AP Biology Math

What Math Should I Know? 1. Chi Square 2. Hardy Weinberg 3. Mean (average) 4. Calculating Rate/Slope on a graph 5. Probability 6. Surface Area to Volume Ratio 7. Gene Linkage/ Percent Recombination 8. Standard Deviation 9. Standard Error 10. Gibbs Free Energy

1. Chi Square Test This test is used to determine the validity of an experiment. For example, if you flip a coin 100 time and get 53 heads and 47 tails, are these results valid? Usually there is a null hypothesis that states the results are valid/good We want to accept the null hypothesis If we reject the null hypothesis, this means that the results are due to something else (bad lab practice, contamination, etc)

Equation: Remember, the X2 does not mean to take the square root at the end, X2 just stands for the chi square value 0 – observed (data given in the experiment) E = expected (usually due to punnett squares)

For example, flipping a coin would have 1 degree of freedom Degrees of freedom: N-1 N refers to the number of phenotypes For example, flipping a coin would have 1 degree of freedom 2 phenotypes (head/tail) – 1 A dihybrid punnett square would have 3 degrees of freedom 4 phenotypes - 1

Using the chi square value and the degrees of freedom, you can determine if the results are due to random chance (which means the data is good) or if the results are bad For the table below, a probability greater that .10 (10%) is considered valid results (good)

Example If you flip a coin 100 times and 53 are heads and 47 are tails, calculate the chi square value and state is these results are valid

Example If you cross a colorblind male with a hetero normal vision female, you get: 23 colorblind females 25 normal visioned males 28 colorblind males 24 normal visioned females Calculate the chi square value and determine if these results are valid

Guaranteed chi square problem on the AP Biology Test!

2. Hardy Weinberg These equations are used to mathematically observe if evolution is taking place. If there is any change in allele frequency, evolution is happening!

Equations p + q = 1 p2 + 2pq + q2 = 1 p is the dominant allele q is the recessive allele If you see the word allele, its referring to either p or q p2 + 2pq + q2 = 1 p2 is the homozygous dominant phenotype 2pq is the heterozygous dominant phenotype q2 is the recessive phenotype If you see any type of phenotype (blue, tall, etc) its referring to this equation

Example: If p is .6, what is q?

Example: If q2 is .36, what is p2?

Example If the white phenotype has a frequency of .81, what is the heterozygous frequency?

Example If there are 70 tall plants and 30 short plants (tall is dominant) what is the dominant allele frequency?

Example If there are 150 blue turtles in a population of 200 individuals, how many are heterozygous blue? Blue is dominant.

Guaranteed Hardy Weinberg problem on the AP Biology Test!

3. Mean (average) Add up all the data points and divide by the number of data points

What is the mean value of the following grade point averages: Example What is the mean value of the following grade point averages: 3.7 3.5 3.9 2.4 3.0 3.2 2.9 3.6

4. Calculating Slope/Rate

Example Calculate the slope of the line

5. Probability Used to predict outcomes Usually used in punnett squares

Adding Probability Example When the 2 data points are mutually exclusive Basically not predicting 2 things happening in a row The first data point is dependent on the second data point Example If you cross 2 heterozygous tall people, what is the probability of having two heterozygous genotypes?

Multiplying probability By far the more common application This happens when 2 data points are independent of each other Example What are the odds of having 4 boys in a row? What is the probability of flipping a coin 3 times and getting tails each time?

Example If you cross a hetero A+ and a hetero B+, what is the probability (odds) of having a B- kid?

6. Cell Surface to Volume Ratio Used in the cell lab The smaller the cell, the greater the ratio is It’s a little counter intuitive, but just think about it Example Calculate the surface area to volume ratio for a cube that is .6cm and a cube that is 4cm

7. Gene Linkage / Percent Recombination This type of math is used to determine the distance between 2 genes on the same chromosome. You can tell if the problem is a gene linkage problem if the results are random and not expected Basically doesn’t fit a 3:1 ratio or 1:2:1 ratio, etc Always assume the the dominant are linked and the recessive are linked – these are the expected Therefore the heterozygous would be considered the recombinants Generally, the 2 biggest numbers are the expected and the 2 smallest numbers are the recombinants

Equation: (total number of recombinants) / (total number of all data) Multiple by 100 to get the % recombination

You cross a hetero tall, black (TtBb) with short blonde (ttbb) Example: tall is dominant, short is rec. Black is dominant, blonde is rec. You cross a hetero tall, black (TtBb) with short blonde (ttbb) You know this is linkage because a dihybrid punnett square would result in 25% for all four phenotypes, that is not represented in the data below Data is: Tall, black = 101 Short, blonde = 103 Tall blonde = 21 Short, black = 23 What is the % recombination

Tall, black and short, blonde are the expected Tall, blonde and short, black are the recombinants (21 + 23) / (21 + 23 + 101 + 103) = 44/248 = .177 x 100 = 17.7% recombination (this is also how far apart the genes are on the chromosome)

8. Standard Deviation Standard deviation is the measure of dispersion/fluctuation of ALL data around the mean

What is the standard deviation for the following data: Example: What is the standard deviation for the following data: 10 7 6 8 9

Solution: Standard deviation = 1.6 The mean (average) is 8 (10-8)2/4 + (7-8) 2/4 + (6-8) 2/4 + (8-8) 2/4 + (9-8) 2/4 =2.5 Take the square root of 2.5 Standard deviation = 1.6

Make a graph: 68% of all data falls within 1 standard dev of the mean Should look like a bell curve

8 (mean) is the highest point 9.6 (8 + 1.6) is one standard dev along with 6.4 (8-1.6) 68% of all data falls between 6.4 and 9.6 11.2 [8 + (2 x 1.6)] and 4.8 [8 – (2 x 1.6)] 95% of all data falls between 4.8 and 11.2 12.8 [8 + (3 x 1.6)] and 3.2 [8 – (3 x 1.6)] 99% of all data falls between 3.2 and 12.8

9. Standard Error This shows how a particular data point varies from the mean How are standard deviation and standard error different? Error refers to a single or couple of data points while deviation refers to many data points They are closely related, thus the equation

Example (large population size) Determine the standard deviation and standard error for the following people with the following weights 150 168 145 190 155 142

hi

Standard deviation = 21 Standard error = 8.4 Standard deviation tells us that 68% of all data falls between 147 lbs and 189 lbs (the mean is 168) Standard error tells us that each individual data point has a potential error of 8.4 lbs

Example (small pop size) Determine the standard deviation and standard error for the following population size 130 155 190

hi

Standard deviation = 30 Standard error = 17.6 Clearly a small population is going to have more fluctuation for its data points