Normal (Gauss) Distribution questions

In summary, the standard normal distribution is an idealized probability distribution that can approximate many practical applications. It is related to the normal distribution through a linear change of variables and is used because of the Central Limit Theorem and its connection to the binomial distribution. In engineering, it is helpful for quickly calculating percentiles using the z value.
  • #1
Master J
226
0
Am I correct in my understanding of the standard normal distribution:

An idealized probability distribution which can be used to approximate many distributions which arise in practical applications.
(WHat are some of these applications?)

How exactly does the standardized normal distribution related to the Normal (Gauss) distribution, as in, why is it needed? Why are the results obtained from the standardized?

As a physics student, a lot of the details of these would be beyond my scope, but I would like to have an idea so to better understand it.

Hope someone can help clear thse up! Thanks!
 
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  • #2
Master J said:
Am I correct in my understanding of the standard normal distribution:

An idealized probability distribution which can be used to approximate many distributions which arise in practical applications.
(WHat are some of these applications?)
I think if the standard deviation is small relative to the mean then it is an okay for a lot of applications. For large standard deviations it won't always be a good choice given that the tails go to infinity and the higher order statistics are determined completely by the mean and standard deviation.

How exactly does the standardized normal distribution related to the Normal (Gauss) distribution, as in, why is it needed? Why are the results obtained from the standardized?
They are related by a linear change of variables.
 
  • #3
One of the main justifications for employing a Gaussian model is given by the Central Limit Theorem. This theorem states that suitable linear combinations of suitably-behaved random variables will, asymptotically, display a Gaussian distribution, regardless of the distributions of the individual random variables being combined.

So, any time you are looking at a random variable that is produced by linearly combining lots of well-behaved random variables (which is common in physics and engineering), then you can justify assuming that the result is Gaussian.

Another justification is that the binomial distribution can, in the limit, be nicely approximated by a Gaussian distribution, so there is also a connection to discrete random variables.
 
  • #4
Master J said:
How exactly does the standardized normal distribution related to the Normal (Gauss) distribution, as in, why is it needed? Why are the results obtained from the standardized?

The standard normal distribution is helpful in engineering practice because it allows textbooks to print a table giving the area under the distribution curve between 0 and a given z value. You can calculate the z value for any normal distribution using the mean and stdev. Z value is interpreted as "the number of standard deviations away from the mean" and can be positive or negative.

For example you can look up the 99 percentile Z value, and then calculate the 99th percentile threshold in your application if you know your mean and stdev.
 

1. What is the Normal (Gauss) Distribution?

The Normal (Gauss) Distribution, also known as the Gaussian Distribution, is a probability distribution that is commonly used to model continuous variables in natural or social sciences. It is a bell-shaped curve that represents the distribution of data around a mean value, with most of the data falling close to the mean and decreasing as it moves away from the mean.

2. How is the Normal (Gauss) Distribution calculated?

The Normal (Gauss) Distribution is calculated using the formula:
f(x) = (1 / σ√(2π)) * e^(-((x-μ)^2 / 2σ^2))
Where μ is the mean, σ is the standard deviation, and e is the base of the natural logarithm. This formula represents the probability density function of the Normal Distribution.

3. What is the importance of the Normal (Gauss) Distribution in science?

The Normal (Gauss) Distribution is important in science because it is used to describe many natural phenomena, such as human height, blood pressure, and test scores. It is also used in statistical inference, where it serves as a key assumption for many statistical methods and tests.

4. What are the characteristics of the Normal (Gauss) Distribution?

The Normal (Gauss) Distribution is characterized by its mean, median, and mode being equal, and the curve being symmetrical around the mean. It also follows the 68-95-99.7 rule, where approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

5. Can the Normal (Gauss) Distribution be used for any type of data?

No, the Normal (Gauss) Distribution is not suitable for all types of data. It is best used for continuous, quantitative data that is normally distributed, meaning the data is not heavily skewed and the mean, median, and mode are similar. If the data is not normally distributed, other types of distributions, such as the Poisson or Binomial distribution, may be more appropriate.

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