Questions related to quantitative skills and biostatistics pop up all over the place. Even if you think you are a whiz, check out the formula pages near the back of this book. Expect at least a couple questions using formulas from these sheets.
It is often easy to see patterns in data, but it is not always easy to determine if a pattern is valid or significant. Quantitative data analysis is the first step in figuring this out. In this chapter, we will review how data can be summarized, presented, and tested for validity. This depends on what kind of question was being asked at the beginning of the experiment. When you’re reading this chapter, pay attention to which techniques are used when. On the AP Biology Exam, you should be able to justify which techniques are best.
This list is a good reference for the types of quantitative questions you should be able to tackle. We will not walk though all of these in detail, but you will find some examples in the end-of-chapter drill and in the practice tests. Use the formula sheets to familiarize yourself with these concepts.
Must-Know Types of Quantitative Problems
☐ Hardy-Weinberg Equilibrium
☐ Water Potential/Osmosis
☐ Energy Pyramids/Biomass
☐ Chi-squared Analysis
☐ Gene Linkage Analysis
☐ Inheritance I Probability
☐ Reading from Graphs
☐ Predicting from Graphs
☐ Dilutions of Solutions
☐ Population Growth
Instead of presenting raw data, scientists use descriptive statistics and graphs to summarize large datasets, present patterns in data, and communicate results. Descriptive statistics summarize and show variation in the data. There are many types of descriptive statistics, including measures of central tendency (such as mean, median, and mode) and measures of variability (such as standard deviation, standard error, and range). The AP Biology Equations and Formulas sheet contains definitions for mean, median, mode, and range. It also includes equations for mean, standard deviation, and standard error. That said, you will not need to calculate standard deviation or standard error. Instead, focus on understanding how these values are used.
Descriptive statistics are used to summarize the data collected from samples, but may also describe the entire population you’re trying to study. Experiments are designed to include a sample, which is a subset of the population being studied. The best experiments use random sampling, which makes sure there is no bias in picking which individuals from the population will be included in the sample. It is important that the sample is big enough that the data from the experiment is a good representation of what would happen if the whole population were measured.
Graphs are visual representations of data and are often used to reveal trends that might not be obvious by looking at a table of numbers. There are six types of graphs you should be familiar with:
Bar graph
Pie graph
Histogram
Line graph
Box-and-whisker plot
Scatterplot
Each of these is described more in the sections that follow. Pay attention to which graph is used when. On the AP Biology Exam, you may need to decide which graph is best and then create it. You may also have to analyze graphs given to you on the exam.
A good graph must include the following things:
a good title that describes what the experiment was and what was measured
axes labeled with numbers, labels and units, and index marks
a frame or perimeter
data points that are clearly marked (e.g., easily identifiable lines or bars)
Data can be quantitative (based on numbers or amounts that can be measured or counted) or qualitative (data that is descriptive, subjective, or difficult to measure). For example, behavioral observations are qualitative. Most data on the AP Biology Exam will be quantitative, with either counts or measurements.
Count data are generated by counting the number of items that fit into a category. For example, you could count the number of organisms with a particular phenotype or the number of animals in one habitat versus another. Count data also include data that is collected as percentages or the results of a genetic cross. This type of data is usually summarized in a bar graph or a pie graph. For example, suppose a mixed population of E. coli were plated on growth plates. Some cells contained a β-galactosidase marker and grew in blue colonies, while others did not contain the marker and grew in white colonies. The colonies were counted after 24 hours of growth.
In another example, suppose fly populations were monitored in northern Maine to determine if population size varies over the warm months of the year.
Count data can be analyzed with hypothesis testing, which will be described later in this chapter.
Measurements are continuous, meaning there is an infinite number of potential measurements over a given range. Size, height, temperature, weight, and response rate are all measurements. There are two types of measurement data: parametric and nonparametric.
Normal, or parametric, data is measurement data that fits a normal curve, or distribution, usually when a large sample size is used. For example, if you took a large sample of 17-year-olds in America and graphed the frequency of heights, the results will be normally distributed.
