Post on 23-Feb-2016
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Big questions in Science…
What do I need to know about statistics to succeed in IB
Biology?
Statisticians…
‘..people who like figures, but don’t have the personality skills to become accountants…’
• do uncertainty, randomness and chance have a place in science?
• How should we react to them?...
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.
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 sufficientrelevant, reliable data so they can testthe 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
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).
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 (±0.1cm) 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.
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 (±0.1cm) 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.
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
A. colubris, 15.9mm
C. latirostris, 18.8mm
Graph 1: Comparing mean bill lengths in two hummingbird species, A. colubris and C. latirostris.
Species of hummingbird
Mea
n Bi
ll le
ngth
(±0
.1cm
) Descriptive title, with graph number (graph number must match table number)
Labeled point
Y-axis clearly labeled, with uncertainty.
Make sure that the y-axis begins at zero.
x-axis labeled
0.0
2.0
4.0
6.0
8.0
10.0
12.0
14.0
16.0
18.0
20.0
A. colubris, 15.9mm
C. latirostris, 18.8mm
Graph 1: Comparing mean bill lengths in two hummingbird species, A. colubris and C. latirostris.
Species of hummingbird
Mea
n Bi
ll le
ngth
(±0
.1cm
)
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.
How can I find the mean and the standard deviation on my calculator?
http://click4biology.info/c4b/1/gcStat.htm
Head over to click4biology for instructions on how to calculate the mean and standard deviation using the TI 83 plus and the TI 84 plus calculator
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 (±0.1cm) 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?
b. The greatest variability in the data?
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 (±0.1cm) 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?
b. The greatest variability in the data?
C. latirostris
A. colubris
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?
b. The greatest variability in the data?
Error bars could represent standard deviation, range or confidence intervals.
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
b. The greatest variability in the data?
Error bars could represent standard deviation, range or confidence intervals.
B
0.0
5.0
10.0
15.0
20.0
A. colubris, 15.9mm
C. latirostris, 18.8mm
Graph 1: Comparing mean bill lengths in two hummingbird species, A. colubris and C.
latirostris. (error bars = standard deviation)
Species of hummingbird
Mea
n Bi
ll le
ngth
(±0
.1cm
) 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?
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’.
-3.0
2.0
7.0
12.0
17.0
22.0
A. colubris, 15.9mm(n=10)
C. latirostris, 18.8mm(n=10)
Graph 1: Comparing mean bill lengths in two hummingbird species, A. colubris and C.
latirostris.(error bars = standard deviation)
Species of hummingbird
Mea
n Bi
ll le
ngth
(±0
.1cm
) 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.
Inferential StatisticsComparing two data sets: The T-test…
• Used to compare two normally distributed data sets (ideally with similar variances)
• A t-test is a statistic that checks if the means of 2 groups are reliably different
• Just looking at the means may show you that they are different, but doesn’t show if the difference is reliable
• We always test the NULL Hypothesis (H0)• T-test…the movie…
So what are degrees of freedom?
Degrees of freedom represent sample size.For only one group, df = n-1, where n = number of samplesUsually we are looking at 2 groups, so df = (n1 + n2) -2
P value = 0.1 0.05 0.02 0.01confidence 90% 95% 98% 99%
degrees of freedom
1 6.31 12.71 31.82 63.66 2 2.92 4.30 6.96 9.92 3 2.35 3.18 4.54 5.84 4 2.13 2.78 3.75 4.60 5 2.02 2.57 3.37 4.03 6 1.94 2.45 3.14 3.71 7 1.89 2.36 3.00 3.50 8 1.86 2.31 2.90 3.36 9 1.83 2.26 2.82 3.25
10 1.81 2.23 2.76 3.17
We can calculate the value of ‘t’ for a given set of data and compare it to critical values that depend on the size of our sample and the level of confidence we need.
Example two-tailed t-table.
What happens to the value of P as the confidence in the results increases?
What happens to the critical value as the confidence level increases?
“critical values”
P value = 0.1 0.05 0.02 0.01confidence 90% 95% 98% 99%
degrees of freedom
1 6.31 12.71 31.82 63.66 2 2.92 4.30 6.96 9.92 3 2.35 3.18 4.54 5.84 4 2.13 2.78 3.75 4.60 5 2.02 2.57 3.37 4.03 6 1.94 2.45 3.14 3.71 7 1.89 2.36 3.00 3.50 8 1.86 2.31 2.90 3.36 9 1.83 2.26 2.82 3.25
10 1.81 2.23 2.76 3.17
We can calculate the value of ‘t’ for a given set of data and compare it to critical values that depend on the size of our sample and the level of confidence we need.
Example two-tailed t-table.
We usually use P<0.05 (95% confidence) in Biology, as our data can be highly variable
“critical values”
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
t was calculated as 2.15 (this is done for you)
t cv 2.15
If t < cv, accept H0 (there is no significant difference)If t > cv, reject H0 (there is a significant difference)
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
0.05
t was calculated as 2.15 (this is done for you)
t cv 2.15
If t < cv, accept H0 (there is no significant difference)If t > cv, accept H0 (there is a significant difference)
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
2.069
0.05
t was calculated as 2.15 (this is done for you)
t cv 2.15 > 2.069
If t < cv, accept H0 (there is no significant difference)If t > cv, reject H0 (there is a significant difference)
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
2.069
0.05
t was calculated as 2.15 (this is done for you)
t cv 2.15 > 2.069
If t < cv, accept H0 (there is no significant difference)If t > cv, accept H0 (there is a significant difference)
Conclusion: “There is a significant difference in the wing spans of the two populations of birds.”
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
2.0452.045
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
“There is no significant difference in the size of shells between north-side and south-side snail populations.”
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
2.0862.086
2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.php
“There is a significant difference in the resting heart rates between the two groups of swimmers.”
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."
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?
Correlation does not imply causality.
Pirates vs global warming, from http://en.wikipedia.org/wiki/Flying_Spaghetti_Monster#Pirates_and_global_warming
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
Question check:
Question check: