An investigation of shell An investigation of shell length variation in a length variation in a mollusc species mollusc species • A marine gastropod ( A marine gastropod ( Thersites bipartita Thersites bipartita ) has ) has been sampled from two different been sampled from two different locations: locations: – Sample A: Shells found in full marine Sample A: Shells found in full marine conditions conditions – Sample B: Shells found in brackish water Sample B: Shells found in brackish water conditions. conditions. • sample size = 10 shells sample size = 10 shells • length of the shell measured as shown length of the shell measured as shown 06/22/22 IB Biology Topic 1 Stat Notes 1
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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.
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An investigation of shell length An investigation of shell length variation in a mollusc speciesvariation in a mollusc species
• A marine gastropod (A marine gastropod (Thersites bipartitaThersites bipartita) has ) has been sampled from two different locations: been sampled from two different locations: – Sample A: Shells found in full marine conditionsSample A: Shells found in full marine conditions– Sample B: Shells found in brackish water Sample B: Shells found in brackish water
conditions. conditions.
• sample size = 10 shellssample size = 10 shells• length of the shell measured as shownlength of the shell measured as shown
04/11/23 IB Biology Topic 1 Stat Notes 1
Analysis of Gastropod DataAnalysis 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• Uncertainty Uncertainty
– all measuring devices! all measuring devices! – reflects the precision of the measurementreflects the precision of the measurement
• There should be no variation in the precision of raw There should be no variation in the precision of raw datadata
must be consistent.must be consistent.
04/11/23 IB Biology Topic 1 Stat Notes 2
1.1.1 Error bars and the 1.1.1 Error bars and the representation of variability in data.representation of variability in data.
• Biological systems are subject to a genetic Biological systems are subject to a genetic program and program and environmental variationenvironmental variation
• collect a set of data collect a set of data it shows variation it shows variation• Graphs: show variation using error barsGraphs: show variation using error bars
– show range of the data or show range of the data or – standard deviationstandard deviation
04/11/23 IB Biology Topic 1 Stat Notes 3
Mean & Range for each groupMean & Range for each group
• MarineMarine
• BrackishBrackish
04/11/23 IB Biology Topic 1 Stat Notes 4
Graph Mean & Range for each groupGraph Mean & Range for each group
• Quick Quick comparison comparison of the 2 of the 2 data setsdata sets
04/11/23 IB Biology Topic 1 Stat Notes 5
1.1.2 Calculation of Mean and Std Dev1.1.2 Calculation of Mean and Std Dev
• 3 classes of data3 classes of data• MeanMean
– arithmetic mean (avg): measure of the central arithmetic mean (avg): measure of the central tendency (middle value) tendency (middle value)
• Std DevStd Dev– Measures spread around the meanMeasures spread around the mean– Measure of variation or accuracy of measurementMeasure of variation or accuracy of measurement
04/11/23 IB Biology Topic 1 Stat Notes 6
1.1.2 Calculation of Mean and Std Dev1.1.2 Calculation of Mean and Std Dev
• Std Dev of sample = sStd Dev of sample = s
• is for the sample is for the sample notnot the the total total populationpopulation
• Pop 1. Mean = 31.4Pop 1. Mean = 31.4
s = 5.7s = 5.7• Pop 2. Mean =41.6 Pop 2. Mean =41.6
s = 4.3s = 4.3
04/11/23 IB Biology Topic 1 Stat Notes 7
Graphing Mean and Std Dev: Error BarsGraphing Mean and Std Dev: Error Bars
• Mean +/- 1 std devMean +/- 1 std dev• no overlap between no overlap between
these two populationsthese two populations
• The question being The question being considered is:considered is:– Is there a significant Is there a significant
difference between the difference between the two samples from two samples from different locations?different locations?
• oror– Are the differences in Are the differences in
the two samples just the two samples just due to chance due to chance selection? selection?
04/11/23 IB Biology Topic 1 Stat Notes 8
Graphing Mean and Std Dev: Error BarsGraphing Mean and Std Dev: Error Bars
StdDev graph compares StdDev graph compares 68% of the population68% of the population % begins to show that % begins to show that they look different.they look different.
Range graph :Range graph : misleads us to think misleads us to think the data may be similarthe data may be similar
04/11/23 IB Biology Topic 1 Stat Notes 9
1.1.3 Standard deviation and the 1.1.3 Standard deviation and the spread of values around the mean.spread of values around the mean.
1.1. StdDev is a measure of how spread out the StdDev is a measure of how spread out the data values are from the mean. data values are from the mean.
2.2. Assume: Assume: 1.1. normal distribution of values around the normal distribution of values around the
mean mean 2.2. data not skewed to either enddata not skewed to either end
3.3. 68%68% of all the data values in a sample can of all the data values in a sample can be found between the mean +/- 1 standard be found between the mean +/- 1 standard deviationdeviation
1.1.3 Standard deviation and the 1.1.3 Standard deviation and the spread of values around the mean.spread of values around the mean.
