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Experimental Design, Describing Data and Statistical Analysis AP Biology Part 1: Experimental Design The goal of scientific investigation is to obtain and interpret reliable scientific data. In biology, we engage in the scientific method in order to answer questions about the natural world. Scientific Method Make observations – Identify an event you’d like to understand more about. Formulate a question – What testable, data-driven question can you ask about the event? Form a hypothesis – Write a statement that could answer the question. The statement should be both testable and falsifiable. Test the hypothesis – Collect data and observations that helps you determine whether your hypothesis is accurate. Data Analysis – Graphs, computations, and/or statistical tests that allow the scientist to determine patterns. Draw conclusions – Interpret your data and observations, attempted to support or refute your hypothesis. In science, nothing can be proven, only supported! Variables Variables – Things that change in an experiment. Changes can be a part of the experimental design OR something you’re measuring in order to answer your question. Independent variables are controlled or chosen by the scientist. On a graph, placed on the x-axis. Dependent variables are measured or observed by the scientist. On a graph, placed on the y-axis. Use the sentence: “The _______ depends on the ______.” to help you figure out which is which. Practice identifying variables Let’s start by considering some biological questions (pay attention to the IV and DV here): How do solutes dissolved in water affect water’s properties of adhesion and cohesion? What temperature is ideal for an enzyme-mediated rate of reaction? How do toxins impact the rate of mitosis in onions? Are mealworms more attracted to grains or meats? How do environmental conditions impact the rate of transpiration? Constants A constant is something that should be kept the same for the entire experiment, especially between different trials. E.g., temperature, humidity, light levels, etc. Might also be called a “controlled variable” Something that, if not maintained at a consistent level throughout the experiment could reasonably impact your data. Controls A control is something used as a standard of comparison in an experiment. It is often the same conditions as your experimental conditions, just without the independent variable. A basis for comparison that allows you to claim that changes in your dependent variable are likely due to your independent variable. Any time you are interpreting data, you should intentionally and explicitly look at the control and state, “… as compared to the control” or use numbers to do this explicitly. Ex: “The number of drops of acetone that could be held on a penny was 6 less than the control.” Positive & Negative Controls There are two types of controls: positive and negative. Some labs have one or the other, some have both; this just depends on your experimental design. A positive control is designed where a “known” response is expected. For example, we expect bacteria to grow on nutrient agar. A negative control is designed where no response is expected. This shows that your experimental setup is working properly. For example, we expect that antibiotics to kill bacteria grown on nutrient agar that has been supplemented with antibiotics. Types of Data & Observations Scientific observations can take two forms: Qualitative or Quantitative. Qualitative data refers to the qualities of something (e.g., color, shape, texture, odor). Ex: The liquid turned orange after we added the iron. Quantitative data refers to the quantity of something (e.g., amount or value) Ex: The mass increased 4 grams. Collecting Data We need to have a sufficiently large sample size. 40 is ideal, but it’s pretty unrealistic for our class. The data need to be normally distributed. In some applications, repetition (multiple trials) is more realistic than collecting many individuals. Trials occur when you do the EXACT same thing repeatedly. No variations at all. Why are large, unbiased samples so important? The goal is that your sample is indicative of the entire population. So the 10 plants you find the transpiration of are indicative of every plant of that species, in those conditions, on the planet. Crazy. The goal: the sample mean and spread (called Standard Deviation) is the same as the population mean and spread. This is where a bell curve comes from: A large, unbiased sample allows us to make inferences of the population. The individuals in the sample are a reasonable approximation of the population. Of course there will be variation, but as long as we have a large, unbiased sample, results should be legitimate and variation should produce a bell curve around the mean. Part 2: Describing Data Visual Descriptions of Data are Graphs Use whatever graph best allows for the visualization of relationships Bar Line And all of their many variations Descriptive Statistics – Complete calculations to understand the intricacies of the data. Your calculator will help! Mean Median Standard Error of the Mean Interquartile Range (IQR) We will often graph a dataset’s descriptive statistics rather than the raw data in order to show relationships more clearly. Graphing Requirements A good title (Y vs. X of _____) Label your axes with units, and use evenly spaced and scaled numbers to spread out the data. Clearly mark data points. When making a line graph, draw a line of best fit Any extrapolation beyond the last data point should be shown with a dashed line. What kind of graph do I make? Bar Graphs… Behold: the return of the BAR