Statistics

Statistical Analysis
IB Diploma Biology

Objectives of this Unit:
 Types of Data, Types of Graphs, Applications and Statistics to match your data
o Bar Graphs, Line Graph, Scatter Plot, Histogram, Pie Chart
o Mean, S.D., Regression, Chi Square Analysis, Anova
 State that error bars are a graphical representation of the variability of data
o Range and standard deviation show the variability/spread in the data
 Calculate the mean and standard deviation of a set of values
 Using Excel formulas
o Given a mean and S.D. state the range for different parameters
 State the term standard deviation is used to summarize the spread of values around the mean
o 68% of all data +/- 1 standard deviation, 95% within 2 SD
 Explain how S.D. is useful for comparing the means and spread of data between two or more
samples
o Greater S.D. shows greater variability of data
o This can be used to inter reliability in methods or results BUT in Biology we also expect
variability
 Deduce the significance of the difference between two sets of data using calculated values for t and
tables
o Using t value and t table and critical values
o Directly calculating P values using excel in lab reports
o Difference between P and T
 Explain that correlation does not establish that there is a causal relationship between two variables
 Proper Lab Format
 Designing Lab Process

What are statistics?
• Statistics are numbers used to:
Describe and draw conclusions about DATA
• These are called descriptive (or “univariate”) and
inferential (or “analytical”) statistics, respectively.

Variables
• A variable is anything we can measure/observe
• Three types:
– Continuous: values span an uninterrupted range (e.g. height)
– Discrete: only certain fixed values are possible (e.g. counts)
– Categorical: values are qualitatively assigned (e.g. low/med/hi)
• Dependence in variables:
“Dependent variables depend on independent ones”
– Independent variable – variable you are changing
– Dependent variable – variable you measure to see result
– Controlled variables – variables that can also impact the
dependent variable that you identify as needed to not vary
*** Experimental Control – NOT the same as controlled variables

Descriptive statistics
Numerical
– Mean
– Variance
• Standard deviation
• Standard error
– Median
– Mode
– Skew
– etc.
Graphical
– Histogram
– Boxplot
– Scatterplot
– etc.
Techniques to summarize data

What graph to use ?
Line Scatter Histogram Bar
Appropriat
e for data
when:
Important
Features
Include
Sample and
other notes
Outlier - An outlier is an observation that lies an abnormal distance from other values in
a random sample from a population. In a sense, this definition leaves it up to the analyst
(or a consensus process) to decide what will be considered abnormal. Before abnormal
observations can be singled out, it is necessary to characterize normal observations.

Numeric Descriptive Statistics

S
The Mean:
Most important measure of “central tendency”
Xi
i=1
N
m =
N
Population Mean

S
The Mean:
Most important measure of “central tendency”
i=1
n
n
Sample Mean
X =
Xi

Additional central tendency measures
M = X(n+1)/2 (n is odd)
Median: the 50th percentile
(n is even)
Xn/2 + X(n/2)+1
2
M =
Mode: the most common value
1, 1, 2, 4, 6, 6, 6, 7, 7, 7, 7, 8, 8, 9, 9, 10, 12, 15
Which to use: mean, median or mode?

Variance:
Most important measure of “dispersion”
s2 = S
N
Population Variance
(Xi - µ)2

Variance:
Most important measure of “dispersion”
s2 = S
n - 1
Sample Variance
(Xi - X)2
From now on, we’ll ignore sample vs. population. But remember:
We are almost always interested in the population, but can measure only a sample.

“Graphical Statistics”
Lets look deeper into graphs now

The Friendly Histogram
• Histograms represent the distribution of data
• They allow you to visualize the mean, median,
mode, variance, and skew at once!

