Two Classes Meet the Bell Curve December 2004 MUPGRET Workshop

Math and Science Mathematics is an integral part of science. Used every day by bench scientists to perform experiments, interpret data, and make predictions.

Statistics and Science Necessity for analyzing datasets Experiment must be well designed to be meaningful Ex. replications and controls Should know how you’ll analyze data before you start the experiment

Data Analysis Data Come in Different Types Testing How Well Data Fit Hypotheses

Data Types Yes or No (Qualitative; Discontinuous) ---Ratios of Two or More Classes How Much? (Quantitative; Continuous) ---Frequencies of Different Measurements But These Two Shade into Each Other ---Depending on Numbers Observed and on Measurement Discreteness

Statistical Testing for “Fit”  For Ratios, Chi-Square tests are often used  For Frequencies, means, standard deviations, and linear regression are often used

Chi-squared Tests if your ratios are statistically different from your expectation. Can be applied to any set of ratios. For example, do your data fit the 3:1 hypothesis? Chi-squared =  [(observed-expected) 2 /expected]

Replications Give a better estimate of the true mean. Help to remove environmental variation from measurements. Reduce noise. Reduce effect of outliers in the dataset.

Outliers

Mean Average of a group of datapoints. Treatment mean Replicate mean Grand mean

Standard Deviation The difference between the mean treatment value and the grand mean. Can think of it as the distance of the mean treatment value from the line of best fit.

Linear regression Line of best fit. Algebraic equation.

Genetic Models, Simple

One Gene, Two Genes, …

Four Genes,

Six Genes, Twelve Genes

Genetic Models, Complex

Genetic Models, What’s This?

Continuous Distributions Test if your distributions are statistically different from hypothetical distributions. For example, do your measured data fit with chance, or are they biased? Mean, Standard Deviation

The Bell Curve

Testing Selection Advance

High Heritability!

Lower Heritability!

Probability Tests the likelihood that something will or will not occur. Used extensively in everyday life. Las Vegas type gaming Lotto Insurance amortization Decisions regarding medical treatment

Everyday examples Rolling the dice 1 in 6 chance that you will roll a one with a single die. (1/6) 2 = 1/36 chance you will roll snake eyes. Playing cards 4 in 52 chance (1/13) of drawing an ace at random from a deck. What’s the chance of a full house?

Biology examples Punnett square Nucleotide frequencies along a gene are used to examine evolutionary forces. Mutation rates Testing limits and sample sizes for transgenics. DNA forensics

Mendel’s Results Parent Cross F 1 Phenotype F 2 data Round x wrinkled Round 5474 : 1850 Yellow x green Yellow 6022 : 2001 Purple x white Purple 705 : 224 Inflated x constricted pod Inflated 882 : 299 Green x yellow pod Green 428 : 152 Axial x terminal flower Axial 651 : 207 Long x short stem Long 787 : 277

Important Observations F 1 progeny are heterozygous but express only one phenotype, the dominant one. In the F 2 generation plants with both phenotypes are observed  some plants have recovered the recessive phenotype. In the F 2 generation there are approximately three times as many of one phenotype as the other.

3 : 1 Ratio The 3 : 1 ratio is the key to interpreting Mendel’s data and the foundation for the the principle of segregation.

Punnett Square A (½)a (½) A (½)AA (½ x ½ = ¼) Aa (½ x ½ = ¼) a(½)Aa (½ x ½ = ¼) aa (½ x ½ = ¼) Male Female ¼ AA :½ Aa : ¼ aa

A Molecular View ParentsF1F1 F 2 Progeny WW ww Ww¼WW ¼Ww ¼wW ¼ww 1: 2 : 1 Genotype = 3: 1 Phenotype

Alleles  People have thousands of genes.  Each gene has one to many alleles.  Each allele has a different DNA sequence.  Some DNA differences are small, some large.  Some allelic differences result in different phenotypes, e.g., brown vs. blue eyes.  Frequencies of alleles vary.

Molecularly Differing Alleles

Using and Predicting  How often is a given allele from a heterozygous parent transmitted to offspring?  How often is an allele in a population, occurring at a frequency of 0.1, found in a sample of individuals of size n?  How large a sample of individuals from a population is needed to be 95% sure of including at least one individual with an allele that is present at frequency p?