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Statistical Analysis IB Diploma Biology Modified by Christopher Wilkinson from Stephen Taylor Image: 'Hummingbird Checks Out Flower' http://www.flickr.com/photos/25659032@N07/7200193254 http://www.flickr.com/photos/25659032@N07/7200193254 Found on flickrcc.net

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Assessment StatementsObj. 1.1.1 State that error bars are a graphical representation of the variability of data. 1 1.1.2 Calculate the mean and standard deviation of a set of values Using Excel Using your graphing calculator 2 1.1.3 State that the term standard deviation (s) is used to summarize the spread of values around the mean, and that 68% of all data fall within (±) 1 standard deviation of the mean. 1 1.1.4 Explain how the standard deviation is useful for comparing the means and the spread of data between two or more samples. 3 1.1.5 Deduce the significance of the difference between two sets of data using calculated values for t and the appropriate tables. We will also do this with P values using Excel in lab reports. 3 1.1.6 Explain that the existence of a correlation does not establish that there is a causal relationship between two variables. 3 Assessment statements from: Online IB Biology Subject GuideOnline IB Biology Subject GuideCommand terms: http://i-biology.net/ibdpbio/command-terms/http://i-biology.net/ibdpbio/command-terms/

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This Excel StatbookStatbook has guidance and ‘live’ examples of tables, graphs and statistical tests.

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Hummingbirds are nectarivores (herbivores that feed on the nectar of some species of flower). 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.wikimedia commonsDick Daniels In return for food, they pollinate the flower. This is an example of mutualism – benefit for all.

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Researchers studying comparative anatomy collect data on bill-length in two species of hummingbirds: Archilochus colubris (red-throated hummingbird) and Cynanthus latirostris (broadbilled hummingbird). Photo: Archilochus colubris (male), wikimedia commons, by Joe Schneidwikimedia commons To do this, they need to collect sufficient relevant, reliable data so they can test the Null hypothesis (H 0 ) that: “there is no significant difference in bill length between the two species.”

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The sample size must be large enough to provide sufficient reliable data and for us to carry out relevant statistical tests for significance. Photo: Broadbilled hummingbird (wikimedia commons).wikimedia commons We must also be mindful of uncertainty in our measuring tools and error in our results.

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Measurement and Uncertainty What was the uncertainty of the temperature from the diffusion lab? (+/- 0.5 degrees Celsius) What was the uncertainty of the mass? (+/-0.01g)

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1.1.2: Calculate the mean and standard deviation of a set of values Using Excel Using your graphing calculator.

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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) nA. colubrisC. latirostris 113.017.0 214.018.0 315.018.0 415.018.0 515.019.0 616.019.0 716.019.0 818.020.0 918.020.0 1019.020.0 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.

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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) nA. colubrisC. latirostris 113.017.0 214.018.0 315.018.0 415.018.0 515.019.0 616.019.0 716.019.0 818.020.0 918.020.0 1019.020.0 Mean 15.918.8 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.

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DELETE X DELETE X

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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

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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.

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1.1.3: State that the term standard deviation (s) is used to summarize the spread of values around the mean, and that 68% of all data fall within (±) 1 standard deviation of the mean.

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1.1.4: Explain how the standard deviation is useful for comparing the means and the spread of data between two or more samples..

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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) nA. colubrisC. latirostris 113.017.0 214.018.0 315.018.0 415.018.0 515.019.0 616.019.0 716.019.0 818.020.0 918.020.0 1019.020.0 Mean 15.918.8 s 1.911.03 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?

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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) nA. colubrisC. latirostris 113.017.0 214.018.0 315.018.0 415.018.0 515.019.0 616.019.0 716.019.0 818.020.0 918.020.0 1019.020.0 Mean 15.918.8 s 1.911.03 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

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1.1.1: State that error bars are a graphical representation of the variability of data.. Error bars can be used to show either the range or the standard deviation.

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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.

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Put the error bars for standard deviation on our graph.

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Delete the horizontal error bars

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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?

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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’.

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1.1.5:Deduce the significance of the difference between two sets of data using calculated values for t and the appropriate tables. We will also do this with P values using Excel in l ab reports.

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Our results show a very small overlap between the two sets of data. The t-test is a statistical test that helps us determine the significance of the difference between the means of two sets of data. So how do we know if the difference is significant or not? We need to use a statistical test.

