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Skills in BiologyHypotheses and PredictionsScientific knowledge grows through a process called the scientific method. This process involves observation and measurement, hypothesizing and predicting, and planning and executing investigations designed to test formulated hypotheses. A scientific hypothesis is a tentative explanation for an observation, which is capable of being tested by experimentation. Hypotheses lead to predictions about the system involved and they are accepted or rejected on the basis of findings arising from the investigation. Rejection of the hypothesis might lead to new alternative explanations for the observations. Acceptance of the hypothesis as a valid explanation is not necessarily permanent: explanations may be rejected at a later date in light of new findings.

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Accept or reject

Based on a hypothesis, predictions (expected, repeatable outcomes) can be generated about the behavior of the system. Predictions may be made on any aspect of the material of interest, e.g. how different variables (factors) relate to each other.

Making Predictions

Investigations are planned so that the predictions about the system made in the hypothesis can be tested. Investigations may be laboratory or field based.

Designing an investigation

A hypothesis based on observations is used to generate the null hypothesis (Ho); the hypothesis of no difference or no effect. Hypotheses are expressed in the null form for the purposes of statistical testing. Ho may be rejected in favor of accepting the alternative hypothesis, HA.

Generating a Null Hypothesis

Features of a sound hypothesis: It is based on observations and

prior knowledge of the system It offers an explanation for an

observation. It refers to only one

independent variable. It is written as a definite

statement and not as a question It is testable by experimentation It leads to predictions about the

system.

Formulating a Hypothesis

The predictions are tested out in the practical part of an investigation.

Testing the Prediction

The observations lead to the formation of questions about the system being studied.

Asking Questions

Making ObservationsThese may involve the observation of behaviors in wild populations, physiological measurements made during previous experiments or accidental results obtained when seeking answers to completely unrelated questions.

Exercise 1.1Questions and Hypotheses

This exercise explores the nature of scientific questions and hypotheses. A hypothesis offers a tentative explanation to questions generated by observations. Hypotheses are often constructed in a form that allows them to be tested statistically. For every hypothesis, there is a corresponding null hypothesis; a hypothesis against the prediction. Predictions are tested with laboratory and field experiments and carefully focused observations. For a hypothesis to be accepted it should be possible for anyone to test the predictions with the same methods and get a similar result each time.

Part A—Asking QuestionsScientists are characteristically curious and creative individuals whose curiosity is directed toward

understanding the natural world. They use their study of previous research or personal observations of natural phenomena as a basis for asking questions about the underlying causes or reasons for these phenomena. For a question to be pursued by scientists, the phenomenon must be well defined and testable. The elements must be measurable and controllable.

There are limits to the ability of science to answer question. Science is only one of many ways of knowing about the world in which we live. Consider, for example, this question: Do excessively high temperatures cause people to behave immorally? Can a scientist investigate this question? Temperature is certainly a well-defined, measurable and controllable factor, but morality of behavior is not scientifically measurable. We probably could not even reach a consensus on the definition. Thus, there is no experiment that can be performed to test the question. Which of the following do you think can be answered scientifically? Place a check beside the questions you think can be answered scientifically.

_______1. Does binge drinking cause more brain damage in teenagers than in adults?

_______2. Is genetically modified corn safe to eat?

_______3. Do children who wash their hands often and bathe daily have a greater risk of asthma than those who

wash their hands less often and bathe every other day?

_______4. Should endangered species be cloned to prevent extinction?

_______5. What is the function of spines on cacti?

_______6. Did the 19 year old college student develop ulcers because of his stress and fast food diet?

How did you decide which questions can be answered scientifically?__________________________

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Part B—Developing HypothesesAs questions are asked, scientists attempt to answer them by proposing possible explanations. Those proposed explanations are called hypotheses. A hypothesis tentatively explains something observed. It proposes an answer to a question. Consider question 5, preceding. One hypothesis based on this question might be “If spines on cacti prevent animals from eating the cacti, then cacti will have spines.” The hypothesis has suggested a possible explanation for the observed spines.

A scientifically useful hypothesis must be testable and falsifiable . To satisfy the requirement that a hypothesis is falsifiable, it must be possible that the test results do not support the explanation. In our example, if spines are removed from test cacti and the plants are not eaten by animals then the hypothesis has been falsified. Even though the hypothesis can be falsified, it can never be proved to be true. The evidence from an investigation can only provide support for the hypothesis. In our example, if cacti without spines were eaten, the hypothesis has not been proved, but has been supported by evidence. Other explanations still must be excluded, and new evidence from additional experiments and observations might falsify this hypothesis at a later date. In science seldom does a single test provide results that clearly support or falsify a hypothesis. In most cases the evidence serves to modify the hypothesis or the conditions of the experiment.

