Exploring the normal distribution

The normal distribution has two parameters: \(\mu\) and \(\sigma\) and is usually indicated as \(N(\mu,\sigma)\). Is this application you can explore several features of the normal distribution. For instance, given a value of k, the probabiliy of observing a value within the interval \(\mu \pm k \times \sigma \) is always the same for any normal, independently of the parameter values. In when \( k=1.96 \) that probability is 0.95.
The area under two values of X indicates the probability of obtaining values within this two values. For example, if the area is equal to 0.69, this meants tha if we observe a subject from this population its measured value of X will be within this interval in the 69% of the cases.
A sample of individuals will show a distribution (histogram) that is in accordance with the normal curve that represents the behaviour of this variable in the population.

Parameters

In a normal distribution \(N(\mu,\sigma)\), the parameter \(\mu\) defines the position of the distribution. This distribution is symmetric with respect \(\mu\).That is: \(P(X>\mu)=P(X<\mu)=0.5\). The parameter \(\sigma\) indicates the amplitude of the possible values of X. The arrow in the graphic show the value of this parameter fixed in the parameters panel. For a fixed value of \(k\), the probability of the interval \(\mu \pm k \times \sigma \) is always the same, independently of the parameter values. The probability of obtaining a value offside the interval \(\mu \pm 3 \times \sigma \) is almost 0.

\(P(\mu - k \times \sigma < X < \mu + k \times \sigma )\)



If \(X\) is a \(N(\mu,\sigma), then:\)

The probability \(P(X<x)\) is obtained with the instruction:

pnorm(x,mu,sigma)

The probability \(P(X>x)\) is obtained with the instruction:

1-pnorm(x,mu,sigma)

The probability of an interval \(P(a<X<b)\) is obtained with the instruction:

pnorm(b,mu,sigma)-pnorm(a,mu,sigma)

The value of \(a\) that fulfils \(P(X<a)=\alpha\) is obtained with the instruction:

qnorm(alpha,mu,sigma)

The normal standard: N(0,1)

Computing probabilities

If \(X\) is a normal distribution \(N(\mu,\sigma)\), the the variable: $$Z=\frac{X-\mu}{\sigma} \rightarrow N(0,1)$$ The \(N(0,1)\) is called normal standard and is denoted by \(Z\)
If \(X\) is a normal distribution \(N(\mu,\sigma)\), then: $$P(X\le x)=P \left( Z \le \frac{x-\mu}{\sigma}\right)$$

Example

Let's \(X\) be a normal distribution \(N(100,3)\)
Then: $$P(X\le 102)=P \left( Z \le \frac{102-100}{3}\right) = P(Z \le 0.67) = 0.7475$$ In R it is equivalent to compute: 


                      

                    

                  

                  

                  
Computing quantiles

                  

                  

                    

                      The value  \(x_\alpha\) that fulfils the equation
                                            $$P(X \le x_\alpha)=\alpha$$
                                            is called  quantile \(\alpha\) .
                      Using the standard normal:
                                            $$P(X \le x_\alpha)=P \left( Z \le \frac{x_\alpha-\mu}{\sigma} \right)=\alpha$$
                      Then 
                                            $$z_\alpha=\frac{x_\alpha-\mu}{\sigma} \rightarrow x_\alpha=\mu+z_\alpha \sigma$$
                    

                    

                      
Example

                      

                      Let's \(X\) be a normal distribution \(N(50,2)\)
                      

                      The quantile 0.92 is computed as
                      qnorm(0.92,50,2), which gives a result of
                                            52.81014
                      The quantile 0.92 for the standard normal, i.e. \(z_{0.92}\) is
                                            obtained as  qnorm(0.92) which gives 1.405072
                      You can check that:
                      50+1.405072*2=52.81014
                    

                  

                

                

                  

                  

                  
                  
Source: (http://en.wikipedia.org/wiki/File:Normal_distribution_and_scales.gif) 

                  

                  

                    

                      
Z-scores

                      

                      The z-score of an observed value \(x\) of a \(N(\mu,\sigma)\) is:
                                           $$z=\frac{x-\mu}{\sigma}$$
                      
                      

                      If \(\bar X\) is the mean of a sample, and \(s\) is
                                                    the correspondiong standard deviation, then the z-score is
                                                    obtained as:
                                                    $$z=\frac{x-\bar X}{s}$$
                      
                    

                    

                      
Interpretation

                      

                      The z-score indicates how many times the observed value \(x\) is
                                             above (or below in case it is negative) the mean in units of 
                                             standard deviations.
                      

                      

                      As z-scores are distributed as a \(N(0,1)\), its value can be
                                             read in terms of this distribution. For example, supose a man that has a weight of
                                             90.3 kg. In the population, we have a mean of 80.3kg. with a standard deviation of
                                             5.3kg. In tha case, the z-socre for this man is:
                                             $$ z =\frac{90.3-80.3}{5.3}=1.89$$
                                             In a \(N(0,1)\) the \(P(Z<1.89)=0.97\) this means this man has a
                                             percentile of weigth of 97 and that few people on this population will
                                             have a higher weight.
                      

                      

                      Z-scores allow comparing variables with different measurement units. For example,
                                             if an individual has a z-score of 2.1 for weight and 0.3 for height, this indicates
                                             that is close to the mean in height but well above the mean in weight!
                    

                  

                

                

                  

                    
                    
                  

                  

                

              

            

          

          

            

              
              

                

                  

                    

                      

                        

                          

                            
                            
                          

                          
                        

                        

                          
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    (c) Albert Sorribas, Ester Vilaprinyo, Rui Alves, Montserrat Rue 

                  Biomodels Group 

                   University of Lleida - Institute of Biomedical Research (IRBLleida)