02_Data_Treatment - For this simple procedure, we have a...

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For this simple procedure, we have a minimum of five sources of error. Each can occur as a + or - error. Our total error becomes: E T = + E wt1 + E vol + E drain + E wt2 + E density And this was for a simple example.
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Identifying sources of error can help you reduce some sources. You can never eliminate all sources of error. The sources will be random in nature. We must rely on statistical treatment of our data to account for these errors. A character associated with an event - its tendency to take place. To see what we’re talking about, let’s use an example - a 10 sided die. If many people each had an identical die, and each gave it a roll, what would be the expected result? For a single roll, each value is equally likely to come up. What if each person had two dice? 0 1 2 3 4 5 6 7 8 9 frequency 1/10 0 1 2 3 4 5 6 7 8 9 Average of two dice - one roll We can continue this trend, using more dice and a single roll. 0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 5 6 7 8 9 Three dice Four dice As the number of dice approaches infinity, the values become continuous. The curve becomes a normal or Gaussian distribution P(x)
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exp [ - ] (x- μ ) 2 2 ! 2 The distribution can be described by: μ f(x) = 1 2 #! 2 - + < X < ! - standard deviation μ - universal mean An infinite data set is required. These terms can be calculated by: μ = $ x i / N N i = 1 ! 2 = variance = $ (x i - μ ) 2 N i = 1 ! = ! 2 These are for infinite data sets but can be used for data sets where N > 100 and any variation is truly random in nature. -3 ! -2 ! -1 ! 0 +1 ! +2 ! +3 ! 2mg 5mg 8mg 11mg 14mg 17mg 20mg When determining ! , you are assuming a normal distribution and relating your data to ! units. -3 ! -2 ! -1 ! 0 +1 ! +2 ! +3 ! The area under any portion of the curve tells you the probability of an event occurring + 1 ! = 68.3% of the data + 2 ! = 95.5% + 3 ! = 99.7% We can use the normal distribution curve to predict the likelihood of an event occurring. This approach is only valid for large data sets and is useful for things like quality control of mass produced products. In the following examples, we will assume that there is a very large data set and that μ and ! have been tracked. Assuming you know μ and ! for a dataset, you can calculate u (the reduced variable) as: u = ( x - μ ) / ! This is simply converting your test value from your normal units (mg, hours, . ..) to standard deviation.
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Assuming that your data is normally distributed, you can use u to predict the probability of an event occurring. The probability can be found by looking up u on a table - Form A and Form B Which form you use is based on the question to be asked. This form will give you the area under the curve from u to |u| area |u| area 0.0 0.5000 2.0 0.0227 0.2 0.4207 2.2 0.0139 0.4 0.3446 2.4 0.0082 0.6 0.2743 2.6 0.0047 0.8 0.2119 2.8 0.0026 1.0 0.1587 3.0 1.3 x 10 -3 1.2 0.1151 4.0 3.2 x 10 -5 1.4 0.0808 6.0 9.9 x 10 -10 1.6 0.0548 8.0 6.2 x 10 -16 1.8 0.0359 10.0 7.6 x 10 -24 This form will give you the area under the curve from 0 to u
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02_Data_Treatment - For this simple procedure, we have a...

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