GRAPH! The only rule to graphing is that you make the one that represents your data the best. Bar graph: Used for categorical independent variables. Since we always do multiple trials, the top of the bar represents the MEAN of the trials of that data. Might have error bars, which show the variation in the data. Box & Whisker Plot Box & Whisker plots are modified bar graphs that show how the data is spread out using quartiles. Your calculator will find your quartiles for you. Line Graphs Line graph: Used for continuous independent variables. Might have two y-axes for multiple scales Might plot the means, and have error bars to show variation in data. Might have a log scale Constructing a Data Table The rules are much the same as graphing: have a good title, have labels and units clearly indicated, and of course, USE A RULER. Depending on your data, your independent variable may go on the left or across the top. Either is appropriate. I have the data. Now what? Descriptive statistics describe your data. Calculate descriptive statistics for each group – control and experimental: Mean (or average) Standard Deviation – How spread out your data is from the mean you just calculated. Standard Error of the Mean – How confident you are that your sample mean includes the population mean. Using your TI-84 to lighten the load: Consider this: I’ve taken my temperature every morning for the last two weeks. Here are my findings: 97.8 98.1 98.7 98.4 98.3 97.9 97.4 97.7 98.2 98.3 97.6 98.9 98.6 98.5 . Calculate by hand the standard deviation Remember the equation is Using your TI-84 to lighten the load: Consider this: I’ve taken my temperature every morning for the last two weeks. Here are my findings: 97.8 98.1 98.7 98.4 98.3 97.9 97.4 97.7 98.2 98.3 97.6 98.9 98.6 98.5 Enter the dataset in a list by pressing “Stat” then “Edit” When you are finished, it will look like this. Using your TI-84 to lighten the load: To have your calculator compute descriptive statistics for this list of data: Press “Stat” Press over to highlight “Calc” along the top. The top option is “1-var Stats” – select that. Your screen will read like this: Ensure the correct List is specified; leave FreqList blank Press down twice to highlight “Calculate” and hit enter. Using your TI-84 to lighten the load: The output is shown at the right, and matches the defined terms on your formula sheet: I highlighted the mean (x) and standard deviation (Sx). I also highlighted n, the sample size. Your calculator does not automatically calculate 𝑆𝐸𝑥 for you – you must do that on your own. But it’s just plug in values from your calculator! When graphing, you will typically graph 𝑥 ± 2𝑆𝐸𝑥 , so be prepared to do a bit more arithmetic. Showing descriptive statistics graphically: box and whisker plot This same output also generates the values necessary to produce a boxplot. Simply scroll down to see it all. Showing descriptive statistics graphically: mean±2SEM How different is different enough? The larger the error bar, the less confident we are that the calculated mean is representative of the entire population. Part 3: Statistical Analysis Answering the question: How different is different enough? “My control data has a mean of 12 cm/year and experimental data has a mean of 15 cm/year.” Is there a biological relationship here? “I flipped this coin 20 times and got 14 heads and 6 tails; that’s pretty different from the 10 heads and 10 tails I was expecting. Is this coin most likely fair? Or “loaded”? How do we go about answering biological questions? Collect data: large, unbiased sample. Describe the data with descriptive statistics (mean, SEM) Write statistical (null and alternative) hypotheses Determine appropriate statistical test Calculate the test statistic, critical value, and determine degrees of freedom. (Calculator!) Interpret statistical test with respect to statistical hypotheses. Draw a scientific conclusion, using support from the statistical test. Now, to statistical tests… All statistical tests work on fancy, calculusbased areas under curves that indicate probabilities… Don’t worry about it. Just be able to interpret it. We’re trying to distinguish between natural variation in a dataset and variation that is the result of the independent variable. “What is the probability that my two data sets are different?” Are boys taller than girls? Some boys are shorter than some girls. How do we answer that question? Taking your data to court: In a court of law, a defendant is found to be either guilty or not guilty. A defendant is never found to be innocent. Why not? Remember, in statistics, we are trying to determine if two datasets are different enough to claim a biological relationship exists with the independent variable. In statistical analysis, you write a null hypothesis (the defendant – that there is not a relationship with the independent variable), and it is find to reject it or fail to reject it. What’s the difference? How different do two datasets need to be in order to be different enough to claim they are different enough and a biological relationship exists? Taking your data to court: Think of yourself like a prosecutor, trying to provide sufficient data to support rejecting the null hypothesis. So, we write statistical hypotheses, two of them for every statistical test. Don’t let the language upset you. Null hypothesis, or H0, just says “There is no significant difference between X and Y.” Means that the distributions of your two datasets (experimental and control) overlap too much for you to claim that the data comes from two different populations. Alternate hypothesis, or HA, just says “There is a significant difference between X and Y” Means that the