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An investigation of shell length variation in a mollusc species A marine gastropod (Thersites bipartita) has been sampled from two different locations:A marine gastropod (Thersites bipartita) has been sampled from two different locations: –Sample A: Shells found in full marine conditions –Sample B: Shells found in brackish water conditions. sample size = 10 shellssample size = 10 shells length of the shell measured as shownlength of the shell measured as shown 10/12/2014IB Biology Topic 1 Stat Notes1

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Analysis of Gastropod Data measured height of shells (ruler)measured height of shells (ruler) Units: mm + / - 1 mm (ERROR)Units: mm + / - 1 mm (ERROR) Significant digitsSignificant digits UncertaintyUncertainty –all measuring devices! –reflects the precision of the measurement There should be no variation in the precision of raw dataThere should be no variation in the precision of raw data must be consistent. 10/12/2014IB Biology Topic 1 Stat Notes2

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1.1.1 Error bars and the representation of variability in data. Biological systems are subject to a genetic program and environmental variationBiological systems are subject to a genetic program and environmental variation collect a set of data it shows variationcollect a set of data it shows variation Graphs: show variation using error barsGraphs: show variation using error bars –show range of the data or –standard deviation 10/12/2014IB Biology Topic 1 Stat Notes3

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Mean & Range for each group MarineMarine BrackishBrackish 10/12/2014IB Biology Topic 1 Stat Notes4

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Graph Mean & Range for each group Quick comparison of the 2 data setsQuick comparison of the 2 data sets 10/12/2014IB Biology Topic 1 Stat Notes5

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1.1.2 Calculation of Mean and Std Dev 3 classes of data3 classes of data MeanMean –arithmetic mean (avg): measure of the central tendency (middle value) Std DevStd Dev –Measures spread around the mean –Measure of variation or accuracy of measurement 10/12/2014IB Biology Topic 1 Stat Notes6

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1.1.2 Calculation of Mean and Std Dev Std Dev of sample = sStd Dev of sample = s is for the sample not the total populationis for the sample not the total population Pop 1. Mean = 31.4Pop 1. Mean = 31.4 s = 5.7 Pop 2. Mean =41.6 s = 4.3Pop 2. Mean =41.6 s = 4.3 10/12/2014IB Biology Topic 1 Stat Notes7

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Graphing Mean and Std Dev: Error Bars Mean +/- 1 std devMean +/- 1 std dev no overlap between these two populationsno overlap between these two populations The question being considered is:The question being considered is: –Is there a significant difference between the two samples from different locations? oror –Are the differences in the two samples just due to chance selection? 10/12/2014IB Biology Topic 1 Stat Notes8

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Graphing Mean and Std Dev: Error Bars StdDev graph compares 68% of the population % begins to show that they look different. Range graph : misleads us to think the data may be similar misleads us to think the data may be similar 10/12/2014IB Biology Topic 1 Stat Notes9

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1.1.3 Standard deviation and the spread of values around the mean. 1. StdDev is a measure of how spread out the data values are from the mean. 2. Assume: 1.normal distribution of values around the mean 2.data not skewed to either end 3. 68% of all the data values in a sample can be found between the mean +/- 1 standard deviation 10/12/2014IB Biology Topic 1 Stat Notes10

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http://www.patana.ac.th/Secondary/ Science/c4b/1/stat1.htm#gastro Animation of mean and standard deviation 10/12/2014IB Biology Topic 1 Stat Notes11

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1.1.3 Standard deviation and the spread of values around the mean. 4. 95% of all the data values in a sample can be found between the mean + 2s and the mean -2s. 10/12/2014IB Biology Topic 1 Stat Notes12

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1.1.4 Comparing means and standard deviations of 2 or more samples. Sample w/ small StdDev suggests narrow variation Sample w/ larger StdDev suggests wider variation Example: molluscs Pop 1. Mean = 31.4 Standard deviation(s)= 5.7 Pop 1. Mean = 31.4 Standard deviation(s)= 5.7 Pop 2. Mean =41.6 Standard deviation(s) = 4.3 Pop 2. Mean =41.6 Standard deviation(s) = 4.3 10/12/2014IB Biology Topic 1 Stat Notes13

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1.1.4 Comparing means and standard deviations of 2 or more samples. Pop 2 has a greater mean shell length but slightly narrower variation. Why this is the case would require further observation and experiment on environmental and genetic factors. http://www.patana.ac.th/Secondary/Science/c4b/1/stat1.htm#gastro 10/12/2014IB Biology Topic 1 Stat Notes14

