Diagrams vs Artistic Renderings
Statements in Results text rather than in Figures
Data Set: 10,8,8,6,6,6,6,4,4,2
Descriptive Statistics: Give a "picture" of the whole set of numbers
average = mean = = 6.0
--if all the values were the same in this dataset, what value would it be?
standard deviation = = 2.30940108
--what is the amount of "scatter" around the average value?
--how should I round off this number? Significant digits...precision + 1
number = n = 10
--how many data points are there?
A histogram is most helpful in getting a "picture" of the data:
You might notice that these data fall into a sort of bell-shaped curve...a Gaussian curve.
This is sometimes called a "normal" distribution. Not all data fall into a normal distribution.
If data fit a normal distribution then about 68% of the data points are within 1 SD of the mean.
95% of the data points are within 2 SD of the mean. In the case of our simple data,
80% are within 1SD and 100% are within 2SD of the mean. Our sample is small and contrived!
Larger datasets are more likely to fit a Gaussian distribution.
Data can also be graphed as a symbol at the mean, a box around the symbol extending ± 1 SD, with lines extending over the range of data values:
Also notice that the box-and-whisker plot above is quite unconventional!
This graph is contrived to fit across a browser...rather than requiring a lot of
scrolling. But in doing this, the graph has the measured variable (the dependent variable)
on the horizontal axis. This is usually NOT done! Fortunately here, we lack an experiment
so perhaps saving space is worth the unconventional graph orientation?
However data are shown, it is critical that the figure legend describe exactly what is being shown!!
Don't forget legends on your figures! Label your graphs! Use a computer or at least a
straightedge and graph paper!
Hypothesis Testing: (comparing two or more data sets)
If data fall into a normal distribution, we can use parametric statistical tests
(t, z, chi-square, regression, ANOVA, etc.).
If data do not fall into a normal distribution, we must go to non-parametric statistical tests (Wilcoxon, Kruskal-Wallis, etc.).
So let's say we have a second data set: 7,7,8,8,8,8,8,8,9,9 x
= 8.0 sd = 0.67
The averages are different, but the deviations encompass both averages. Are the two sets
We first need a hypothesis. An educated guess about our question.
We want to be able to reject it. Generally we are testing either a model for correctness,
or a null-hypothesis (the manipulation has no effect). Our hypothesis is: THE SAMPLES ARE NOT DIFFERENT!
The parametric tests generally assume that the data sets are completely independent
of each other, and are taken from a population distributed in a normal way (Gaussian).
We need to check these assumptions out before starting!
The test we choose will go through some tedious calculations that our friends, the statisticians
have produced. Thank goodness computers can do all of this for us! In the old days we
had to do the calculations by hand; the result was usually a value that had to be
compared with a table of comparison values from a book. If your calculated value
was above a certain table value you could reject your hypothesis. In our lab exercises
we will use the computer to see how to do the work more easily.
The computer can give us our old-fashioned values if we want them, but usually
all you really need beyond the descriptive statistics, is the value of p. The value of p
is the probability that in another repeat of your project you would get more extreme results
that this one if the null hypothesis you are testing accounts for the data observed.
More simply it is often interpreted as the probability that the differences between your data and the
null hypothesis are due to chance alone.
When p approaches 1.0 you become more sure of the null hypothesis.
As p approaches 0.0 you begin to have doubts or reject the null hypothesis.
How low does p have to go before we reject the hypothesis? This is somewhat problematic...
but the value that it must go below is called α. Convention sets alpha
to 0.05, but this is a "one-size-fits-all" kind of value. It is good for some
experiments, but is the wrong value for others. In most work, we allow 5% error
in our testing. We are willing to be wrong one-time in twenty times...and still
stick to our hypothesis!
What kind of errors are we talking about? Statistical errors! NOT biology errors! NOT measurement errors! NOT human errors! If you have biology, measurement, or human errors, you simply do the trial over. But no matter how careful you are...even if you could be perfect in the handling of the project, you will still make statistical errors...these are errors due to chance and chance alone. They come in two types.
Type I: you reject a true null-hypothesis (convicting the innocent)
Type II: you fail to reject a false null-hypothesis (acquitting the guilty)
If you set α (reasonable doubt) too low, you will make lots of Type II errors, but will not make many Type I errors. Our justice system has been guilty of this; some claim that it happened in the O.J. Simpson case.
If you set α (reasonable doubt) too high, you will make lots of Type I errors,
but will not make many Type II errors. This is unthinkable in our justice system,
but recently, right here in Connecticut, an innocent man was set free when DNA
evidence altered the balance of "reasonable doubt" and the real criminal was found and confessed!.
If you were screening pesticides, you might use a high α so that you do not miss
any of the possibilities (Type I errors are OK...type II errors are fatal). But for final
testing of the pesticide for a virulent and otherwise untreatable pest, you would
choose to use a low α to be sure that your
pesticide beats nothing (Type II errors OK, but want to avoid Type I)!
OK! Suppose your p is less than α. You can now reject the null-hypothesis.
You can say your pesticide responses are statistically significant.
But what if your p is very low? Is it more significant?
A test with p=0.001 is not more significant than a test with p=0.02 when your critical value is α=0.05; both are statistically significant. Many plant physiology articles will show numbers in tables with superscript symbols as found in the key below:
Because of the various kinds of errors involved at different levels, you
should not use the "meanings" for error levels less than whatever α
you have chosen. Think of significant in the same way as pregnant.
You are or you are not pregnant; you cannot be very pregnant or extremely
pregnant. So only the two first rows in the chart have correct statistical
meaning (if alpha is 0.05).
What test should I use?
Comparing means of two samples: t-test or Wilcoxon test
Comparing two proportions or percentages: z-test
Comparing counted outcomes in classes: Chi-Squared
Testing a dynamic model: Regression (Linear, Logistic, Polynomial, Non-linear)
Testing multiple variables on a process: ANOVA