Several descriptive statistics can be used to summarize normal data.
The sample size (n) refers to the number of members of the population that are included in the study. Sample size is an important consideration when you’re trying to determine how well the data in the study represents a population. Large sample sizes are always better, but for technical reasons, experiments can’t be infinitely large.
The mean (x) is the average of the sample, calculated by adding all of the individual values and dividing by the number of values you have. The mean is not necessarily a number provided in the sample. You should be able to recognize what the mean of a given dataset is and be able to calculate the mean using the equation on the AP Biology Equations and Formulas sheet. One limitation of mean is that it is influenced by outliers, or numerical observations that are far removed from the rest of the observations.
The standard deviation (s) can determine if numbers are packed together or dispersed, because it is a measure of how much each individual number differs from the mean. A low standard deviation means the data points are all similar and close to the mean, while a high standard deviation means the data are more spread out. Here is the relationship between a normal distribution and standard deviation (SD).
This means that about 70 percent of the data is within one standard deviation of the mean in a normally distributed dataset. Standard error (SE) can also be used to report how much a given dataset varies and is calculated by dividing the standard deviation by the square root of the sample size.
Nonparametric data often includes large outliers and do not fit a normal distribution. In order to determine if data is parametric or not, you can construct a histogram, or frequency diagram. These graphs give information on the spread of the data and the central tendencies. Making a histogram is like setting up bins, or intervals with the same range, that cover the entire dataset (the x-axis). The measurements in each bin are graphed on the y-axis. For example, suppose you counted the number of pine needles on each branch of a Pinus strobus tree. A histogram of this data may look like this.
This dataset does not match a normal distribution, meaning it is a nonparametric dataset. Keep in mind that even if a certain dataset is not normal, the population itself could be. There could have been sampling bias or errors in the data collection, which can lead to sample data that doesn’t match population data. The best way to fix this is usually to increase the sample size. Histograms differ from bar graphs in that they show ranges instead of just categories.
Nonparametric data requires a different set of descriptive statistics than parametric data.
The median is the middle number in a dataset and is determined by putting the numbers in consecutive order and finding the middle number. If there is an odd number of numbers, there will be a single number that is the median. If there is an even number of numbers, the median is determined by averaging the two middle numbers. Therefore, the median is not necessarily one of the numbers in the dataset. The median is useful in gauging the midpoint of the data, but will not necessarily tell you much about outliers.
The mode is the most frequently recurring number in the dataset. If there are no numbers that occur more than once, there is no mode. If there are multiple numbers that occur most frequently, each of those numbers is a mode. The mode must be one of the numbers in the sample, and modes are never averaged.
The range is the difference between the smallest and largest number in a sample. This value is less useful than standard deviation, because it does not give any information on individual values or the majority of values.
Example 1: Eleven male laboratory mice were weighed at 32 weeks of age. The data collected was: 32 g, 28 g, 29 g, 34 g, 30 g, 28 g, 32 g, 31 g, 30 g, 32 g, 33 g. What is the mean, median and mode of this dataset?
Solution: Let’s start by putting the 11 numbers in the dataset in order, from smallest value to largest value: 28 g, 28 g, 29 g, 30 g, 30 g, 31 g, 32 g, 32 g, 32 g, 33 g, 34 g. The mean is
The median is the middle value, or 31 g. The mode is the most frequent number, or 32 g.
Example 2: In the following set of values, what is the range?
Values: –5, 8, 11, –1, 0, 4, 14
Solution: The smallest value in the set above is –5, and the largest is 14. The difference between these two is the range, which is 19.
You will be tested on scientific experiments and the scientific reasoning behind different aspects of the experiments a LOT. You should be clear with the steps of the scientific method and know how to properly design an experiment and recognize a poorly designed experiment. The following list should remind you:
Hypothesis: A well-thought-out prediction of what the outcome of the experiment will be.
