# Making the most of Standard Deviation and P value — graphing?

I have a question or two in regard to getting the most out of the statistics of my data. By way of background, my qualitative observations of two distinct populations that are uniformly regarded to be unrelated led me to believe that they are closely related (by way of formation process). I began looking for a metric to test this assertion quantitatively and found one in the ratio between two measured parameters, a ratio that scales with size of the structures under study. Tabulating these ratios for 119 samples in each of the two populations (a and b), I derived means $x_a = 0.64092437$ and $x_b = 0.615966387$, standard deviations $\sigma_a = 0.067220754$ and $\sigma_b = 0.062833103$ and a $t$-test p value of 0.003401814.

In my extremely limited familiarity with statistical analysis, these look like very promising numbers viz. degree of variance and acceptance of my Null Hypothesis (that two populations that are considered to be unrelated are actually closely related) at > 99% CI..

So, my questions are i) do I have this right (given that my numbers are the output of the Excel 'black box') and ii) if so, is there any way to graphically chart these data (raw, mean, or stats) to highlight this closeness visually? Excel provides little of use that I have found, and it seems that a y-axis graph of three-standard deviations to show the entire possible variation would show how tight the data are, but produce a graph with what looks like a single vertical line on it.

How would a statistical adept tackle making the most of his/her data in this respect?

• What formal hypothesis were you testing with a t-test and how does it relate to a notion about whether they're unrelated or "closely related"? – Glen_b -Reinstate Monica Apr 28 '15 at 16:35
• I don't know that one tests a 'formal hypothesis' with a t-test, but rather attempts to refute the Null, whereupon the Alternate gains support. If the Null is that they are not (related) and the Alternate that they are, then the t test quantifies that, no? – Dave Apr 28 '15 at 20:49
• I'm basically asking what are your null and alternative hypothesis, expressed in symbols. I'm not aware of any t-test that tests the hypotheses you mention, so I'm trying to figure out what you actually did so I can explain to you what you actually tested instead of the hypothesis you mention. Did you just do an ordinary two sample location test? – Glen_b -Reinstate Monica Apr 28 '15 at 23:05
• Glenn, Brent, thanks for your input. Underscoring my aforementioned neophyte status, the first elements of both of your responses have lost me. For instance, what does "...I'm not aware of any t-test that tests the hypotheses you mention" -- sorry, hit carriage return while editing -- please ignore for present – Dave Apr 29 '15 at 6:37
• mane? Similarly, "...What is meant by related vs unrelated?" Perhaps I can illustrate what I'm trying to do by extreme example: few would consider that mountains and molehills are related by formation mechanism, but if I had cause to believe that they were I would want to test the Null Hypothesis that they are genetically unrelated in seeking support for my Alternate Hypothesis that both mountains and molehills have a common origin. That is what I actually want to test and I did struggle to fit this into the model of hypothesis testing, which seems to require that one can only test for... – Dave Apr 29 '15 at 6:48

What is meant by related vs unrelated? For the two-sample t test which you have applied, the appropriate null hypothesis is that the two populations both have the same mean (of whatever ratio you are measuring). A small P-value is evidence that the mean of the first population is not the same as the mean of the second population.

As far as graphing the data, Excel is probably not going to be a very convenient tool. You might consider importing the data into R. A "bee swarm" plot (a version of a "dot plot" or "strip chart") may be a convenient way of visualizing your data: here is a way to do this in R (here I'll use simulated, random data restricted to have the same sample means and standard deviations that you gave):

library(MASS);
library(beeswarm);

# Simulate random data conditioned on given summary statistics
xa=.64092437
xb=.615966387
sa=.067220754
sb=.062833103
na=119
nb=119
a=mvrnorm(na,xa,sa,empirical=TRUE)
b=mvrnorm(nb,xb,sb,empirical=TRUE)

# Create "bee swarm" plot
dev.new(width=9,height=4);
beeswarm(list(a,b),vertical=FALSE,method="hex"); 