4. 4. 95%95% of all the data values in a sample can of all the data values in a sample can be found between the mean + 2s and the be found between the mean + 2s and the mean -2s.mean -2s.
04/11/23 IB Biology Topic 1 Stat Notes 12
1.1.4 Comparing means and standard 1.1.4 Comparing means and standard deviations of 2 or more samples.deviations of 2 or more samples.
Example: molluscsExample: molluscs Pop 1. Mean = 31.4 Standard deviation(s)= 5.7Pop 1. Mean = 31.4 Standard deviation(s)= 5.7
Pop 2. Mean =41.6 Standard deviation(s) = 4.3Pop 2. Mean =41.6 Standard deviation(s) = 4.3
04/11/23 IB Biology Topic 1 Stat Notes 13
1.1.4 Comparing means and standard 1.1.4 Comparing means and standard deviations of 2 or more samples.deviations of 2 or more samples.
Pop 2 has a Pop 2 has a greater mean shell lengthgreater mean shell length but but slightly narrower variationslightly narrower variation. .
WhyWhy this is the case would require further this is the case would require further observation and experiment on observation and experiment on environmental and genetic factors.environmental and genetic factors.
1.1.5 Comparing 2 samples with t-Test1.1.5 Comparing 2 samples with t-Test
For the examples you'll use in biology, tails is always 2 , and type can be:For the examples you'll use in biology, tails is always 2 , and type can be:1, paired1, paired2,Two samples equal variance2,Two samples equal variance3, Two samples unequal variance 3, Two samples unequal variance
04/11/23 IB Biology Topic 1 Stat Notes 16
Good idea to graph itGood idea to graph it
• Bar chartBar chart
• Error barsError bars
• Stats Stats
04/11/23 IB Biology Topic 1 Stat Notes 17
T-test: Are the mollusc shells from the T-test: Are the mollusc shells from the two locations significantly different? two locations significantly different?
• T-test tells you the probability (P) that the 2 T-test tells you the probability (P) that the 2 sets are basically the same. (null hypothesis) 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 higher P = more likely that the two sets are the
samesame, and that any differences are just due to , and that any differences are just due to random chancerandom chance. .
– lower P = more likely that that the two sets are lower P = more likely that that the two sets are significantly differentsignificantly different, and that any differences , and that any differences are are realreal. .
04/11/23 IB Biology Topic 1 Stat Notes 18
T-test: Are the mollusc shells from the T-test: Are the mollusc shells from the two locations significantly different? two locations significantly different?
• In biology the critical P is usually 0.05 (5%) In biology the critical P is usually 0.05 (5%) (biology experiments are expected to (biology experiments are expected to produce quite varied results)produce quite varied results)– If P > 5% then the two sets are the same 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 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 > 5At least > 5
04/11/23 IB Biology Topic 1 Stat Notes 19
Drawing Conclusions Drawing Conclusions
1. State null hypothesis & alternative hypothesis 1. State null hypothesis & alternative hypothesis (based on research ?)(based on research ?)
2. Set critical P level at P=0.05 (5%)2. Set critical P level at P=0.05 (5%)3. Write the decision rule—3. Write the decision rule— If P > 5% then the two sets are the same (i.e. accept If P > 5% then the two sets are the same (i.e. accept
the null hypothesis).the null hypothesis). If P < 5% then the two sets are different (i.e. reject If P < 5% then the two sets are different (i.e. reject
the null hypothesis).the null hypothesis).4. Write a summary statement based on the decision.4. Write a summary statement based on the decision. The null hypothesis is rejected since calculated The null hypothesis is rejected since calculated
P = 0.003 (< 0.05; two-tailed test).P = 0.003 (< 0.05; two-tailed test).5. Write a statement of results in standard English.5. Write a statement of results in standard English. There is a significant difference between the height There is a significant difference between the height
of shells in sample A and sample B. of shells in sample A and sample B. 04/11/23 IB Biology Topic 1 Stat Notes 20
• Data are not continuous Data are not continuous & normally distributed& normally distributed
• Usually scatterplot for Usually scatterplot for either type of correlationeither type of correlation
• both correlation both correlation coefficients indicate a coefficients indicate a strong + corr.strong + corr.– large females pair with large females pair with
large maleslarge males– Don’t know why, but it Don’t know why, but it
shows there is a shows there is a correlation to investigate correlation to investigate further.further.
04/11/23 IB Biology Topic 1 Stat Notes 23
Causative: Use linear regressionCausative: Use linear regression
• Fits a Fits a straight line straight line to datato data
• Gives slope Gives slope & intercept& intercept– m and c in m and c in
the equation the equation y = mx + cy = mx + c
Doesn’t PROVE causation, but Doesn’t PROVE causation, but suggests it...need further investigation! suggests it...need further investigation!