Independent Variable: The factor that you, as the experimenter, will change between the different groups in the experiment. If one plant is grown in pH 5 and the other in pH 7, then pH is the independent variable.
Dependent Variable: The data, or the thing that you measure during the experiment. The height of the plant you are growing might be the dependent variable.
Constants (Controlled Variables): The things that are the same between all your groups. Hint: Everything should be constant except the independent variable.
Control Groups: Any group that is needed simply so you can compare your interesting experimental groups to it. These are often the “no treatment” group.
Statistical Significance: The trustworthiness of the results and the certainty you have in your conclusions. This can always be increased by including more individuals in the groups or including more trials.
Most biological experiments do one of three things:
look at how something changes over time
compare groups of some sort
test for an association
How you present and summarize data depends on what kind of experiment you perform.
Time-course experiments look at how something changes over time. A line graph is usually used to present this type of data. Several lines can be plotted on the same graph, but these must be clearly labeled. Also, each dot on a line graph could represent one data point or a mean of values. If mean values are plotted, standard deviation or standard error can also be shown. More generally, line graphs can also be used to compare the way in which a dependent variable (y-axis) changes in relation to an independent variable (x-axis).
When making line graphs, the intervals on the axes must be consistent. Also make sure it is clear whether the data starts at the origin (0, 0) or not. The same guidelines apply to scatterplots, which will be discussed below.
Data points should be connected with a solid line. You can extrapolate (extend) the line past the data points, but then you must use a broken line. Finally, the slope on a line graph tells you the rate of change.
Suppose we want to determine how stress hormone levels change after the end of an important exam. If we measure cortisol and epinephrine in the serum of several people every hour for five hours after the end of an exam, we would have several values for each time point. We could plot the mean of each measurement and also show data variance using standard error bars. In this example, the distance above the data point is the standard error, and the distance below the data point is the standard error. We call this “±SE.”
Comparative experiments compare populations, groups, or events. There are several options for graphing data from these experiments:
Bar graphs are helpful to compare categories of data if the data is parametric. It is also a good idea to show variance of the data by including the standard error. This gives information on how different the two means are from each other. Suppose you compare two strains of fission yeast before and after treatment with a drug that inhibits cell wall synthesis. As a measure of viability, you measure how much Myc mRNA was made in each strain with and without treatment. In this case, each bar is a mean of values and the error bars are ±SE, as described above.
Box-and-whisker plots should be used for nonparametric data. These graphs show median (horizontal line), quartiles (top and bottom of the box), and largest and smallest values (ticks at the tops and bottoms of the lines). In other words, the middle 50% of plant weights is represented by the box, with the median shown with the line, and the highest and lowest values are marked with the top and bottom extenders (whiskers). Remember, the median is not the average, so the line doesn’t have to be in the middle. Suppose the weights of two types of Arizona plants are measured, to determine which is more plentiful in a given niche:
In order to conclude that the samples or groups being compared are different or not, hypothesis testing should be performed. This will be discussed later in this chapter.
Association experiments look for associations between variables. They attempt to determine if two variables are correlated, and additional tests can demonstrate causation. Scatterplots are used to present data from association experiments. Each data point is plotted as a dot. Suppose different substrate concentrations are used in an enzymatic reaction (using the enzyme Kozmase III), and the enzyme efficiency is measured (in percent of maximum). The data could be presented like this.
If the relationship looks linear, a linear regression line can be added. There are a few common shapes of scatterplots you should be familiar with:
A bell-shaped curve is associated with random samples and normal distributions.
An upward curve (concave in shape) is associated with exponentially increasing functions. A common example of this is the lag phase of bacterial growth in a liquid culture.
A curve that looks like the sine wave is common when biological rhythms are being studied.
You already saw an introduction to probability calculations in Chapter 8 of this book. The probability (P) that an event will occur is the number of favorable cases (a) divided by the total number of possible cases (n).
P = a / n
This can be determined experimentally by observation (such as when population data is being collected) or by the nature of the event. For example, the probability of getting a two when rolling a die is 1/6, since there are six sides on a die.
