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What is the benefit of transforming standard normal distributions to conform to the standard distribution?
There are no benefits in doing something that cannot be done. The standard normal distribution is not transformed to the standard distribution because the latter does not exist.
Do some normal probability distributions have different means and different standard deviations?
Yes. Normal (or Gaussian) distribution are parametric distributions and they are defined by two parameters: the mean and the variance (square of standard deviation). Each pair of these parameters gives rise to a different normal distribution. However, they can all be "re-parametrised" to the standard normal distribution using z-transformations. The standard normal distribution has mean 0 and variance 1.
What is the difference between t-distribution and standard normal distribution?
the t distributions take into account the variability of the sample standard deviations. I think that it is now common to use the t distribution when the population standard deviation is unknown, regardless of the sample size.
Why normality is required for standard deviation application?
Because the z-score table, which is heavily related to standard deviation, is only applicable to normal distributions.
What is a z score and what its used for?
The Normal probability distribution is defined by two parameters: its mean and standard deviation (sd) and, between them, these two can define infinitely many different Normal distributions. The Normal distribution is very common but there is no simple way to use it to calculate probabilities. However, the probabilities for the Standard Normal distribution (mean = 0, sd = 1) have been calculated numerically and are tabulated for quick reference. The z-score is a linear transformation of a Normal variable and it allows any Normal distribution to be converted to the Standard Normal. Finding the relevant probabilities is then a simple task.
What role do z-scores play in the transformation of data from multiple distributions to the standard normal distribution?
None.z-scores are linear transformations that are used to convert an "ordinary" Normal variable - with mean, m, and standard deviation, s, to a normal variable with mean = 0 and st dev = 1 : the Standard Normal distribution.
What is the benefit of transforming standard normal distributions to conform to the standard distribution?
There are no benefits in doing something that cannot be done. The standard normal distribution is not transformed to the standard distribution because the latter does not exist.
In what ways is the t distribution similar to the standard normal distribution?
Check the lecture on t distributions at StatLect. It is explained there.
Do some normal probability distributions have different means and different standard deviations?
Yes. Normal (or Gaussian) distribution are parametric distributions and they are defined by two parameters: the mean and the variance (square of standard deviation). Each pair of these parameters gives rise to a different normal distribution. However, they can all be "re-parametrised" to the standard normal distribution using z-transformations. The standard normal distribution has mean 0 and variance 1.
What is the difference between t-distribution and standard normal distribution?
the t distributions take into account the variability of the sample standard deviations. I think that it is now common to use the t distribution when the population standard deviation is unknown, regardless of the sample size.
Why standard deviation is best measure of dispersion?
standard deviation is best measure of dispersion because all the data distributions are nearer to the normal distribution.
Why normality is required for standard deviation application?
Because the z-score table, which is heavily related to standard deviation, is only applicable to normal distributions.
What is a z score and what its used for?
The Normal probability distribution is defined by two parameters: its mean and standard deviation (sd) and, between them, these two can define infinitely many different Normal distributions. The Normal distribution is very common but there is no simple way to use it to calculate probabilities. However, the probabilities for the Standard Normal distribution (mean = 0, sd = 1) have been calculated numerically and are tabulated for quick reference. The z-score is a linear transformation of a Normal variable and it allows any Normal distribution to be converted to the Standard Normal. Finding the relevant probabilities is then a simple task.
What requirements are necessary for a normal probability distribution to be a standard normal probability distribution?
The normal distribution, also known as the Gaussian distribution, has a familiar "bell curve" shape and approximates many different naturally occurring distributions over real numbers.
What does it mean normal distribution?
The Normal distribution is a probability distribution of the exponential family. It is a symmetric distribution which is defined by just two parameters: its mean and variance (or standard deviation. It is one of the most commonly occurring distributions for continuous variables. Also, under suitable conditions, other distributions can be approximated by the Normal. Unfortunately, these approximations are often used even if the required conditions are not met!
When it comes to comparing data from different distributions what is the benefit of normal standard distribution?
There may or may not be a benefit: it depends on the underlying distributions. Using the standard normal distribution, whatever the circumstances is naive and irresponsible. Also, it depends on what parameter you are testing for. For comparing whether or not two distributions are the same, tests such as the Kolmogorov-Smirnov test or the Chi-Square goodness of fit test are often better. For testing the equality of variance, an F-test may be better.
What happens in a normal distribution when the means are equal but the standard deviation changes?
The two distributions are symmetrical about the same point (the mean). The distribution where the sd is larger will be more flattened - with a lower peak and more spread out.
