2.
The second manipulation is to subtract the mean value from each control value. This term is called the difference score. Individual difference scores can be positive or negative and the sum of the difference scores is always zero.
3.
The third manipulation is to square the difference score to make all the terms positive. Next the squared difference scores are summed.
Flight Operations & Line Assistance Getting to Grips with Aircraft Performance Monitoring
CRUISE PERFORMANCE ANALYSIS
4. Finally, the predictable dispersion or standard deviation (σ) can be calculated as follows:
measurement number i
x mean value of all the measurement points
n number of measurement points
2.5.4. Degrees of freedom
The "n-1" term in the above expression represents the degrees of freedom. Loosely interpreted, the term "degrees of freedom" indicates how much freedom or independence there is within a group of numbers. For example, if you were to sum four numbers to get a total, you have the freedom to select any numbers you like. However, if the sum of the four numbers is supposed to be 92, the choice of the first 3 numbers is fairly free (as long as they are low numbers), but the last choice is restricted by the condition that the sum must equal 92. For example, if the first three numbers chosen at random are 28, 18, and 36, these numbers add up to 82, which is 10 short of the goal. For the last number there is no freedom of choice. The number 10 must be selected to make the sum come out to 92. Therefore, the degrees of freedom have been reduced by 1 and only n-1 degrees of freedom remain. In the standard deviation formula, the degrees of freedom are n minus 1 because the mean value of the data has already been calculated (which imposes one condition or restriction on the data set).
2.5.5. Variance
Another statistical term that is related to the distribution is the variance, which is the standard deviation squared (variance = σ2 ). The STANDARD DEVIATION may be either positive or negative in value because it is calculated as a square root, which can be either positive or negative. By squaring the STANDARD DEVIATION, the problem of signs is eliminated. One common application of the variance is its use in the determination whether there is a statistically significant difference in the imprecision between different methods.
In many applications (especially in the APM program), the STANDARD DEVIATION is often preferred because it is expressed in the same units as the data. Using the STANDARD DEVIATION, it is possible to predict the range of control values that should be observed if the method remains stable. The STANDARD DEVIATION is often used to impose "gates" on the expected normal distribution of control values. Additional gates can also be defined thanks to the APM program.
2.5.6. Normal or Gaussian distribution
Traditionally, after the discussion of the mean, standard deviation, degrees of freedom, and variance, the next step is to describe the normal distribution (a frequency polygon) in terms of the standard deviation "gates”.
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