Normal Distribution: Properties & Formula Explained

[disregardthat]

What properties does some measurement possesses such that they fit along a normal curve? And how was the general formula found? Wikipedia says very little on this.

 

[mathman]

What properties does some measurement possesses such that they fit along a normal curve? And how was the general formula found? Wikipedia says very little on this.

The normal distribution comes out of one of the fundamental theorems in probability theory "Central Limit Theorem". The general idea is that when adding up, and properly normalizing, a large number of independent random variables, the distribution of the result is approximately normal.

 

[disregardthat]

The normal distribution comes out of one of the fundamental theorems in probability theory "Central Limit Theorem". The general idea is that when adding up, and properly normalizing, a large number of independent random variables, the distribution of the result is approximately normal.

What assumptions are we making about the nature of our measurements when we assume they will fit the normal curve?

 

[mathman]

What assumptions are we making about the nature of our measurements when we assume they will fit the normal curve?

The errors are random in nature, independent from each other. If there is a systematic error, it will show up as an error in the mean (assuming you have a theoretical mean to compare).

 

[disregardthat]

The errors are random in nature, independent from each other. If there is a systematic error, it will show up as an error in the mean (assuming you have a theoretical mean to compare).

Could you describe it in another way? I am aware of the "errors" from the mean in nature, but what is characteristic for the distribution of these error which makes the normal curve an appropriate model?

The distribution curve for a binomial experiment fits the normal curve. Are we in some sense assuming that the measurements have the same characteristics? If so, in what sense?

 

[mathman]

The main point is that the errors be independent. The binomial approaches the normal because the assumptions of the central limit theorem hold.

 

[disregardthat]

How does the independence of errors imply that it is normally distributed?

 

[mathman]

How does the independence of errors imply that it is normally distributed?

I suggest that you look up the central limit theorem. If you google "Central Limit Theorem Proof" you will get a wealth of information.