Science is a way of knowing about the natural world that involves testing hypotheses or explanations. The scientific method can be applied to the unusual and the commonplace. You use the scientific method when you investigate why your once white socks are now blue. Your hypothesis might be that your blue jeans and socks were washed together, an assertion that can be tested through observations and experimentation.

Students often think that controlled experiments are the only way to test a hypothesis. The test of a hypothesis may include experimentation, additional observations or the synthesis of information from a variety of sources. Many scientific advances have relied on other procedures and information to test hypotheses. For example, James Watson and Francis Crick developed a model that was their hypothesis for the structure of DNA. Their model could only be supported if the accumulated data from a number of other scientists were consistent with the model. Actually, their first model (hypothesis) was falsified by the work of Rosalind Franklin. Their final model was tested and supported not only by the ongoing work of Franklin and Maurice Wilkins but also by research previously published by Erwin Chargaff and others. Watson and Crick won the Nobel Prize for their scientific work. They did not perform a controlled experiment in the laboratory but tested their powerful hypothesis through the use of existing evidence from other research. Methods other than experimentation are acceptable in testing hypotheses. Think about other areas of science that require comparative observations and the accumulation of data from a variety of sources, all of which must be consistent with and support hypotheses or else be inconsistent and falsify hypotheses.

The information in your biology textbook is often thought of as a collection of facts, well understood and correct. It is true that much of the knowledge of biology has been derived through scientific investigations, has been thoroughly tested, and is supported by strong evidence. However, scientific knowledge is always subject to novel experiments and new technology, any aspect of which may result in modification of our ideas and a better understanding of biological phenomena. The structure of the cell membrane is an example of the self-correcting nature of science. Each model of the membrane has been modified as new results have negated one explanation and provided support for an alternative explanation.

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ApplicationBefore scientific questions can be answered, they must first be converted to hypotheses, which can be tested. For each of the following questions, write an explanatory hypothesis. Recall that the hypothesis is a statement that explains the phenomenon you are interested in investigating. For the purposes of our class, you will need to try to state the hypothesis as an “if…then” statement.

1. Does regular interaction with pets improve the health of the elderly?

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2. What effect do high concentrations of the industrial pollutant PCB (polychlorinated biphenyl) have on killer

whale reproduction?

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Scientists often propose and reject a variety of hypotheses before they design a single test. Which of the following statements would be useful as scientific hypotheses and could be investigated using scientific procedures? Give the reason for each answer by stating whether it could possibly be falsified and what factors are measurable and controllable.

1. The number of fungiform papillae (bumps on the tongue) affects taste sensitivity.

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2. Inflated self-esteem in young males increases the odds of aggression.

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3. Exposure to environmental pollutants produces feminization in newly hatched male

alligators.

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___________________________________________________________________________

Exercise 1.2Designing Experiments to Test Hypotheses

The most creative aspect of science is designing a test of your hypothesis that will provide unambiguous evidence to falsify or support a particular explanation. Scientists often design, critique and modify a variety of experiments and other tests before they commit the time and resources to perform a single experiment. In this exercise, you will follow the procedure for experimentally testing hypotheses, but it is important to remember that other methods, including observation and the synthesis of other sources of data, are acceptable in scientific investigations. An experiment involves defining variables, outlining a procedure and determining controls to be used as the experiment is performed. Once the experiment is defined, the investigator predicts the outcome of the experiment based on the hypothesis.

Read the following description of a scientific investigation of the effects of sulfur dioxide on soybean reproduction. Then in Lab Study A you will determine the types of variables involved, and in Lab Study B, the experimental procedure for this experiment and others.

