z-Transform
Sometimes one has the problem to make two samples comparable, i.e. to compare measured values of a sample with respect to their (relative) position in the distribution. An often used aid is the z-transform which converts the values of a sample into z-scores:
Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history. Next: Z-Transform of Typical Signals Up: ZTransform Previous: Properties of ROC Properties of Z-Transform. The z-transform has a set of properties in parallel with that of the Fourier transform (and Laplace transform). The difference is that we need to pay special attention to the ROCs. In the following, we always assume.
with
zi ... z-transformed sample observations
xi ... original values of the sample
... sample mean
s ... standard deviation of the sample
The z-transform is also called standardization or auto-scaling. z-Scores become comparable by measuring the observations in multiples of the standard deviation of that sample. The mean of a z-transformed sample is always zero. If the original distribution is a normal one, the z-transformed data belong to a standard normal distribution (μ=0, s=1).
Z Transform Table E^at
The following example demonstrates the effect of the standardization of the data. Assume we have two normal distributions, one with mean of 10.0 and a standard deviation of 30.0 (top left), the other with a mean of 200 and a standard deviation of 20.0 (top right). The standardization of both data sets results in comparable distributions since both z-transformed distributions have a mean of 0.0 and a standard deviation of 1.0 (bottom row).
| Hint: | In some published papers you can read that the z-scores are normally distributed. This is wrong - the z-transform does not change the form of the distribution, it only adjusts the mean and the standard deviation. Pictorially speaking, the distribution is simply shifted along the x axis and expanded or compressed to achieve a zero mean and standard deviation of 1.0. |
z-Transform
Sometimes one has the problem to make two samples comparable, i.e. to compare measured values of a sample with respect to their (relative) position in the distribution. An often used aid is the z-transform which converts the values of a sample into z-scores:
Z Transform Table Pdf
with
Z Transform Table Laplace
Z Transform Table Pdf
zi ... z-transformed sample observations
xi ... original values of the sample
... sample mean
s ... standard deviation of the sample
The z-transform is also called standardization or auto-scaling. z-Scores become comparable by measuring the observations in multiples of the standard deviation of that sample. The mean of a z-transformed sample is always zero. If the original distribution is a normal one, the z-transformed data belong to a standard normal distribution (μ=0, s=1).
Z Transform Pdf
The following example demonstrates the effect of the standardization of the data. Assume we have two normal distributions, one with mean of 10.0 and a standard deviation of 30.0 (top left), the other with a mean of 200 and a standard deviation of 20.0 (top right). The standardization of both data sets results in comparable distributions since both z-transformed distributions have a mean of 0.0 and a standard deviation of 1.0 (bottom row).
Z Transform Table Discrete Time
| Hint: | In some published papers you can read that the z-scores are normally distributed. This is wrong - the z-transform does not change the form of the distribution, it only adjusts the mean and the standard deviation. Pictorially speaking, the distribution is simply shifted along the x axis and expanded or compressed to achieve a zero mean and standard deviation of 1.0. |