Two Classes Meet the Bell Curve December 2004 MUPGRET Workshop
Dec 19, 2015
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 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.
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.
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
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
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.
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?