Constructing a Histogram is Easy
X (data)
7.4
7.6
8.4
8.9
10.0
10.3
11.5
11.5
12.0
12.3
Histogram of X
Value
6 8 10 12 14
0
1
2
3
Frequency
(count)

Interpreting Histograms
• Mean?
• Median?
• Mode?
• Standard
deviation?
• Variance?
• Skew?
(which way does the
“tail” point?)
• Shape?
Value
0 20 40 60
0
20
40
Frequency

Interpreting Histograms
• Mean?
= 9.2
• Median? = 6.5
• Mode?
= 3
• Standard
deviation? = 8.3
• Variance?
• Skew?
(which way does the
“tail” point?)
• Shape?
Value
0 20 40 60
0
20
40
Frequency

An alternative:
Boxplots
0 20 40
0
20
40
Value
Frequency

Boxplots also
summarize a lot
of information...
Within each sample:
6
5
4
3
2
1
SC Ba Se Es Da Is Fe
Island
Weight(kg)
Compared across samples:
25% percentile
75% percentile
Median
“Outliers”

The Normal Distribution
aka “Gaussian” distribution
• Occurs frequently in nature
• Especially for measures that
are based on sums, such as:
– sample means
– body weight
– “error”
• Many statistics are based on
the assumption of normality
– You must make sure your data
are normal, or try something
else!
Sample normal data:
Histogram + theoretical distribution
(i.e. sample vs. population)

Properties of the Normal Distribution
• Symmetric
Mean = Median = Mode
• Theoretical percentiles can be computed exactly
~68% of data are within 1 standard deviation of the mean
>99% within 3 s.d.
“skinny tails”

>99%
~95%
~68%
Important!
Handy!
Amazing!

What if my data aren’t Normal?
• It’s OK!
• Although lots of data are Gaussian (because of the CLT),
many simply aren’t.
– Example: Fire return intervals
Time between fires (yr)
• Solutions:
– Transform data to make it
normal (e.g. take logs)
– Use a test that doesn’t
assume normal data
• Don’t worry, there are plenty
• Especially these days...
• Many stats work OK as long as data are “reasonably” normal

Inference: the process by which we draw
conclusions about an unknown based on
evidence or prior experience.
In statistics: make conclusions about a
population based on samples taken from
that population.
Important: Your sample must reflect the
population you’re interested in, otherwise
your conclusions will be misleading!

Statistical Hypotheses
• Should be related to a scientific hypothesis!
• Very often presented in pairs:
– Null Hypothesis (H0):
the “boring” hypothesis of “no difference”
– Alternative Hypothesis (HA)
the interesting hypothesis of “there is an effect”
• Statistical tests attempt to (mathematically)
reject the null hypothesis

Significance
• Your sample will never match H0 perfectly,
even when H0 is in fact true
• The question is whether your sample is
different enough from the expectation under
H0 to be considered significant
• If your test finds a significant difference, then
you reject H0.

p-Values Measure Significance
The p-value of a test is the probability of observing data
at least as extreme as your sample, assuming H0 is true
• If p is very small, it is unlikely that H0 is true
(in other words, if H0 were true, your observed sample would be unlikely)
• How small does p have to be?
– 0.05 is a common cutoff
• If p<0.05, then there is less than 5% chance that you would observe
your sample if the null hypothesis was true.

‘Proof’ in statistics
• Failing to reject (i.e. “accepting”) H0 does not
prove that H0 is true!
• And accepting HA doesn’t prove that HA is true
either!
Why?
• Statistical inference tries to draw conclusions
about the population from a small sample
– By chance, the samples may be misleading
– Example: if you always accept H0 at p=0.05, then
1 in 20 times you will be wrong!

Play it Safe
Avoid using the term Prove in your labs
Instead say “the data accepts or supports” the
hypothesis
Watch out for reaching – classic student error,
stick to the scope of your lab data in your
conclusions, this is not your life work.

“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.
Image: 'Transverse section of part of a stem of a Dead-nettle (Lamium sp.) showing+a+vascular+bundle+and+part+of+the+cortex'
http://www.flickr.com/photos/71183136@N08/6959590092 Found on flickrcc.net

“Which medicine should I prescribe?”
Image from: http://www.msf.org/international-activity-report-2010-sierra-leone
Donate to Medecins Sans Friontiers through Biology4Good: http://i-biology.net/about/biology4good/

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?