Example: What is the probability of drawing the nine of hearts from a deck of cards?
Solution: Since there are 52 cards in a deck and this question is asking about one particular card, P = 1/52.
Many genetics problems involve several probabilities, all being considered together to answer a larger biological question. There are three rules that are often used.
The product rule is used for independent events and is also called the “AND rule.” The probability of independent events occurring together is the product of their separate probabilities. This rule is used when the order of the events matters.
Example: What is the probability of drawing a heart and a spade consecutively from a deck of cards if the first draw was replaced before the second draw?
Solution:
P(heart) = 
= 
P(spade) = =
P(heart and spade) = P(heart) × P(spade)
P(heart and spade) = × = 
The sum rule is used for studying two mutually exclusive events, and can be thought of as the “EITHER” rule. The probability of either of two events occurring is the sum of their individual probabilities.
Example: What is the probability of drawing a diamond or a heart from a deck of cards?
Solution:
P(diamond) = =
P(heart) = =
P(diamond or heart) = P(diamond) + P(heart)
P(diamond or heart) = + = 
= 
If the events you are studying are not mutually exclusive, such as having brown hair and brown eyes then you could have both occur at the same time. If you are trying to determine the likelihood of EITHER occurring but NOT BOTH, then you must calculate using the sum rule (odds of either occurring) and the product rule (odds of both occurring). Then, take the result using the product rule and subtract it from the result when using the sum rule since you want the odds of either occurring but not both.
Many experiments involve comparing two datasets or two groups, and a t-test can be used to calculate whether the means of two groups are significantly different from each other. This test is most often applied to datasets that are normally distributed. A p-value equal to or below 0.05 is considered significant in most biology-related fields. T-test calculations involve comparing each data point to the group’s mean and also take standard deviation and sample size into consideration.
Understanding mathematically how p-values are calculated and t-tests are performed is not important here. Instead, it’s important that you understand how these values are used to interpret data.
Let’s work through an example to demonstrate how common statistics are used in biological labs. A researcher has sections of two different types of skin cancers from human patients. She stains them for the protein CD31, which is a marker of endothelial cells. She then takes digital pictures of the immunofluorescent sections and counts the number of blood vessels present in each picture. She takes five pictures of each slide.
|
Slide A |
Slide B |
|
10 |
3 |
|
8 |
2 |
|
7 |
4 |
|
8 |
4 |
|
11 |
4 |
(a) What is the mean, median, and mode for each dataset?
(b) The standard error of group A is 0.735, and the standard error for group B is 0.400. What does this tell you about the data?
(c) What assumptions are made about the data in performing these tests?
(d) What could the researcher do to increase her confidence in the results?
Solutions:
(a) The means are
Remember, the median is the middle number, so it’s best to put the data in order first.
|
Slide A |
Slide B |
|
7 |
2 |
|
8 |
3 |
|
8 |
4 |
|
10 |
4 |
|
11 |
4 |
The median of group A is 8, and the median of group B is 4.
The mode is the most frequent value. For group A, this is 8. For group B, the mode is 4.
(b) Standard error is the standard deviation divided by the square root of the sample size. The sample size is five for both group A and group B because five pictures were taken for each slide. Because the standard error of group A is larger than that of group B and because the two groups have the same sample size, the standard deviation of group A must also be larger than that of group B. Note that this may or may not be the case if the sample sizes were different. A larger standard deviation means the data is more variable, so you can conclude that the data from group A has a larger spread than the data from group B.
(c) As with most datasets, the researcher is assuming the data fits a normal distribution and that her sample size is large enough to be meaningful.
(d) More data allows for more confident conclusions. The researcher could therefore take more pictures from each slide to increase the sample size.
