# How to calculate the maximum value of a normal distribution given the std dev?

#### Jennifer Murphy

##### Member
Is there a formula for the maximum value of a normal distribution given the std dev?

I generated several normal distributions in Excel with various means and std deviations. The means did not seem to have any affect on the maximums. Herte are a few maximums based on the std devs:

StdDev Max
0.1 3.9894
0.5 0.7979
1.0 0.3989
1.5 0.2660
2.0 0.1995
5.0 0.0074

Does the max make the total area under the curve = 1?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
This all depends on what you are doing. Mean is the central location, so yeah it won't mess with the dispersion. Obviously, if you can keep the std the same but change the mean it will likely just shift the distribution around.

I would imagine there is no max value possible, because any value is possible - its probability is just real low. The max value you see in an individual sample of the random variable is then based on it's probability given the curve, so if you are sampling a large number of draws, you will see larger values at times. What you also get is based on the seed used in the process, so you could run the same thing twice but one time see a slightly larger value.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
PS, I don't get your above data. As the std increases I would imagine the possible max for a sample should go up, but your values are the other way - which doesn't make sense to me.

#### Dason

Yeah you really need to explain what you're talking about more because your results don't make any sense at all. But even so you can talk about what the cdf of the max should be for a given sample size, mean, and standard deviation but in theory there is no actual maximum value so you'd have to talk about 99th percentile of the maximum or something like that.

#### Jennifer Murphy

##### Member
OK, let me try again. Here's some data. The Columns (D-G) are generated by the Excel Norm.Dist function. The means are all 0. The std devs vary.

I would like a formula (f(stddev)) that would calculate these maximums. Is that any clearer?

#### Jennifer Murphy

##### Member
PS: The maximum Y value of any Normal distribution is at X=the mean, No?

So if we have a function to generate that Y value, it ought to give us that maximum value, No?

#### Dason

Nope. Still makes no sense. Let's start from the beginning. What exactly do you think you mean when you're talking about the max of the distribution?

#### Dason

...

...

Are you referring to the maximum value the pdf takes on? Because I guess I don't understand why you would ever care about that but yes that always will achieve the maximum at x={the mean} and yes it will decrease as the standard deviation increases.

But yes there is a very easy way to achieve that value because the normal pdf has a closed form solution. So plug in x=mu and then yeah... that's it - you'll get the "maximum" value of the pdf if that's what you care about.

Maybe you could explain what you're actually interested in and why the pdf value at a particular point seems to matter to you - because I don't know why you would care about that but I can imagine situations where somebody might think this is relevant but what they really want is something else.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Yeah, I thought you were trying to get at the largest value in the sample or the maximum value, - but I see you meant the value with the highest probability. As mentioned, the mean is the measure of centrality for the standard normal!!

I will echo @Dason - are you playing around or do you have an underlying goal?

#### katxt

##### Well-Known Member
But yes there is a very easy way to achieve that value because the normal pdf has a closed form solution. So plug in x=mu and then yeah... that's it - you'll get the "maximum" value of the pdf if that's what you care about.
The cut down version of the formula that Dason mentions becomes =1/SD/SQRT(2*PI()) for the Max row in Excel.

#### Jennifer Murphy

##### Member
The cut down version of the formula that Dason mentions becomes =1/SD/SQRT(2*PI()) for the Max row in Excel.
Thanks. I eventually worked it out as:

Sorry, I don't see how to resize it.

#### katxt

##### Well-Known Member
The very formula. A surprising place for pi to appear, since there are no circles around.

#### Dason

The very formula. A surprising place for pi to appear, since there are no circles around.
Circles are involved.

If you would see the proof that the normal distribution integrates to 1 you wouldn't be surprised about that.

Edit: it involves considering the joint distribution of two independent normal random variables and then converting to polar coordinates to do the actual integration.

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#### hlsmith

##### Less is more. Stay pure. Stay poor.
I just figured curves were involved

#### katxt

##### Well-Known Member
Sorry, Dason. Most regulars here know that. I should have said that it is surprising for those just embarking on statistics.