Investigation of the Effect of Sulfur Dioxide on Soybean ReproductionAgricultural scientists were concerned about the effect of air pollution, sulfur dioxide in particular, on soybean production in fields adjacent to coal powered plants. Based on initial investigations, they proposed that sulfur dioxide in high concentrations would reduce reproduction in soybeans. They designed an experiment to test this hypothesis (figure 1.1). In this experiment, 48 soybean plants, just beginning to produce flowers, were divided into two groups, treatment and no treatment. The 24 treated plants were divided into four groups of six. One group of 6 treated plants was placed in a fumigation chamber and exposed to 0.6 ppm of sulfur dioxide for 4 hours to simulate sulfur dioxide emissions from a power plant. The experiment was repeated on the remaining three treated groups. The no treatment plants were placed similarly in groups of 6 in a second fumigation chamber and simultaneously exposed to filtered air for 4 hours. Following the experiment, all plants were returned to the greenhouse. When the beans matured, the number of bean pods, the number of seeds per pod, and the weight of the pods were determined for each plant.

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Part A—Determining the VariablesRead the description of each category of variable; then identify the variable described in the preceding investigation. The variables in an experiment must be clearly defined and measurable. The investigator will identify and define dependent, independent and controlled variables for a particular experiment.

The Dependent VariableWithin the experiment, one variable will be measured or counted or observed in response to the experimental conditions. This variable is the dependent variable. For soybeans, several dependent variables are measured, all of which provide information about reproduction. What are they?______________________________________________________________________________________________

______________________________________________________________________________________________

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The Independent VariableThe scientist will choose one variable, or experimental condition, to manipulate. This variable is considered the most important variable by which to test the investigator’s hypothesis and is called the independent variable. What was the independent variable in the investigation of the effect of sulfur dioxide on soybean reproduction?________________________________________________________________________________________________________________________________________________________________________________Can you suggest other variables that the investigator might have changed that would have had an effect on the

dependent variables?____________________________________________________________________________

______________________________________________________________________________________________

Although other factors, such as light, temperature, time and fertilizer, might affect the dependent variables, only one independent variable is usually chosen. Why is it important to have only one independent variable? __________________________________________________________________________________________________________________________________________________________________________________________

Why is it acceptable to have more than one dependent variable?________________________________________

______________________________________________________________________________________________

______________________________________________________________________________________________

The Controlled Variable or ConstantsConsider the variables that you identified as alternative independent variables. Although they are not part of the hypothesis being tested in this investigation, they would have significant effects on the outcome of this experiment. These variables must, therefore, be kept constant during the course of the experiment. They are known as the controlled variables. The underlying assumption in experimental design is that the selected independent variable is the one affecting the dependent variable. This is only true if all other variables are controlled.

What are the controlled variables in this experiment?

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What variables other than those you may have already listed can you now suggest?

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Part B—Choosing or Designing the ProcedureThe procedure is the stepwise method, or sequence of step, to be performed for the experiment. It should be recorded in a laboratory notebook before initiating the experiment and any exceptions or modifications should be noted during the experiment. The procedures may be designed from research published in scientific journals, through collaboration with colleagues in the lab or other institutions, or by means of one’s own novel and creative ideas. The process of outlining the procedure includes determining control treatment(s), levels of treatments and numbers of replications.

Level of TreatmentThe value set for the independent variable is called the level of treatment. For this experiment, the value was determined based on previous research and preliminary measurements of sulfur dioxide emissions. The scientists may select a range of concentrations from no sulfur dioxide to an extremely high concentration. The levels should be based on knowledge of the system and the biological significance of the treatment level. What was the level of treatment in the soybean experiment?____________________________________________________________________________________________

____________________________________________________________________________________________

ReplicationScientific investigations are not valid if the conclusions drawn from them are based on one experiment with one or two individuals. Generally, the same procedure will be repeated several times (replication), providing consistent results. Notice that scientists do not expect exactly the same results inasmuch as individuals and their responses will vary. Results from replicated experiments are usually averaged and may be further analyzed using statistical tests. Describe replication in the soybean experiment. _____________________________________________________________________________________________

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Control or Control GroupThe experimental design includes a control in which the independent variable is held at an established level or is omitted. The control or control treatment serves as a benchmark that allows the scientist to decide whether the predicted effect is really due to the independent variable. In the case of the soybean experiment, what was the control treatment?

______________________________________________________________________________________________

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What is the difference between the control and the controlled variables discussed previously?

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Part C—Writing HypothesesWhen rigorous science is conducted, experimental design generally includes a Null Hypothesis as well as Alternative Hypotheses. A Null Hypothesis is one that can be tested statistically and generally states that there is no effect on the experimental organisms when exposed to the treatment. The Alternative Hypothesis predicts what the investigator thinks will happen in the experiment. The prediction is always based on the particular experiment designed to test a specific hypothesis. Hypotheses/Predictions are written in the form of if/then statements: “If [the independent variable] does this, then the [dependent variable] does this” For example, “if cactus spines prevent herbivory, then removal of the spines will result in cacti being eaten by animals.” Making a prediction provides a critical analysis of the experimental design. If the predictions are not clear, the procedure can be modified before beginning the experiment. For the soybean experiment, the idea being tested was: “Exposure to sulfur dioxide reduces reproduction.” State the Null Hypothesis and an Alternative Hypothesis.Null (HO):

______________________________________________________________________________________________

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Alternative (HA):

______________________________________________________________________________________________

______________________________________________________________________________________________

To evaluate the results of the experiment, the investigator always returns to the prediction. If the results match the prediction, then the hypothesis is supported . If the results do not match the prediction, then the hypothesis is rejected. Either way, the scientist has increased knowledge of the process being studied. Many times the rejection of a hypothesis can provide more information than confirmation, since the ideas and data must be critically evaluated in light of new information. In the soybean experiment, the scientist may learn that the prediction is true (sulfur dioxide does reduce reproduction at the concentration tested). As the next step, the scientist may now wish to identify the particular level at which the effect is first demonstrated.

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Case Study: Catalase ActivityCatalase is an enzyme that converts hydrogen peroxide (H2O2) to oxygen and water. An experiment investigated the effect of temperature on the rate of the catalase reaction. Small (10 cm3) test tubes were used for the reactions, each containing 0.5 cm3 of enzyme and 4 cm3 of hydrogen peroxide. Reaction rates were assessed at four temperatures (10° C, 20° C, 30° C and 60° C). For each temperature there were two reaction tubes (i.e. tubes 1 and 2 were both kept at 10° C). The height of oxygen bubbles present after one minute of reaction was used as a measure of the reaction rate; a faster reaction rate produced more bubbles. The entire experiment, involving eight tubes, was repeated on two separate days.

1. What is the purpose of this experiment?

2. Construct good hypotheses for this experiment.a. Null Hypothesis

b. Alternative Hypothesis

3. What is the independent variable?________________________________________

4. What is the range of values for the independent variable?_____________________

5. What is the dependent variable?_________________________________________

6. Each temperature represents a: (circle one) treatment/sample/trial

7. Why would it have been desirable to have included an extra tube containing no enzyme?

8. Identify three variables that might have been controlled in this experiment and how they could have been monitored.

a. _______________________________________

b. _______________________________________

c. _______________________________________

9. Explain why controlled variables should be monitored carefully.

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Recording ResultsDesigning a table to record your results is part of planning your investigation. Once you have collected all your data, you will need to analyze and present it. To do this, it may be necessary to transform your data first, by calculating a mean or a rate. An example of a table for recording results is presented below. This example represents a fairly standard layout. The labels on the columns and rows are chosen to represent the design features of the investigation. The first column contains the entire range chosen for the independent variable. There are spaces for multiple sampling units, repeats (trials) and averages.

Trial 1 (plant mass in grams) Trial 2 (plant mass in grams) Trial 3 (plant mass in grams)Day no. Day no. Day no.

Plant # 0 2 4 6 8 10 0 2 4 6 8 10 0 2 4 6 8 10pH 3 1 0.5 1.1

2 0.6 1.23 0.7 1.3Av. 0.6 1.2

pH 5 1 0.6 1.42 0.8 1.73 0.5 1.9Av. 0.6 1.7

pH 7 1 0.7 1.32 0.8 1.33 0.4 1.7Av. 0.6 1.4

pH 9 1 0.5 0.62 0.9 1.13 0.8 1.0Av. 0.7 0.9

Table 1.1 Plant mass in grams for three trials of plants exposed to four different pH levels.

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Dependent variable and its units

Space for repeats of the experimental design (in this case, three trials).

Space for 3 plants at each pH

Range of values for independent variable

Case StudyCarbon Dioxide levels in a respiration chamberA datalogger was used to monitor the concentrations of carbon dioxide in respiration chambers containing five green leaves from one plant species. The entire study was performed in conditions of full light and involved three identical set ups. The CO2 concentrations were measured every minute, over a period of ten minutes, using a CO2

sensor. A mean CO2 concentration (for the three set ups) was calculated. The study was carried out two more times, two days apart.

Design a table to collect data from the case study described. Include space for individual results and averages from the three set ups.

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Graphing1. Every graph presented should have a good title that tells what the graph is about. A good title

tells exactly what information the author is trying to present with the graph and usually includes both the dependent and independent variable.

2. Axes should be carefully labeled with numbers, labels and units. Intervals marked on line graphs should be equal in size and the scale should appropriately fit data.

3. Only plot points that have been measured. Plot points accurately!

4. Guidelines for Bar Graphs5. Bar graphs are appropriate for data that are non-

numerical and discrete for at least one variable, i.e. they are grouped into separate categories. There are no dependent or independent variables. a. Data are collected for discontinuous, non-

numerical categories (i.e. place, color and species) so the bars do not touch.

b. Data values may be entered on or above the bars if you wish

c. Multiple sets of data can be displayed side by side for direct comparison (i.e. males and females in the same age group).

d. Axes may be reversed so that the categories are on the x axis, i.e. the bars can be vertical or horizontal.

6. Guidelines for Histogramsa. Histograms are plots of continuous data

and are often used to represent frequency distributions where the y axis shows the number of times a particular measurement or value was obtained. For this reason, they are often called frequency histograms.

b. The data are numerical and continuous (i.e. height or weight) so the bars touch.

c. The x axis usually records the class interval. The y axis usually records the number of individuals in each class interval (frequency).

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7. Guidelines for Circle Graphsa. Circle graphs can be used instead of bar graphs,

generally in cases where there are six or fewer categories involved. A circle graph provides strong visual impact of the relative proportions in each category, particularly where one of the categories is very dominant.

b. The data for one variable are discontinuous (non-numerical or categories)

c. The data for the dependent variable are usually in the form of counts, proportions or percentages.

8. Guidelines for Line Graphsa. Line graphs are used when one variable (the independent variable) affects another, the

dependent variable. b. Data must be continuous for both variables.c. The dependent variable is usually the

biological response.d. The independent variable is often time or

the experimental treatment. e. In cases where there is an implied trend

(e.g. one variable increases with the other), a line of best fit is usually plotted through the data points to show the relationship.

f. If fluctuations in the data are likely to be important (e.g. with climate and other environmental data) the data points are usually connected directly (point to point).

g. Take care to NOT extend the line past the last data point; if the line extends past the last data point, it should be a dashed line or have an arrow point on the end to indicate that data is being extrapolated.

h. Can be drawn with measure of error. The data are presented as points (the calculated means), with bars above and below, indicating a measure of variability or spread in the data (e.g. standard error, standard deviation or 95% confidence intervals).

i. Where no error value has been calculated, the scatter can be shown by plotting the individual data points vertically above and below the mean. By convention, bars are not used to indicate the range of raw values in a data set.

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9. Plotting Multiple Data Setsa. A single figure can be used to show two or more data sets, i.e. more than one curve can be

plotted per set of axes. This type of presentation is useful when you want to visually compare the trends for two or more treatments, or the response of one species against the response of another.

b. If the two data sets use the same measurement units and a similar range of values for the independent variable, one scale on the Y axis is used.

c. If the two data sets use different units and/or have a very different range of values for the independent variable, two scales for the y axis are used. The scales can be adjusted if necessary to avoid overlapping plots.

d. The two curves must be distinguished with a key.

10. Guidelines for Scatter Graphsa. The data for this graph must be continuous for

both variables. b. There is no independent variable, but the

variables are often correlated, i.e. they vary together in some predictable way.

c. Scatter graphs are useful for determining the relationship between two variables.

d. The points on the graph need not be connected, but a line of best fit is often drawn through the points to show the relationship between the variables.

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11. Graph each of the sets of data below . Be sure to select the correct type of graph and follow the guidelines listed above. Each graph should be hand drawn on a separate sheet of graph paper and stapled together in the order listed below.

Graph ACounts of eight mollusk species were made from a series of quadrat (1 square meter) samples at two sites on a rocky shore. The summary data are presented below.

Table 1. Average abundance of 8 molluscan species from two sites along a rocky shore.

Species Mean (no. m2)Site 1 Site 2

Ornate limpet 21 30Radiate limpet 6 34Limpet sp. A 38 0Cats-eye 6 2Top shell 2 4Limpet sp. B 57 39Limpet sp. C 0 2Chiton 1 3

Graph BA census of a deer population on an island indicated a population of 2000 animals in 1960. In 1961, ten wolves (natural predators of deer) were brought to the island in an attempt to control deer numbers. Over the next nine years, the number of deer and wolves were monitored. The results of these population surveys are presented below.

