Understanding normal distribution graphs

[SeanC]

Hey,

What do the x and y values on a normal distribution graph refer to? X=standard deviation and y=mean?

Thanks.

[SeanC]

Okay, I found that the horizontal numbers refer to SD, but what about the vertical? Are they just arbitrary numbers assigned based on the height of the curve for use in calculations?

[DrVanNostrin]

I'm not sure exactly what you're talking about, but I'll take a stab. You were probably looking at a normal(0,1) distibution. This a normal distribution with a mean of 0 and SD of 1. If this is the case the x-values do correspond to z-scores. The y-values are a little more complicated.

Since a normal distribution is continous the probability of getting any exact value is 0 (there are an infinite amount of possible outcomes). However the area under the curve is equal to the probability of falling in that region. For example, the region bounded by x = -1, x = 1, y=0 and the distribution itself would have an area of 0.68. If you were to integrate from x = -1 to x = 1 the result would be 0.68 (you can't integrate a normal distribution using standard techniques). For any continous distribution the integral from -infinity to infinity must equal 1.

[SeanC]

Hey Dr,

Thanks for the reply. I actually do understand that and this whole thing is starting to come into focus. The only thing left that I don't quite get is this idea of "normal (0,1) distribution." What if you're calculating for something that has a mean of, let's say, 100, and an SD of 50?

Thanks.

[DrVanNostrin]

That's what z-scores are for.

[Paul2432]

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Since a normal distribution is continous the probability of getting any exact value is 0 (there are an infinite amount of possible outcomes).

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If the probability of any particular outcome is zero, how does anything ever happen? (assuming something with a zero probability cannot happen)

Paul

[DrVanNostrin]

[ QUOTE ]
If the probability of any particular outcome is zero, how does anything ever happen? (assuming something with a zero probability cannot happen)

[/ QUOTE ]
I may not know enough about the subject to answer this correctly. But this is how I think about it. Let me know if I'm wrong.

Since a continious distribution is made up of an infinite number of mutually exclusive possibilies, the probably of one of those events occuring is infinity*0. Which is not equal to zero.

let C be some real finite number greater than 0

C/infinity = 0

infinity*0 = C > 0

[Grizwold]

Hi Paul,

[ QUOTE ]
[ QUOTE ]
If the probability of any particular outcome is zero, how does anything ever happen? (assuming something with a zero probability cannot happen)

[/ QUOTE ]

...a continuous distribution is made up of an infinite number of mutually exclusive possibilities…

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A loose example to illustrate this statement:

Assume a normal distribution represents the distance you walk every day. The mean is 5 miles and the standard deviation is 1.01 miles.

What are the chances you will walk exactly 5.53 miles tomorrow? 0. Why? This is where the continuous distribution comes in. Believing there is some chance you walked exactly 5.53 miles that day, you measure the distance on your magic. The ruler reads 5.53 miles, but how do you know it was not 5.531 or 5.529? If you determine it’s definitely between 5.531 and 5.529 on your magic ruler, how do you it’s not 5.5301 or 5.53000000000001? It will never be exactly 5.53. All you can say is that for sure it is between 5.31 and 5.529, and give the probability of that.

Clark

[LarryLaughs]

It may also help to understand that the area below the normal distribution curve is 1. It represents all possible results of a given normally distributed variable. Any single case is an infinitely "thin" slice of the area (along the y-axis direction), which is why there is no real probability of that specific value occuring, just what is the likelihood of a smaller or larger value appearing.