Two Classes Meet the Bell Curve

December 2004MUPGRET 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

0

10

20

30

40

50

60

Frequency

One Gene, Two Genes, …

Four Genes,

Six Genes, Twelve Genes

Genetic Models, Complex

0

5

10

15

20

25

30

35

40

45

50

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 ResultsParent CrossParent Cross FF11

PhenotypePhenotypeFF22 data data

Round x Round x wrinkledwrinkled

RoundRound 5474 : 5474 : 18501850

Yellow x greenYellow x green YellowYellow 6022 : 6022 : 20012001

Purple x whitePurple x white PurplePurple 705 : 224705 : 224

Inflated x Inflated x constricted podconstricted pod

InflatedInflated 882 : 299882 : 299

Green x yellow Green x yellow podpod

GreenGreen 428 : 152428 : 152

Axial x terminal Axial x terminal flowerflower

AxialAxial 651 : 207651 : 207

Long x short Long x short stemstem

LongLong 787 : 277787 : 277

Important Observations F1 progeny are heterozygous but express

only one phenotype, the dominant one. In the F2 generation plants with both

phenotypes are observedsome plants have recovered the recessive phenotype.

In the F2 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 SquareA (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 aa¼ AA :½ Aa : ¼ aa

A Molecular View

Parents F1 F2 Progeny

WW ww Ww ¼WW 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?