1. AP Biology Math

2. 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

3. 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)

4. 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)

5. 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

6. 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)

7. 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

8. 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

9. Guaranteed chi square problem on the AP Biology Test!

10. 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!

11. 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

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

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

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

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

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

17. Guaranteed Hardy Weinberg problem on the AP Biology Test!

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

19. 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

20. 4. Calculating Slope/Rate

21. Example
Calculate the slope of the line

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

23. 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?

24. 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?

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

26. 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

27. 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

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

29. 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

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

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

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

33. 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

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

35. 8 (mean) is the highest point
9.6 ( ) 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

37. 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

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

39. hi

40. 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

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

42. hi

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