cruise are low and when we can be satisfied with a 2 times out of 3 chance of arriving at a 
figure that will vary within one SD plus or minus from the Mean.\\
\\
Use "2" for "t" for more accuracy. This variation of 2 SD's plus or minus makes gambling odds of
95 in 100 that we'll not exceed 2 Standard Deviations. This will take 4 times as many samples as
for 1 SD. Here's and example:\\
\\
--To make a cruise of ±10% LE with a gambling chance of ± 2 SD (95 times in 100) CV = 30%.\\
\\
Use: \\
N = (2<sup>2</sup>×.30<sup>2</sup>)/(.10<sup>2</sup>) = (4(.09))/(.01) = 36 plots\\
For 1 SD, (t)<sup>2</sup> = 1 and N would be 9 plots.\\
For 3 SD, (t)<sup>2</sup> = 9 and N would be 81 plots.\\
If CV were 15%, N would be one-quarter as many.\\
If SD were 1 and LE were .20, N would be 9 plots.\\
\\
Example: Tree counts on 9 sampling points--
^point no. ^tree count ^Deviation from mean ^(dev)^2    |
|1         |3          |2                   |4          |
|2         |3          |2                   |4          |
|3         |10         |5                   |25         |
|4         |5          |0                   |0          |
|5         |5          |0                   |0          |
|6         |4          |1                   |1          |
|7         |3          |2                   |4          |
|8         |5          |0                   |0          |
|9         |8          |3                   |9          |
^n = 9     ^9/46       ^                    ^∑(D)^2 = 47|
M = 5.1 (call it 5)\\
SD(Std. Dev) = √(∑(D)<sup>2</sup>)/(n-1) = √(47/8) = 2.42\\
\\
SE(std. Error) = SD/√n = 2.42/√9 = .81 \\
\\
LE (Limit of Error) = SE/M = .081/5 = .16 = 16%\\
\\
CV (Coef. of Var.) = SD/M = 2.42/5 = .48 = 48%\\
\\
N (No of plots): \\
N (For 1 SD) = (1)<sup>2</sup>(CV)<sup>2</sup>/(% Acc)<sup>2</sup> = 
(.48)<sup>2</sup>/(.10)<sup>2</sup> = 23 plots\\
N (For 2 SD) = (2)<sup>2</sup>(CV)<sup>2</sup>/(% Acc)<sup>2</sup> =
(4×.23)<sup>2</sup>/(.01)<sup>2</sup> = 92 plots\\
N (For 1 SD ±20%)= (CV)<sup>2</sup>/(% Acc)<sup>2</sup> = .23/.04 = 6 plots.

===No of Samples to be Taken from Infinite Population = n===
^CV%    ^Specified % Limit - SE        ^^^|
^       ^± 1-1 1/2% ^±5%  ^±10%  ^±20%    |
^       ^N                             ^^^|
|10     |45         |4    |1     |1       |
|20     |178        |16   |4     |1       |
|30     |400        |36   |9     |3       |
|40     |712        |64   |16    |4       |
|50     |1,112      |100  |25    |7       |
|60     |1,600      |144  |36    |9       |
|70     |2,178      |196  |49    |13      |
|80     |2,845      |256  |64    |16      |
|90     |3,600      |324  |81    |21      |
|100    |4,445      |400  |100   |25      |
|150    |10,000     |900  |225   |57      |
Service Forester's Handbook

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