Transcript statistical significance of the difference between two

Addressing scientific uncertainty
How scientists address these:
Unpredictability -> Projected scenarios
Data -> Statistics
Models -> Probabilities
Scientists also:
– Obtain review by peers and the scientific community
– Do the best they can given the methods they have, by
explicitly stating the strengths and weakness of their
data, models, and arguments.
Statistics
• Accuracy vs. Precision – language to talk about
our ball rolling results
• Mean, Median, and Mode – ability to talk about
data sets, rather than individual results
• Graphing – ways of displaying data sets
• Standard Deviation – a quantitative way of
characterizing the spread of a single data set
(and making error bars)
• T-test - the simplest quantitative way of
comparing two different data sets
William Sealy Gosset (1876–1937):
An English chemist and statistician,
best known by his pen name
Student.
He was employed by Guinness
beer company to quality control
their beer .
Photo source: Wikipedia, courtesy of Aaron Keys.
Creamy pint of Guinness beer
"Student“ in 1908. Image source: Wikipedia.
So, here is the problem: Gosset wanted to figure out what ingredients (from
what merchant) made a good beer. There were two problems:
1) You had to drink a lot of beer (with all the attendant problems - see
below) or you had to have a relatively small sample size; and
2) As with most natural samples, you do not know the distribution before
you start.
Image removed due to copyright:
Man jumps into pile of empty beer
cans.
To deal with these problems, Gosset (aka Student) publishes anonymously a
way of distinguishing different, small populations.
He calls it the t-test.
It uses probability functions.
I much prefer tea (t), after all.
Image source: Wikipedia.
A Probability distribution
These were discussed last time – discuss with a group:
Which line is most accurate?
While line is most precise?
A Probability distribution
All lines are equally accurate
The dark blue line is the least precise; the yellow line is the most precise.
So, the lower the number of objects, the less statistically precise is your
result.
In statistics, the t-distribution or Student's tdistribution is a probability distribution that
arises in the problem of estimating the mean
of a normally distributed population when the
sample size is small.
Student's t-tests is the basis for determing the
statistical significance of the difference
between two sample means, and for
confidence intervals for the difference
between two population means.
t-test
It is not so bad. X is what you are measuring. S is the grand
standard deviation. 1 = group one, 2 = group two. The
denominator is the standard error of the difference between two
means.
If the calculated t value is above the threshold chosen for statistical
significance (usually the 0.05 level), then the two groups differ.
May look like a graph to
you, looks like a beer to
me.
Image source: Wikipedia.
Marble Rolling
• Does a small marble roll down the slide at
a different speed than a larger marble?
An important point is that Student’s t-test is only
the most basic of all statistical tests. It is the
hammer of a toolbox: The simplest and the most
used.
Galileo Galilei (1564 – 1642)
All of this statistical
stuff finally gets us
back to interpretting
some graphs – the
ones that Galileo
created with his rolled
ball experiments.
Image from Wikimedia Commons
Voila!
Pictures from Museum of Science, Florence. Source: G. H. Rieke lecture.
With respect to experiments…….
Error - random variability in research
Bias - non-random or directed effects
caused by a factor or factors unrelated by
the independent variable
Uncertainty - a state of having limited
knowledge where it is impossible to
exactly describe existing state or future
outcome (more than one possible
outcome)
There is not a linear relationship between time and distance, thus the
two quantities are not proportional to each other.
Note: You probably have the axes reversed (distance on x axis), which
is better
2
There is a linear relationship between time and distance, thus the
two quantities are proportional to each other.
Galileo also noted that the angle of the ramp didn’t matter
at all, as the speed at the end of the ramp would be the
same (as the cartoon experiments that you did for
homework showed you). You can also think of the
effect, as Galileo did, with respect to a ball rolling down a
gradual slope and up a steep slope and vice versa –
except for friction, it goes up the same distance and thus
the speed must be the same.
Salviati says this in the Two Dialogues:
We pass now to . . . naturally accelerated motion, such as that generally
experienced by heavy falling bodies . . . . in the investigation of naturally
accelerated motion we were led, by hand as it were, in following the habit
and custom of nature herself, in all her various other processes, to employ
only those means which are most common, simple and easy . . . When,
therefore, I observe a stone initially at rest falling from an elevated position
and continually acquiring new increments of speed, why should I not believe
that such increases take place in a manner which is exceedingly simple and
rather obvious to everybody? If now we examine the matter carefully we find
no addition or increment more simple than that which repeats itself always
in the same manner. This we readily understand when we consider the
intimate relationship between time and motion; for just as uniformity of
motion is defined by and conceived through equal times and equal spaces
(thus we call a motion uniform when equal distances are traversed during
equal time-intervals), so also we may, in a similar manner, through equal
time intervals, conceive additions of speed as taking place without
complication ....
OR, A motion is said to be uniformly accelerated when, starting from rest,
it acquires during equal time intervals, equal increments of speed.
But your graph can tell the story better than Salviati:
Because there is a linear relationship between time2 and distance, and the only force
that acts on marble is vertical acceleration, then:
acceleration (constant) = distance / time2.
Technically, acceleration is proportional to distance / time2, but you get the idea.
What Galileo figured out was that all objects are
subject to a constant vertical acceleration,
which pulls all objects downward at the same
rate.
Image removed due to copyright: A page
from Galileo's notebooks, showing an
experiment such as the one described
here. See Stillman Drake, Galileo's Notes
on Motion, monograph 5, Annali dell'Istituto
e Museo di Storia della Scienza (Florence,
1979), p. 79. Available from Rice
University.
Galileo’s notes
So, a bullet shot horizontally
from a gun hits the ground
the same time a dropped
bullet does. It is strange,
different from what Aristotle
said, and true.
Did Galileo know what a big deal
this was?
Yes, he did. He realized that in
the same way he ended the
Ptolemaic (originally Aristotle’s)
view of the universe, he also
ended the Aristotelian
authoritative view of how the
world works.
I can shoot
cannons
real good
now.
This latter influence was largely
confined to those interested in
scientific matters, but is going to
reach its impact with Newton……
to be continued next week.
Image from Wikimedia Commons