Means (averages) in Biology are almost
never good enough. Biological systems
(and our results) show variability.
Which would you choose now?

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 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

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 (±0.1mm)
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.

The mean is a measure of the central tendency
of a set of data.
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.

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.
x-axis labeled

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
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.

http://click4biology.info/c4b/1/gcStat.htm

http://mathbits.com/MathBits/TINSection/Statistics1/Spreadsheet.html

Standard deviation is a measure of the spread of
most of the data.
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?

most of the data.
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).
a. The longest mean bill length?
C. latirostris
A. colubris

most of the data. Error bars are a graphical
representation of the variability of data.
a. The highest mean?
A
B
Error bars could represent standard deviation, range or confidence intervals.

Put the error bars for standard deviation on our graph.

Put the error bars for standard deviation on our graph.
Delete the horizontal error bars

A. colubris,
15.9mm
C. latirostris,
18.8mm
0.0
5.0
10.0
15.0
20.0
(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?

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’.

A. colubris,
15.9mm
(n=10)
C. latirostris,
18.8mm
(n=10)
-3.0
2.0
7.0
12.0
17.0
22.0
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.

The Null Hypothesis (H0):
“There is no significant
difference.”
This is the ‘default’ hypothesis that we always test.
In our conclusion, we either accept the null hypothesis or reject it.
A t-test can be used to test whether the difference between two means is significant.
• If we accept H0, then the means are not significantly different.
• If we reject H0, then the means are significantly different.
Remember:
• We are never ‘trying’ to get a difference. We design carefully-controlled experiments and
then analyse the results using statistical analysis.

P value = 0.1 0.05 0.02 0.01
confidence 90% 95% 98% 99%
degreesoffreedom
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.
“Degrees of Freedom (df)” is
the total sample size minus
two.
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 0.01
confidence 90% 95% 98% 99%
degreesoffreedom
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.
“Degrees of Freedom (df)” is
the total sample size minus
two*.
We usually use P<0.05 (95%
confidence) in Biology, as our
data can be highly variable
*Simple explanation: we are working in
two directions – within each population
and across populations.
“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)

0.05
t cv
2.15

2.069
0.05
t cv
2.15 > 2.069

2.069
0.05
t cv
2.15 > 2.069
Conclusion:
“There is a significant difference in the wing spans of
the two populations of birds.”

2.0452.045
“There is no significant difference in the size of shells
between north-side and south-side snail populations.”

2.086
2.086
“There is a significant difference in the resting heart
rates between the two groups of swimmers.”

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.

Two tails: we assume data are normally distributed, with two ‘tails’ moving away from mean.
Type 2 (unpaired): we are comparing one whole population with the other whole population.
(Type 1 pairs the results of each individual in set A with the same individual in set B).

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)

Error bars can have very different purposes.
Standard deviation
• You really need to know this
• Look for relative size of bars
• Used to indicate spread of most
of the data around the mean
• Can imply reliability of data
95% Confidence Intervals
• Adds value to labs where we are
looking for differences.
• Look for overlap, not size
• Overlap  no sig. diff.
• No overlap  sig. dif.

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).
n=21 n=21

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

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

http://diabetes-obesity.findthedata.org/b/240/Correlations-between-diabetes-obesity-and-physical-activity
Interpreting Graphs: See – Think – Wonder
See: What is factual about the graph?
• What are the axes?
• What is being plotted
• What values are present?
Think: How is the graph interpreted?
• What relationship is present?
• Is cause implied?
• What explanations are possible and
what explanations are not possible?
Wonder: Questions about the graph.
• What do you need to
know more about?
See – Think - Wonder
Visible Thinking Routine

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

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."
Check out these funny “Correlations”

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

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