This document discusses a study that compared bill length in two species of hummingbirds: Archilochus colubris and Cynanthus latirostris. Researchers took measurements of 10 individuals of each species and calculated the mean and standard deviation. The mean bill length was 15.9mm for A. colubris and 18.8mm for C. latirostris. A. colubris had greater variability in its data (standard deviation of 1.91) compared to C. latirostris (1.03). A t-test showed a statistically significant difference between the means, allowing the researchers to reject the null hypothesis that there is no difference in bill length between the species.

1. Statistics (0.0)
IB Diploma Biology

2. 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.
Photo: Archilochus colubris, from wikimedia commons, by Dick Daniels.

3. Researchers studying comparative anatomy collect
data on bill-length in two species of hummingbirds:
Archilochus colubris
(red-throated hummingbird) and
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.”
Photo: Archilochus colubris (male), wikimedia commons, by Joe Schneid

4. 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.
Photo: Broadbilled hummingbird (wikimedia commons).

6. The mean is a measure of the central tendency
of a set of 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
s
Calculate the mean using:
• Your calculator
(sum of values / n)
• Excel
=AVERAGE(highlight raw data)
n = sample size. The bigger the better.
In this case n=10 for each group.
All values should be centred in the cell, with
decimal places consistent with the measuring
tool uncertainty.

7. The mean is a measure of the central tendency
of a set of 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
Raw data and the mean need to have
consistent decimal places (in line with
uncertainty of the measuring tool)
Uncertainties must be included.
Descriptive table title and number.

8. A. colubris,
15.9mm
C. latirostris,
18.8mm
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
MeanBilllength(±0.1mm)
Species of hummingbird
Graph 1: Comparing mean bill lengths in two
hummingbird species, A. colubris and C. latirostris.
Descriptive title, with graph
number.
Labeled point
Y-axis clearly labeled, with
uncertainty.
Make sure that the y-axis
begins at zero.
x-axis labeled

9. A. colubris,
15.9mm
C. latirostris,
18.8mm
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
MeanBilllength(±0.1mm)
Species of hummingbird
Graph 1: Comparing mean bill lengths in two
hummingbird species, A. colubris and C. latirostris.
From the means alone
you might conclude
that C. latirostris has a
longer bill than A.
colubris.
But the mean only tells
part of the story.

20. 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 Standard deviation can have one more
decimal place.=STDEV (highlight RAW data).
Which of the two sets of data has:
a. The longest mean bill length?
a. The greatest variability in the data?

21. 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 Standard deviation can have one more
decimal place.=STDEV (highlight RAW data).
Which of the two sets of data has:
a. The longest mean bill length?
a. The greatest variability in the data?
C. latirostris
A. colubris

22. Standard deviation is a measure of the spread of
most of the data. Error bars are a graphical
representation of the variability of data.
Which of the two sets of data has:
a. The highest mean?
a. The greatest variability in the data?
A
B
Error bars could represent standard deviation, range or confidence intervals.

23. A. colubris,
15.9mm
C. latirostris,
18.8mm
0.0
5.0
10.0
15.0
20.0
MeanBilllength(±0.1mm)
Species of hummingbird
Graph 1: Comparing mean bill lengths in two
hummingbird species, A. colubris and C. latirostris.
(error bars = standard deviation)
Title is adjusted to
show the source of the
error bars. This is very
important.
You can see the clear
difference in the size of
the error bars.
Variability has been
visualised.
The error bars overlap
somewhat.
What does this mean?

24. 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.
Results are not likely to be
significantly different from
each other.
Any difference is most likely
due to chance.
No (or very few) shared data
points within each data set.
Results are more likely to be
significantly different from
each other.
The difference is more likely
to be ‘real’.

28. A. colubris,
15.9mm
(n=10)
C. latirostris,
18.8mm
(n=10)
-3.0
2.0
7.0
12.0
17.0
22.0
MeanBilllength(±0.1mm)
Species of hummingbird
Graph 1: Comparing mean bill lengths in two
hummingbird species, A. colubris and C.
latirostris.(error bars = standard deviation)
Our results show a very small overlap
between the two sets of data.
So how do we know if the difference is
significant or not?
We need to use a statistical test.
The t-test is a statistical
test that helps us determine
the significance of the
difference between the
means of two sets of data.

31. Excel can jump straight to a value of P for our results.
One function (=ttest) compares both sets of data.
As it calculates P directly (the
probability that the difference is due
to chance), we can determine
significance directly.
In this case, P=0.00051
This is much smaller than 0.005, so
we are confident that we can:
reject H0.
The difference is unlikely to be due to
chance.
Conclusion:
There is a significant difference in bill
length between A. colubris and C.
latirostris.

32. 95% Confidence Intervals can also be plotted as error bars.
These give a clearer indication of the significance of a result:
• Where there is overlap, there is not a significant difference
• Where there is no overlap, there is a significant difference.
• If the overlap (or difference) is small, a t-test should still be carried out.
no overlap
=CONFIDENCE.NORM(0.05,stdev,samplesize)
e.g =CONFIDENCE.NORM(0.05,C15,10)

33. Interesting Study: Do “Better” Lecturers Cause More Learning?
Find out more here: http://priceonomics.com/is-this-why-ted-talks-seem-so-convincing/
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).

34. Interesting Study: Do “Better” Lecturers Cause More Learning?
Find out more here: http://priceonomics.com/is-this-why-ted-talks-seem-so-convincing/
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?

39. From MrT’s Excel Statbook.

40. http://diabetes-obesity.findthedata.org/b/240/Correlations-between-diabetes-obesity-and-physical-activity
Diabetes and obesity are ‘risk factors’ of each other.
There is a strong correlation between them,
but does this mean one causes the other?

41. Correlation does not imply causality.
Pirates vs global warming, from http://en.wikipedia.org/wiki/Flying_Spaghetti_Monster#Pirates_and_global_warming

42. Correlation does not imply causality.
Pirates vs global warming, from http://en.wikipedia.org/wiki/Flying_Spaghetti_Monster#Pirates_and_global_warming
Where correlations exist, we must then design solid scientific experiments to determine the
cause of the relationship. Sometimes a correlation exist because of confounding variables –
conditions that the correlated variables have in common but that do not directly affect each
other.
To be able to determine causality through experimentation we need:
• One clearly identified independent variable
• Carefully measured dependent variable(s) that can be attributed to change in the
independent variable
• Strict control of all other variables that might have a measurable impact on the
dependent variable.
We need: sufficient relevant, repeatable and statistically significant data.
Some known causal relationships:
• Atmospheric CO2 concentrations and global warming
• Atmospheric CO2 concentrations and the rate of photosynthesis
• Temperature and enzyme activity

43. Cartoon from: http://www.xkcd.com/552/
Correlation does not imply causation, but it does waggle its eyebrows
suggestively and gesture furtively while mouthing "look over there."

44. Bibliography / Acknowledgments