Quantifying Noise
- Page ID
- 77466
All data contains some level of uncertainty due to random fluctuations in the measurement process. We will focus on describing random fluctuations that may be described mathematically using a Gaussian distribution shown below.
In this relationship:
- y is the frequency that a value x will occur
- μ is the population mean
- σ is the standard deviation of the population
\[\mathrm{y = \dfrac{exp\left\{\dfrac{-(x - μ)^2}{2σ^2}\right\}}{σ \sqrt{2π}}}\]
Of course, there are such a myriad of samples and measurement methods that each case yields a unique distribution with a unique mean and standard deviation.
In order to generally describe the Gaussian distribution, one must represent the Gaussian distribution in a standardized format. This can be done in two steps:
- Mean-Centering
subtracting the population mean from all the members of the data set so that μ = 0
- Normalization
dividing each member of the data set by the distribution standard deviation so that σ = 1
The x-axis is now represented by a unitless quantity, z
\[\mathrm{z = (x-μ)/σ}\]