Normal Distribution: Properties & Formula Explained

In summary: In short, the theorem states that the distribution of the sum of n random variables is approximately normal.
  • #1
disregardthat
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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.
 
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  • #2
Jarle said:
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.
 
  • #3
mathman said:
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?
 
  • #4
Jarle said:
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).
 
  • #5
mathman said:
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?
 
  • #6
The main point is that the errors be independent. The binomial approaches the normal because the assumptions of the central limit theorem hold.
 
  • #7
How does the independence of errors imply that it is normally distributed?
 
  • #8
Jarle said:
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.
 

What is a Normal Distribution?

A Normal Distribution is a type of probability distribution that is symmetrical and bell-shaped. It is often used to model natural phenomena such as heights, weights, and test scores.

What are the properties of a Normal Distribution?

The properties of a Normal Distribution include:

  • The mean, median, and mode are all equal and located at the center of the distribution.
  • The distribution is symmetrical, meaning that the left and right sides are mirror images of each other.
  • About 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.
  • The area under the curve is equal to 1, making it a valid probability distribution.

What is the formula for a Normal Distribution?

The formula for a Normal Distribution is: where:

  • x is a random variable
  • μ is the mean
  • σ is the standard deviation

What is the Central Limit Theorem and how does it relate to Normal Distribution?

The Central Limit Theorem states that as the sample size increases, the sampling distribution of the sample mean approaches a Normal Distribution, regardless of the shape of the population distribution. This means that Normal Distribution is a fundamental concept in statistics and can be used to model many real-world phenomena.

What is the z-score in Normal Distribution?

The z-score, also known as the standard score, is a measure of how many standard deviations a data point is away from the mean in a Normal Distribution. It is calculated by subtracting the mean from the data point and then dividing by the standard deviation. A z-score of 0 represents the mean, a z-score of 1 represents one standard deviation above the mean, and a z-score of -1 represents one standard deviation below the mean.

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