Transcript AP Stats 2015

Statistical
Analysis
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“Why is this Biology?”
Variation in populations.
Variability in results.
affects
Confidence
in conclusions.
The key methodology in Biology is hypothesis
testing through experimentation.
Carefully-designed and controlled
experiments and surveys give us quantitative
(numeric) data that can be compared.
We can use the data collected to test our
hypothesis and form explanations of the
processes involved… but only if we can be
confident in our results.
We therefore need to be able to evaluate the
reliability of a set of data and the significance
of any differences we have found in the data.
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“Which medicine should I prescribe?”
Generic drugs are out-of-patent,
and are much cheaper than the
proprietary (brand-name)
equivalents. Doctors need to
balance needs with available
resources. Which would you
choose?
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“Which medicine should I prescribe?”
Means (averages) in Biology are
almost never good enough.
Biological systems (and our
results) show variability.
Which would you choose now?
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Why do we need to do
statistical analysis?

Evaluate data

Make sense from the numbers
Hummingbirds are nectarivores (herbivores
that feed on the nectar of some species of
flower).
In return for food, they pollinate the flower.
This is an example of mutualism –
benefit for all.
As a result of natural selection,
hummingbird bills have evolved.
Birds with a bill best suited to
their preferred food source have
the greater chance of survival.
Researchers studying comparative anatomy collect
data on bill-length in two species of hummingbirds:
1. Archilochus colubris
(red-throated hummingbird)
2. Cynanthus latirostris
(broadbilled hummingbird).
To do this, they need to collect sufficient
relevant, reliable data so they can test
the Null hypothesis (H0) that:
“there is no significant difference
in bill length between the two species.”
The sample size must
be large enough to provide
sufficient reliable data and for us
to carry out relevant statistical
tests for significance.
We must also be mindful of
uncertainty in our measuring tools
and error in our results.
III. Measurements & Uncertainty
IV. Mean (Average)
- The mean is a measure of the central
tendency
of a set of data.
Table 1: Raw measurements of bill length in Descriptive table title and number.
A. colubris and C. latirostris.
Bill length (±0.1mm)
Uncertainties must be included.
n
A. colubris
C. latirostris
1
13.0
17.0
2
14.0
18.0
3
15.0
18.0
4
15.0
18.0
5
15.0
19.0
6
16.0
19.0
7
16.0
19.0
8
18.0
20.0
9
18.0
20.0
10
19.0
20.0
Mean
s
n = sample size. The bigger the better.
In this case n=10 for each group.
Raw data and the mean need to have
consistent decimal places (in line with
uncertainty of the measuring tool)
=AVERAGE (highlight raw data)
Calculating the Mean (Average)
Formula
* n = number of pieces of datum;
* x = each piece of datum
Graph 1: Comparing mean bill lengths in two
hummingbird species, A. colubris and C. latirostris.
Descriptive title, with
graph number.
20.0
C. latirostris,
18.8mm
18.0
Mean Bill length (±0.1mm)
16.0
A. colubris,
15.9mm
Labeled point
14.0
12.0
Y-axis clearly labeled,
with uncertainty.
10.0
8.0
6.0
Make sure that the yaxis begins at zero.
4.0
2.0
0.0
Species of hummingbird
Graph 1: Comparing mean bill lengths in two
hummingbird species, A. colubris and C. latirostris.
20.0
C. latirostris,
18.8mm
18.0
Mean Bill length (±0.1mm)
16.0
A. colubris,
15.9mm
14.0
12.0
10.0
But the mean only tells
part of the story…
8.0
6.0
4.0
2.0
0.0
From the means alone
you might conclude
that C. latirostris has a
longer bill than A.
colubris.
Species of hummingbird
What % of the population is
found within one standard
deviation of the mean?
68%
What % fall within two
standard deviations of the
mean?
96%
What % fall with three
standard deviations of the
mean?
99.8%
Standard deviation is a measure of the spread of
most of the data.
Table 1: Raw measurements of bill length in
A. colubris and C. latirostris.
Bill length (±0.1mm)
n
A. colubris
C. latirostris
1
13.0
17.0
2
14.0
18.0
3
15.0
18.0
4
15.0
18.0
5
15.0
19.0
6
16.0
19.0
7
16.0
19.0
8
18.0
20.0
9
18.0
20.0
10
19.0
20.0
Mean
15.9
18.8
s
Which of the two sets of data has:
b.
The greatest variability in the data?
- calculate the standard deviation
Standard deviation can have one more
decimal place.
Standard deviation is a measure of the spread of
most of the data.
Table 1: Raw measurements of bill length in
A. colubris and C. latirostris.
Bill length (±0.1mm)
n
A. colubris
C. latirostris
1
13.0
17.0
2
14.0
18.0
3
15.0
18.0
4
15.0
18.0
5
15.0
19.0
6
16.0
19.0
7
16.0
19.0
8
18.0
20.0
9
18.0
20.0
10
19.0
20.0
Mean
15.9
18.8
s
1.91
1.03
=STDEV
Which of the two sets of data has:
a. The longest mean bill length?
C. latirostris
a. The greatest variability in the data?
A. colubris
Standard deviation can have one more
decimal place.
Standard deviation is a measure of the spread of
most of the data. Error bars are a graphical
representation of the variability of data.
Error bars could represent standard deviation, range or confidence
intervals.
Which of the two sets of data has:
a. The highest mean?
A
a. The greatest variability in the data?
B
The overlap of a set of error bars gives a clue as to the
significance of the difference between two sets of data.
Large overlap
No overlap
Lots of shared data points
within each data set.
No (or very few) shared data
points within each data set.
Results are not likely to be
significantly different from
each other.
Results are more likely to be
significantly different from
each other.
Any difference is most likely
due to chance.
The difference is more likely
to be ‘real’.
Graph 1: Comparing mean bill lengths in two
hummingbird species, A. colubris and C.
latirostris.(error bars = standard deviation)
22.0
C. latirostris,
18.8mm
(n=10)
Mean Bill length (±0.1mm)
17.0
A. colubris,
15.9mm
(n=10)
The data has a greater chance of being
significant.
12.0
7.0
2.0
-3.0
Our results show a very small overlap
between the two sets of data.
Species of hummingbird
Interesting Study: Do “Better” Lecturers Cause More Learning?
Students watched a one-minute video of a lecture. In one video, the lecturer was
fluent and engaging. In the other video, the lecturer was less fluent.
They predicted how much they would learn on the topic
(genetics) and this was compared to their actual score.
(Error bars = standard deviation).
Is there a significant difference in the actual learning?
n=21
n=21
Find out more here: http://priceonomics.com/is-this-why-ted-talks-seem-so-convincing/
Interesting Study: Do “Better” Lecturers Cause More Learning?
Evaluate the study:
1. What do the error bars (standard deviation) tell us about reliability?
2. How valid is the study in terms of sufficiency of data (population sizes (n))?
n=21
n=21
Find out more here: http://priceonomics.com/is-this-why-ted-talks-seem-so-convincing/
VII. What does Significantly Different
mean?
 Most
scientists agree that 2 deviations
above or below the mean indicates that
you are significantly different.
 Significantly
different means the
difference is not due to chance
 The
Null Hypothesis would be rejected.
Are any of these people
significantly different?
10, 13, 15, 20, 24, 28, 30
No!
Only individual with scores
above 34.22 or below 5.78
would be considered
significantly different.