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P value =0.10.050.020.01 confidence 90%95%98%99% degrees of freedom 16.3112.7131.8263.66 22.924.306.969.92 32.353.184.545.84 42.132.783.754.60 52.022.573.374.03 61.942.453.143.71 71.892.363.003.50 81.862.312.903.36 91.832.262.823.25 101.812.232.763.17 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. “critical values” 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?

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P value =0.10.050.020.01 confidence 90%95%98%99% degrees of freedom 16.3112.7131.8263.66 22.924.306.969.92 32.353.184.545.84 42.132.783.754.60 52.022.573.374.03 61.942.453.143.71 71.892.363.003.50 81.862.312.903.36 91.832.262.823.25 101.812.232.763.17 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. “critical values” We usually use P<0.05 (95% confidence) in Biology, as our data can be highly variable

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2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php

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t was calculated as 2.15 (this is done for you) t cv 2.15 If t < cv, accept H 0 (there is no significant difference) If t > cv, accept H 0 (there is a significant difference) 2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php

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0.05 t was calculated as 2.15 (this is done for you) t cv 2.15 If t < cv, accept H 0 (there is no significant difference) If t > cv, accept H 0 (there is a significant difference) 2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php

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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 H 0 (there is no significant difference) If t > cv, accept H 0 (there is a significant difference) 2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php

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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 H 0 (there is no significant difference) If t > cv, accept H 0 (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.phphttp://www.medcalc.org/manual/t-distribution.php

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2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php

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2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php

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2.045 2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://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.”

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2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php

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2.086 2-tailed t-table source: http://www.medcalc.org/manual/t-distribution.phphttp://www.medcalc.org/manual/t-distribution.php “There is a significant difference in the resting heart rates between the two groups of swimmers.”

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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 H 0. The difference is unlikely to be due to chance. Conclusion: There is a significant difference in bill length between A. colubris and C. latirostris.

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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)

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

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

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P value =0.10.050.020.010.005 confidence 90%95%98%99%99.50% degrees of freedom 16.3112.7131.8263.66127.34 22.924.306.969.9214.09 32.353.184.545.847.45 42.132.783.754.605.60 52.022.573.374.034.77 61.942.453.143.714.32 71.892.363.003.504.03 81.862.312.903.363.83 91.832.262.823.253.69 101.812.232.763.173.58 degrees of freedom 111.802.202.723.113.50 121.782.182.683.053.43 131.772.162.653.013.37 141.762.142.622.983.33 151.752.132.602.953.29 161.752.122.582.923.25 171.742.112.572.903.22 181.732.102.552.883.20 191.732.092.542.863.17 201.722.092.532.853.15 degrees of freedom 211.722.082.522.833.14 221.722.072.512.823.12 231.712.072.502.813.10 241.712.062.492.803.09 251.712.062.492.793.08 261.712.062.482.783.07 271.702.052.472.773.06 281.702.052.472.763.05 291.702.052.462.763.04 301.702.042.462.753.03 degrees of freedom 311.702.042.452.743.02 321.692.042.452.743.02 331.692.032.442.733.01 341.692.032.442.733.00 351.692.032.442.723.00 361.692.032.432.722.99 371.692.032.432.722.99 381.692.022.432.712.98 391.682.022.432.712.98 401.682.022.422.702.97

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Dog fleas jump higher that cat fleas, winner of the IgNobel prize for Biology, 2008. http://www.youtube.com/watch?v=fJEZg4QN760

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1.1.5:Explain that the existence of a correlation does not establish that there is a causal relationship between two variables.

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Cartoon from: http://www.xkcd.com/552/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.” Correlation-cause conundrum: Link between contraceptive pill use in women and prostate cancer in men. Huh?Link between contraceptive pill use in women and prostate cancer in men. Huh?

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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?

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

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Correlation does not imply causality. Pirates vs global warming, from http://en.wikipedia.org/wiki/Flying_Spaghetti_Monster#Pirates_and_global_warminghttp://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 CO 2 concentrations and global warming Atmospheric CO 2 concentrations and the rate of photosynthesis Temperature and enzyme activity

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Video: Choosing which Statistical test to use

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Flamenco Dancer, by Steve Corey http://www.flickr.com/photos/22016744@N06/7952552148

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