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1.1.5 Comparing 2 samples with t-Test Null Hypothesis: There is no significant difference between the two samples except as caused by chance selection of data. There is no significant difference between the two samples except as caused by chance selection of data. OR OR Alternative hypothesis: There is a significant difference between the height of shells in sample A and sample B. There is a significant difference between the height of shells in sample A and sample B. http://www.patana.ac.th/Secondary/Science/c4b/1/stat1.htm#gastro 10/12/2014IB Biology Topic 1 Stat Notes15

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1.1.5 Comparing 2 samples with t-Test For the examples you'll use in biology, tails is always 2, and type can be: 1, paired 2,Two samples equal variance 3, Two samples unequal variance 10/12/2014IB Biology Topic 1 Stat Notes16

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Good idea to graph it Bar chartBar chart Error barsError bars StatsStats 10/12/2014IB Biology Topic 1 Stat Notes17

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T-test: Are the mollusc shells from the two locations significantly different? T-test tells you the probability (P) that the 2 sets are basically the same. (null hypothesis)T-test tells you the probability (P) that the 2 sets are basically the same. (null hypothesis) P varies from 0 (not likely) to 1 (certain).P varies from 0 (not likely) to 1 (certain). –higher P = more likely that the two sets are the same, and that any differences are just due to random chance. –lower P = more likely that that the two sets are significantly different, and that any differences are real. 10/12/2014IB Biology Topic 1 Stat Notes18

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T-test: Are the mollusc shells from the two locations significantly different? In biology the critical P is usually 0.05 (5%) (biology experiments are expected to produce quite varied results)In biology the critical P is usually 0.05 (5%) (biology experiments are expected to produce quite varied results) –If P > 5% then the two sets are the same (i.e. accept the null hypothesis).(i.e. accept the null hypothesis). –If P < 5% then the two sets are different (i.e. reject the null hypothesis).(i.e. reject the null hypothesis). For t test, # replicates as large as possibleFor t test, # replicates as large as possible –At least > 5 10/12/2014IB Biology Topic 1 Stat Notes19

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Drawing Conclusions 1. State null hypothesis & alternative hypothesis (based on research ?) 2. Set critical P level at P=0.05 (5%) 3. Write the decision rule— If P > 5% then the two sets are the same (i.e. accept the null hypothesis). If P > 5% then the two sets are the same (i.e. accept the null hypothesis). If P < 5% then the two sets are different (i.e. reject the null hypothesis). If P < 5% then the two sets are different (i.e. reject the null hypothesis). 4. Write a summary statement based on the decision. The null hypothesis is rejected since calculated P = 0.003 (< 0.05; two-tailed test). The null hypothesis is rejected since calculated P = 0.003 (< 0.05; two-tailed test). 5. Write a statement of results in standard English. There is a significant difference between the height of shells in sample A and sample B. There is a significant difference between the height of shells in sample A and sample B. 10/12/2014IB Biology Topic 1 Stat Notes20

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1.1.6 Correlation & Causation Sometimes you’re looking for an association between variables.Sometimes you’re looking for an association between variables. Correlations see if 2 variables vary togetherCorrelations see if 2 variables vary together +1 = perfect positive correlation 0 = no correlation 0 = no correlation -1 = perfect negative correlation -1 = perfect negative correlation Relations see how 1 variable affects anotherRelations see how 1 variable affects another 10/12/2014IB Biology Topic 1 Stat Notes21

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Pearson correlation (r) Data are continuous & normally distributedData are continuous & normally distributed 10/12/2014IB Biology Topic 1 Stat Notes22

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Spearman’s rank-order correlation (r s) Data are not continuous & normally distributedData are not continuous & normally distributed Usually scatterplot for either type of correlationUsually scatterplot for either type of correlation both correlation coefficients indicate a strong + corr.both correlation coefficients indicate a strong + corr. –large females pair with large males –Don’t know why, but it shows there is a correlation to investigate further. 10/12/2014IB Biology Topic 1 Stat Notes23

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Causative: Use linear regression Fits a straight line to dataFits a straight line to data Gives slope & interceptGives slope & intercept –m and c in the equation y = mx + c Doesn’t PROVE causation, but suggests it...need further investigation! 10/12/2014IB Biology Topic 1 Stat Notes24

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