A chi-square test is a statistical tool used to measure the difference between observed and expected data. For example, suppose you roll a die 120 times. You would expect a 1:1:1:1:1:1 ratio between all the numbers that come up. In other words, you would expect to see each side 20 times. However, because this is an experiment and the sample size is only 120, you will probably see some variation in the data. If you rolled 24 sixes for example, the disagreement between observed (24) and expected (20) is small, and could have occurred by chance. If you rolled 40 sixes, however, there is a large disagreement from the expected. Maybe it is a weighted die. The region in between these two values is trickier. Where is the line drawn between expected/normal variations and results that are so different that they must tell a different story altogether?
Chi-square tests are one way to make this decision. They start with a null hypothesis (Ho), which represents a set numerical outcome that you want to compare your data with. The chi-square test will determine if your actual data is close to the null hypothesis outcome and could have occurred due to a fluke (i.e., rolling a six 24 times) or if it is so different that it is probably not a fluke (i.e., 40 rolled sixes) and the null hypothesis outcome must not correctly describe what is occurring. In the dice case, rolling a six 40 times would cause the null hypothesis of expecting a 1:1:1:1:1:1 to be rejected and the die can be considered to be weighted. In the calculations, expected results if the null hypothesis were true are compared to the actual values, and an x2 value is calculated. This value is compared to a critical value, and a decision is made to reject or accept the null hypothesis.
For example, suppose data was collected from 100 families with three children each. Fourteen families had three female children, 36 families had two females and a male, 30 families had one female and two males, and 20 families had three male children. We would expect offspring sex to segregate independently. In other words, the sex of a given child shouldn’t be affected by the sex of previous children. Let’s test whether this is true for this dataset.
Since we expect sex to segregate independently, this is the null hypothesis (Ho). Each family has three children, so there are four possible outcomes for each family:
three daughters
two daughters and a son
one daughter and two sons
three sons
Using some advanced math (that you don’t need to worry about), the probabilities for each of these is:
P(three females) = 12.5%
P(two females, one male) = 37.5%
P(one female, two males) = 37.5%
P(three males) = (1)(0.50)3 = 12.5%
Because there are 100 families, we expect 12.5 to have three females, 37.5 to have two females and a male, 37.5 to have one female and two males, and 12.5 to have three males. This can be calculated by multiplying the probability of each event by the total (100 in this case).
Next, we compare this expected data (E) to what actually happened (the observed data, O). Note that the total numbers for observed and expected columns must be the same. To calculate an χ2 value, the formula (O – E)2/E is calculated for each row of the table, and summed.
The calculated χ2 value is 6.24. Next, this value needs to be compared to a critical value (CV). This is obtained from a table like this one.
This chart is included on the AP Biology Equations and Formulas sheet. In order to use this table, you must know the degrees of freedom (DF), which represent the number of independent variables in the data. In most cases, this is the number of possibilities being compared minus one. In this example, DF = 4 – 1, because we are comparing four different family types.
You also need a reference p-value. This is the probability of observing a deviation from the expected results due to chance. For most biological tests, a p-value of 0.05 is used.
Since we are using DF = 3 and p = 0.05, the critical value is 7.82 (according to the chart above). Next, the critical value is compared to the χ2 value; if χ2 < CV, you accept Ho. If χ2 > CV, you reject Ho. We calculated an χ2 value of 6.24, which is smaller than 7.82. Since χ2 < CV, we accept Ho and conclude that offspring sex could, indeed, be segregating independently in this dataset.
descriptive statistics
sample
population
random sampling
quantitative data
qualitative data
count data
bar graph
pie graph
normal, or parametric, data
normal curve, or distribution
sample size
mean
outliers
standard deviation
standard error
nonparametric data
histogram, or frequency diagram
median
mode
range
hypothesis independent variable
dependent variable
contants (contolled variables)
control groups
statistical significance
time-course experiment
line graph
comparative experiments
bar graph
box-and-whisker plot
association experiments
scatterplot
probability
product rule
sum rule
t-test
p-value
chi-square test
null hypothesis
critical value
degrees of freedom
Descriptive statistics (such as mean, standard deviation, standard error, median, mode, range) and graphs are used to summarize data and show patterns and conclusions:
Normal (parametric) measurement data are summarized by using mean and standard deviation or standard error.