Table 2. Results of a population survey on an islandTime (yr) Wolf Numbers Deer Numbers

1961 10 20001962 12 23001963 16 25001964 22 23601965 28 22441966 24 20941967 21 19681968 18 19161969 19 1952

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Graph CThe results shown below were collected in a study investigating the effect of temperature on the activity of an enzyme.

Table 3. An enzyme’s activity at different temperatures.Temperature °C Rate of reaction

(mg of product formed per minute)

10 1.020 2.130 3.235 3.740 4.145 3.750 2.760 0

Graph DIn a sampling program, the number of perch and trout in a hydro-electric reservoir were monitored over a period of time. A colony of black shag was also present. Shags take large numbers of perch and (to a lesser extent) trout. In 1960-61, 424 shags were removed from the lake during the nesting season and nest counts were made every spring in subsequent years. In 1971, 60 shags were removed from the lake, and all existing nests dismantled. Use a vertical arrow on the graph to indicate the point at which the shags & nests were removed.

Table 4. Results of a population survey of trout and perch at a reservoir between 1960 and 1978.

TimeFish Number

(average per haul) Shag nest numbersTrout Perch

1960 -- -- 161961 -- -- 41962 1.5 11 51963 0.8 9 101964 0 5 221965 1 1 251966 1 2.9 351967 2 5 401968 1.5 4.6 261969 1.5 6 321970 1.5 6 351971 0.5 0.7 421972 1 0.8 01973 0.2 4 01974 0.5 6.5 01975 0.6 7.6 21976 1 1.2 101977 1.2 1.5 321978 0.7 2 28

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Graph EMetabolic measurements were taken from seven Antarctic fish Pagothenia borchgrevinski. The fish

are affected by a gill disease, which increases the thickness of the gas exchange surfaces and affects oxygen uptake. The results of oxygen consumption of fish with varying amounts of affected gill (at rest and swimming) are tabulated below.

Table 5. Oxygen consumption of fish with affected gills.Fish

NumberPercentage of gill

affectedOxygen consumption (cm3g-1h-1)

At rest Swimming1 0 0.05 0.292 95 0.04 0.113 60 0.04 0.144 30 0.05 0.225 90 0.05 0.086 65 0.04 0.187 45 0.04 0.20

Graph FImpact failure threshold of 1018 cold rolled steel

Temperature (deg C) Impact Energy (joules) Mean Standard Deviation

-195 1.4 0.55 0 62.2 20.79

20 70.4 14.45 100 77.3 4.32

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Analysis Questions to answer about graphs:Graph A1. What can you conclude about this set of data?

Graph B1. Explain the relationship between wolf numbers and deer numbers as depicted by this graph.

Graph C1. Estimate the rate of reaction at 15°C.

Graph D1. Describe the evidence suggesting that the shag population is exercising some control over

perch numbers.

2. Describe evidence that the fluctuations in shag numbers are related to fluctuations in trout numbers.

Graph E1. Describe the relationship between the amount of gill affected and oxygen consumption in the

fish for the at rest data.

2. Describe the relationship between the amount of gill affected and oxygen consumption in the fish for the swimming data set.

3. Describe how the gill disease affects oxygen uptake in resting fish.

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Interpreting Line Graphs

For each of the graphs above, give a description of the slope and an interpretation of how one variable changes with respect to the other. For the purposes of your description, call the independent variable in each example “variable X” and the dependent variable “variable Y”. Be aware that the existence of a relationship between two variables does not necessarily mean that the relationship is causative (although it may be).

(a) Slope: Positive linear relationship, with constantly rising slopeInterpretation: Variable Y (transpiration) increases regularly with increase in variable X (wind speed)

(b) Slope: __________________________________________________________________________

Interpretation: ___________________________________________________________________

(c) Slope: __________________________________________________________________________

Interpretation: ___________________________________________________________________

(d) Slope: __________________________________________________________________________

Interpretation: ___________________________________________________________________

(e) Slope: __________________________________________________________________________

Interpretation: ___________________________________________________________________

(f) Slope: __________________________________________________________________________

Interpretation: ___________________________________________________________________

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Descriptive Statistics For most investigations, measures of the biological response are made from more than one sampling unit. The sample size (the number of sampling units) will vary depending on the resources available. In lab based investigations, the sample size may be as small as two or three (e.g. two test tubes in each treatment). In field studies, each individual may be a sampling unit, and the samples size can be very large (e.g. 100 individuals). It is useful to summarize the data collected using descriptive statistics. Descriptive statistics, such as mean, median, mode, and range can help to highlight trends or patterns in the data. Each of these statistics is appropriate to certain types of data or distributions, e.g. a mean is not appropriate for data with a skewed distribution. Frequency graphs are useful for indicating the distribution of data. Standard deviation and standard error are statistics used to quantify the amount of spread in the data and evaluate the reliability of estimates of the true (population) mean.