distributions of your two datasets are sufficiently different for you to claim that the data comes from two different populations. It’s your goal to attack the null hypothesis, showing you have significant evidence against it. In doing so, you lend support to (BUT DO NOT PROVE!) the alternate hypothesis. Ultimately, you obtain a probability, or pvalue: the probability of getting this data by chance alone; that your two datasets are from the same population, or your data are within reason for an expected outcome. Biologists are typically willing to allow 5% error. So, we look for a p-value of 0.05 or less in order to reject the null hypothesis and accept the alternate hypothesis. Two non-biological questions: Do vehicles get better highway MPG when using cruise control compared with when not using cruise control? Null: There is not a significant difference in the gas mileage using cruise control vs. not. Alternate: There is a significant different in gas mileage using cruise control vs. not. Is this weird looking coin I just found in the hallway a fair coin? Or is it “loaded”? Null: There is not a significant difference in the number of heads vs. tails flipped on this coin compared to a fair coin. Alternate: There is a significant….. So, now we test the hypotheses we wrote: There are loads of different statistical tests. You need to calculate two this year: One when you are comparing two datasets to one another (like the gas mileage question) two sample t-test. One when you are comparing a single dataset to expected values (like the fair coin question: Heads vs. Tails) Chi-Square test. You will need to be able to interpret p-values of all tests, but these are the only two I’ll ask you to calculate. Practice! From earlier: Do vehicles get better highway MPG when using cruise control compared with when not using cruise control? Null: There is not a significant difference in the gas mileage using cruise control vs. not. Alternate: There is a significant different in gas mileage using cruise control vs. not. Enter your data into two lists (Stat; 1:Edit) in your calculator – one list for each treatment. MPG with cruise control MPG without cruise control 26 27 22 19 25 26 23 24 28 24 22 21 24 26 24 22 28 27 21 23 Two Sample T-test: Now, perform the t-test: Stat, over to TESTS 4:2-SampTTest… Data List 1: L1 List 2: L2 Freq1: 1 Freq2: 1 µ1:≠ µ2 Pooled: No Calculate, hit enter. T-test: There are two pieces of information you need from this list: T= is the t-statistic P= is the p-value (BE CAREFUL with Scientific Notation: 1.1E-5) P-value will always be between 0 and 1 (remember, it’s a probability) In general, the larger the t-statistic, the smaller the p-value. If the p-value is less than 0.05 we reject the null hypothesis and accept the alternate hypothesis. If the p-value is greater than 0.05 we fail to reject the null hypothesis. This is usually not what we “want” because it means there is no significance between our control and experimental conditions. More on a t-test So, what is our statistical conclusion? What is our scientific conclusion? Shift gears to the other test: Chisquare Goodness of Fit (GOF) Use this test for data when you are comparing one (observed) data set to predicted (expected) values. The coin flip example. 1:2:1 ratios of progeny – think Punnett squares! Evaluating whether organisms are attracted or repelled by something in their environment. Does this population meet the predictions of the Hardy-Weinberg equilibrium? Statistical hypotheses are generally the same! Null: There is no difference between the expected and observed data. Alternate: There is a difference between the expected and observed data. Calculating Chi-square Is this weird looking coin I just found in the hallway a fair coin? Or is it “loaded”? Null: There is not a significant difference in the number of heads vs. tails flipped on this coin compared to a fair coin. Alternate: There is a significant….. Enter the data and do the test: Flip Observed Expected Heads 12 15 Tails 18 15 How to enter data for X2GOF… Enter your data into two lists: Observed data in List 1 Expected data in List 2 Then: STAT, over to TESTS Down to D:X2GOF-Test Observed: L1 Expected: L2 df:? (enter the number of options -1) Calculate and hit enter. More on Chi-square So, what is our statistical conclusion? What is our scientific conclusion? Consider sample size: add a zero to everything and rerun the test: The proportions have NOT changed. What happened to the results? What does this indicate about the importance sample size? When could patterns in data be overlooked? Flip Observed Expected Heads 120 150 Tails 180 150 Doing Chi-square by hand: Looks scary, just algebra. And you don’t have to be very precise. Use the table if you want! Categories (outcome) Expected (e) Observed (o) Heads 15 12 Tails 15 18 o-e (o-e)2 Sum: So, now what? How many degrees of freedom? What’s the critical value? Assume p=0.05 (𝒐 − 𝒆)𝟐 𝒆 But what does it all mean?!?! If the Chi-square value is greater than the critical value, p is less than 0.05 and the H0 may be rejected. The expected and observed values are sufficiently different to reject the null hypothesis. If the Chi-square value is less than the critical value, p is greater than 0.05 and the H0 may not be rejected: the expected and observed values are too similar to reject the null hypothesis. Why? Such torture… Remember, the goal is to answer questions about the natural world. Quantitative evidence helps us come to justifiable answers to our questions. Just claiming two experimental conditions yield “different” results is insufficient. How different? Different enough? Statistics allows us to defensibly answer questions.