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?”

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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?

“Which medicine should I prescribe?”

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Means (averages) in Biology are almost never good enough. Biological systems (and our results) show variability.

Which would you choose now?

Evaluate dataEvaluate data Make sense from the numbersMake sense from the numbers

Why do we need to do Why do we need to do statistical analysis?statistical analysis?

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 sufficientrelevant, reliable data so they can testthe 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

- 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

　 　 　 　

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.

IV. Mean (Average)

n = sample size. The bigger the better. In this case n=10 for each group.

=AVERAGE (highlight raw data)

Formula

* n = number of pieces of datum;* x = each piece of datum

Calculating the Mean (Average)

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

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…

What % of the What % of the population is found population is found within within oneone standard standard

deviation of the mean?deviation of the mean?

68%68%

What % fall within What % fall within twotwo standard deviations of standard deviations of

the mean? the mean?

96%96%

What % fall with What % fall with threethree standard deviations of standard deviations of

the mean?the mean?

99.8%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

　 　 　 　

Standard deviation can have one more decimal place.

Which of the two sets of data has:

b.The greatest variability in the data? - calculate the standard deviation

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

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

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.

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’.

Our results show a very small overlap between the two sets of data.

The data has a greater chance of being significant.

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?

n=21 n=21

Interesting Study: Do “Better” Lecturers Cause More Learning?

Find out more here: http://priceonomics.com/is-this-why-ted-talks-seem-so-convincing/

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

VII.VII. What does Significantly What does Significantly Different mean?Different mean?

Most scientists agree that 2 Most scientists agree that 2 deviations above or below the mean deviations above or below the mean indicates that you are significantly indicates that you are significantly different.different.

Significantly different means the Significantly different means the difference is difference is notnot due to chanc due to chancee

The Null Hypothesis would be The Null Hypothesis would be rejected.rejected.

Are any of these people Are any of these people significantly different?significantly different?

10, 13, 15, 20, 24, 28, 30

No!No!

Only individual with Only individual with scores above scores above 34.2234.22 or or below below 5.785.78 would be would be

considered significantly considered significantly different.different.