Variation in DataWhether they are obtained from observation or experiments, most biological data show variability. In a set of data values, it is useful to know the value about which most of the data are grouped; the center value. This value can be the mean, median or mode depending on the type of variable involved. The main purpose of these statistics is to summarize important trends in your data and to provide the basis for statistical analyses.

Statistic Definition and Use Method of CalculationMean The average of all data entries

Measure of central tendency for normally distributed data

Add up all the data entries Divide by the total number of data

entriesMedian The middle value when data

entries are placed in rank order A good measure of central

tendency for skewed distributions

Arrange the data in increasing rank order

Identify the middle value For an even number of entries, find the

midpoint of the two middle valuesMode The most common data value

Suitable for bimodal distributions and qualitative data

Identify the category with the highest number of data entries using a tally chart or a bar graph

Range The difference between the smallest and largest data values

Provides a crude indication of data spread

Identify the smallest and largest values and find the difference between them

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Distribution of DataVariability in continuous data is often displayed as a frequency distribution. A frequency plot will indicate whether the data have a normal distribution (A), with a symmetrical spread of data about the mean, or whether the distribution is skewed (B), or bimodal (C). The shape of the distribution will determine which statistic (mean, median or mode) best describes the central tendency of the sample data.

When to NOT calculate a mean:a. Do NOT calculate a mean from values that are

already means (averages) themselves.b. Do NOT calculate a mean of ratios (e.g. percentages)

for several groups of different sizes; go back to the raw values and recalculate

c. Do NOT calculate a mean when the measurement scale is not linear (e.g. pH units are not measured on a linear scale).

Measuring SpreadThe standard deviation is a frequently used measure of the variability (spread) in a set of data. It is usually presented in the form x± s. In a normally distributed set of data, 68% of all data values will lie within one standard deviation (s) of the mean (x ¿and 95% of all data values will lie within two standard deviations of the mean. A large standard deviation indicates that the data have a lot of variability. A small sample standard deviation indicates that the data are clustered close to the sample mean.

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Two different sets of data can have the same mean and range, yet the distribution of data within the range can be quite different. In both the data sets pictured in the histograms below, 68% of the values lie within the range x±1 s and 95% of the values lie within x±2 s. However, in B, the data values are more tightly clustered around the mean.

Calculating Standard Deviation:

Set up a table like the one below to easily calculate standard deviation.

Data: 2, 5, 9, 12, 15, 17Calculate mean: 2 + 5 + 9 + 12 + 15 + 17 = 60 60/6=10Use value from table to calculate s:

s=√ 1686−1=√33.6=5.8

x± s10 ± 5.8

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x x−x ( x−x )2

2 2-10 (2-10)2 645 5-10 (5-10)2 259 9-10 (9-10)2 1

12 12-10 (12-10)2 415 15-10 (15-10)2 2517 17-10 (17-10)2 49

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Calculation of Descriptive Statistics

1. A survey of the number of spores found on the fronds of a fern plant was conducted. The data is listed below:

Raw data: Number of spores per frond64 60 64 62 68 66 6369 70 63 70 70 63 6271 69 59 70 66 61 7067 64 63 64

Calculate each of the following—show work for all!a. Mean

b. Median

c. Mode

d. Range

e. Standard deviation

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Reliability of the MeanYou have already seen how to use the standard deviation (s) to quantify the spread or dispersion in your data. The variance (s2) is another such measure of dispersion, but the standard deviation is usually the preferred of these two measures because it is expressed in the original units. Usually you will also want to know how good your sample mean (x ¿ is an estimate of the true population mean (µ). This can be indicated by the standard error of the mean (or just standard error—SE). SE is often used as an error measurement simply because it is small, rather than for any good statistical reason. However, it does allow you to calculate the 95% confidence interval (95% CI).