Parametric data are summarized by using median, mode, and range.
Histograms can be used to see if a dataset has a standard deviation or not.
Count data is summarized in a bar graph or a pie graph.
Time course experiments look at how something changes over time; summarize these using a line graph.
Experiments that compare groups are summarized in a bar graph (if the dataset has a standard deviation) or a box-and-whisker plot (if it does not).
Scatterplots summarize association experiments, and regression lines can be used to determine if the relationship is linear.
Probability can be calculated with the sum rule or the product rule.
Hypothesis testing is used to determine if two groups are significantly different from each other. They start with a null hypothesis, which is rejected or accepted, depending on how a calculated p-value or chi-square value compares to a standard value.
Answers and explanations can be found in Chapter 15.
1. Five subjects were weighed before and after an 8-week exercise program. What is the average amount of weight lost in pounds for all five subjects, rounded to the nearest pound?
|
Subject |
Starting Weight (pounds) |
Final Weight (pounds) |
|
1 |
184 |
176 |
|
2 |
200 |
190 |
|
3 |
221 |
225 |
|
4 |
235 |
208 |
|
5 |
244 |
225 |
(A) 12 pounds
(B) 13 pounds
(C) 14 pounds
(D) 15 pounds
2. The height of six trees is measured. Is plant 6 taller than the median for all six trees?
|
Plant |
Height (inches) |
|
1 |
67 |
|
2 |
61 |
|
3 |
72 |
|
4 |
71 |
|
5 |
66 |
|
6 |
68 |
(A) Yes, the median is 67.3.
(B) No, the median is 67.3.
(C) Yes, the median is 67.5.
(D) No, the median is 67.5.
3. In the following set of test scores, what is the mode and what is the range?
Test Scores: 71, 67, 75, 65, 66, 32, 69, 70, 72, 82, 73, 68, 75, 68, 75, 78
(A) Mode: 68; Range: 75
(B) Mode: 69; Range: 50
(C) Mode: 75; Range: 70.5
(D) Mode: 75; Range: 50
4. Given the cross AaBbCc × AaBbCc, what is the probability of having an AABbCC offspring?
(A) 
(B) 
(C) 
(D) 
5. Given the cross AaBb × aabb, what is the probability of having an Aabb or aaBb offspring?
(A)
(B)
(C)
(D) 0
6. Two pea plants are crossed, and a ratio of 3 yellow plants to 1 green plant is expected in the offspring. It is found that out of 100 plants phenotyped, 84 are yellow and 16 are green. Do the experimental data match the expected data?
(A) Yes, the χ2 value is greater than 3.84.
(B) Yes, the χ2 value is smaller than 3.84.
(C) No, the χ2 value is greater than 3.84.
(D) No, the χ2 value is smaller than 3.84.
7. A mating is set up between two pure breeding strains of plants. One parent has long leaves and long shoots. The other parent has short leaves and stubby shoots. F1 plants are collected, and all have long leaves and long shoots. F1 plants are self-crossed, and 1,000 F2 plants are phenotyped. The data is as follows:
|
Phenotype |
# of F2 |
|
Long leaves, long shoots |
382 |
|
Long leaves, stubby shoots |
109 |
|
Short leaves, long shoots |
112 |
|
Short leaves, stubby shoots |
397 |
|
Total |
1,000 |
Are the genes for leaf and shoot length segregating independently?
(A) Yes; the degrees of freedom are 3, and the calculated χ2 value is small.
(B) No; the degrees of freedom are 3, and the calculated χ2 value is large.
(C) Yes; the degree of freedom is 1, and the calculated χ2 value is small.
(D) No; the degree of freedom is 1, and the calculated χ2 value is large.
Respond to the following questions:
Which topics from this chapter do you feel you have mastered?
Which content topics from this chapter do you feel you need to study more before you can answer multiple-choice questions correctly?
Which content topics from this chapter do you feel you need to study more before you can effectively compose a free response?
Was there any content that you need to ask your teacher or another person about?