Cooper City High School
Broward Community College
Statistics
Comparing the tstatistics with the zstatistics :
Normal Distribution
(Gauss Distribution): This is a probability, symmetric distribution that can be represented
by the mathematical function:
sometimes, it is better to work with the standard normal distribution that can be obtained using the transformation;
where this variable is called zvalue. This
transformation is equivalent to use the values μ=0, and σ=1. Using
this transformation we get the standard normal distribution:
Student or tDistribution: This is another probability, symmetric
distribution that can be represented by the mathematical function:
as you can see, we have a new parameter (ν) usually called degrees of freedom(d.f).
The gamma function Γ(ν) is related to the factorial function this way:
Γ(ν+1) = νΓ(ν)=ν!
It is possible to show that when the number of degrees of freedom increases the tdistribution gets closer and closer to the standard normal distribution.
We want to compare the results from the experiment with the calculations using the z statistic when we assume that we know the standard deviation for the population σ, and the tstatistic when we use the standard deviation from each sample S_{x}. First, let's perform an experiment with samples obtained from a skewed distribution. We select samples of 30 individuals from a randBin(5,0.1) which is a totally skewed distribution with mean value µ=0.5, standard deviation σ=0.671, and skewness ⅟=1.192 (totally skewed to the right). We repeat the experiment 500 times. The sampling distribution for z and t were calculated using the mean value and the standard deviation from each repetition. The results are shown in the following table:
Mean Value 
Standard Deviation 
Skewness 

z 
0.028 
1.005 
0.262 
t 
0.099 
1.081 
0.469 
Let's compare the graph for the zstatistic sampling distribution with the graphs for the normal and student distributions.
Let's compare the graphs for the tstatistic sampling distribution with the graphs for the normal and student distributions.
As you can see, there is not a big difference when the dimension of the sample is not small. Let's repeat these experiments using smaller samples. In this case, we have to select samples from a symmetric population, in another case we know the sampling distribution will be skewed where neither the normal nor the t distributions work.
We will select samples of 5 individuals from a randNorm(0,1) which is a totally symmetric distribution with mean value μ=0, standard deviation σ=1, and skewness γ_{1}=0. We will repeat the experiment 500 times. The parameters for the sampling distributions for the zstatistic(z) and tstatistic(t) where: z= (x  μ) / (σ/√n) and t= (x  μ) / (S_{x}/√n) are calculated using the values of x and S_{x} from the experiment. The results are shown in the following table:
Mean Value 
Standard Deviation 
Skewness 

z 
0.0931 
0.984 
0.251 
t 
0.134 
1.383 
0.164 
Let's compare the graph for the zstatistic sampling distribution with the graphs for the normal and student distributions.
In this case, the Normal Distribution describes the experimental results much better thant the tDistribution. Now, let's compare the graphs for the tstatistic sampling distribution with the graphs for the normal and student distributions.
In this case, it is clear that the tDistribution describes the experimental data much better than the Normal Distribution.