## Normal Distribution Part II

### October 15, 2006

Ok, so in the last post I described what a Normal Distribution was:

*It’s a graph of the frequency of occurrence of values of a specific trait.*

So here’s some more nuggets of information.

### A Normal Distribution can be described using two values; the mean and the ‘standard deviation’ are all that’s needed to describe a normal distribution.

Here a picture showing what the mean and the standard deviation describe:

So the mean described the *position* of the normal distribution on the X-axis, while the standard deviation describes the *width* of the normal distribution.

Mathematicians and statisticians love the normal distribution because you can describe the whole thing with just two numbers.

So here are two normal distributions with the same mean, but different standard deviations:

.. and here are to normal distributions with the same standard deviation but different means:

It is important to understand that evolution can affect *both* of these values either independently, or at the same time.

### The normal distribution is a probability function.

There is a set of statistical tests that are known as the Parametric tests. These are all based on the assumption that the data being tested come from a normally distributed set. This basically allows the tests to make mathematical shortcuts by using the mean and standard deviation to produce probability curves.

There is a probability formula for the normal distribution, but I’m not going to show it, because we don’t need to memorise it to understand statistics.

If you’re interested in writing software, check out my other blog: Coding at The Coal Face

If You Find Yourself Caught in Love by Belle & Sebastian from the album Dear Catastrophe Waitress

Plug In Baby by Muse from the album Origin Of Symmetry

October 4, 2007 at 8:38 pm

Really nice site you have here. IÃ¢â‚¬â„¢ve been reading for a while but this post made me want to say 2 thumbs up. Keep up the great work

November 6, 2008 at 7:24 pm

I believe there is a small error in your definition of f(x): Shouldn’t the 2*sigma^2 term be inside the exponent?

November 7, 2008 at 6:34 am

Gah! You’re right! I can’t change it right now, but I’ll get onto it!

April 15, 2009 at 2:52 pm

After reading this article, I feel that I really need more information on the topic. Can you suggest some more resources ?

August 6, 2009 at 1:09 pm

Dear Dr. Herbie,

This is Vincent Chieh-ying Chang, a finishing PhD student. I am currently working on my PhD thesis where the following online image from your website (https://drherbie.files.wordpress.com/2006/10/WindowsLiveWriter/NormalDistributionPartII_D10A/same%20mean%20different%20stddev%5B2%5D.gif) is used in my thesis body purely for academic use with a full citation of the online source in my references. I was wondering if you could please grant me the permission to use the online image in my thesis with its copyright cleared on my end. Many thanks in advance for any help you can give; I look forward to your reply to my email please!!!

Warm regards, Vincent

August 7, 2009 at 6:21 am

Hi Vincent,

I am more than happy for you to use copies of images from this blog with citations. I make no claim to copyright for these images and hereby grant anybody the right to use these images in any way they desire.

Herbie

January 6, 2010 at 1:57 pm

Hi,

I was recently trying to do some evolutionary modeling (mainly as an outsider), and there is a question that has struck me. As you have said, any phenotypic characteristic is typically distributed normally. Now consider a particular trait, say the running speed of rabbits in a rabbit population. Since a faster rabbit is better at escaping a predator than a slower one, the rabbits with a higher speed will leave more offspring than the slower ones. Off-hand it seems that this should make the initial normal distribution of speeds skewed; so there must be some mechanism that restores the normal distribution. Can you tell me what that mechanism is? I would also be glad if you could give me some references. I’m really just a starter and need some reading material.

January 6, 2010 at 5:21 pm

Well, you’ve very astutely figured out how to recognise a characteristic under selective pressure! As long as the selective pressure is active, the distribution will be skewed (I think of it as the normal distrubution being made of jelly and a giant hand pushing it in one direction). When selective pressure relaxes, mutation and crossover will restore the normal distrubution over time.

As for references? I’m afraid I don’t really have any these days (it’s been over a decade since I last did this stuff!).

July 17, 2011 at 5:47 pm

This is an unrelated question, but I was wondering if you could help me answer it? :

To find a probability of a specific value in a normal distribution, we can standardize the distribution, so that later we can find the probability of that precise value by finding the z-score (standard score) and looking it up in the standard normal probability tables.

My question is: Correct me if I’m wrong – I think that you can have two normal distributions with same mean and standard deviation, yet have different area under their curves, which implies different probability densities. Which essentially means, you have different heights for the bell curves. If you standardize both the distributions ie. Z1 ~ N(0,1) and Z2 ~ N(0,1), how do the normal probability tables that give the probabilities (P(Z<z)) account for the different areas under these standardized curves?