When we measure a particular attribute from a sample of a larger population and calculate a mean for that attribute, we can calculate how closely our sample mean (the statistic) is to the true population mean for that attribute (the parameter). For example: if we calculated the mean number of carapace spots from a sample of six ladybird beetles, how reliable is this statistic as an indicator of the mean number of carapace spots in the whole population? We can find out by calculating the 95% confidence interval.

Reliability of the Sample MeanWhen we take measurements from samples of a large population, we are using those samples as indicators of the trends in the whole population. Therefore, when we calculate a sample mean, it is useful to know how close that value is to the true population mean. This is not merely an academic exercise; it will enable you to make inferences about the aspect of the population in which you are interested. For this reason, statistics based on samples and used to estimate population parameters are called inferential statistics.

The Standard ErrorThe standard error is simple to calculate and is usually a small value. SE is given by:

SE= s√n

Where s = standard deviation and n = sample size.Standard errors are sometimes plotted as error bars on graphs, but it is usually more meaningful to plot the 95% CI.

95% Confidence IntervalThis value is usually written as mean ± 95%CI. A 95% confidence limit tells you that, on average, 95 times out of 100, the limits will contain the true population mean.

Once researchers have developed a hypothesis, designed an experiment, collected data and applied a number of descriptive statistics that summarize the data visually, they can apply the standard error statistic as an inference to describe the confidence they have that the means of the sample represent the true means.

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SUMMARY INFO

Standard DeviationThis quantifies the spread of the data around the mean.The larger the standard deviation, the greater the spread of data around the mean.

Standard Error and 95% confidence limitsUse this test when:

You wish to find out if there is a significant difference between two means;The data are normally distributed; when plotted as a graph it forms a bell shaped

curve.The sizes of the samples are at least 30.Example: Measuring a continuous variable such as height of males and females in a

population.The measured values always show a range from a minimum to a maximum value.The size of the range is determined by such factors as:

Precision of the measuring instrumentIndividual variability among the objects being measured.

Biologists like to be confident that the data they collect is within acceptable limits of variance from the mean.

Standard Error of the MeanUsed to provide confidence limits around the mean.Providing no other factors other than chance influence the results, the means of

future sets of data should fall within these.

95% Confidence LimitsBiologists like to be 95% confident that the mean of any data achieved only varies

from the mean of previously recorded data by chance alone.This range is roughly between -2 and +2 times the standard error.The probability (p) that the mean value lies outside those limits is less than 1 in 20 (p = <0.05 ).Multiply the standard error by 2 then add this to and subtract it from the mean.Do this for both sets of data.If the 95% confidence limits do not overlap there is a 95% chance that the two

means aredifferent.

You can reject the null hypothesis and say:There is a significant difference between the means of the two samples at the 5% level of probability

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Practice

A student investigated the variation in the length of bivalve shells at two locations on a rocky shore.

State the Null Hypothesis:

State an Alternative Hypothesis:

Data CollectedShell Length in mmGroup A Group B

46 2350 2845 4145 3163 2657 3365 3573 2155 3879 3062 3659 3871 4568 2877 42

Complete the table below:Group A Group B

MeanMedianModeRangeSE**95% CI

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Show ALL WORK!!!

**CI = mean ± (SE x 2)Based on the statistics on the previous page, should you accept or reject the Null Hypothesis? Explain.

Based on the statistics calculated on the previous page, what can you conclude?

What does this data and the statistics tell us about the two sets of bivalves and their environment?

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The Chi Square Test

This is a statistical test used to compare observed results with expected results. The calculation generates a X2

value. The higher the value of X2 the greater the difference between the observed and expected results.

Use this test when:The measurements relate to the number of individuals in a particular categoryThe observed number can be compared with an expected number which is calculated from theory

How to use a Chi Squared Test:1. State the Null Hypothesis

a. This statement says that there is no statistical difference between the observed and expected results

2. Calculate the expected valuea. This may be the mean of the expected valuesb. When you study inheritance, you would add up the expected values and apply a ratio

3. Calculate the Chi Squared value

∑ (o−e ) 2e

Where O = observed valueE = Expected value

4. Determine the degrees of freedoma. This is calculated from the formula n-1 where n = the number of sets of resultsb. This basically measures how many classes of results can freely vary in their numbers. For example,

if you were tossing two coins at a time and had an accurate count of how many 2 heads and 2 tails tosses were observed, then you already know how many of the 100 tosses ended up as mixed head-tails, so the third measurement provides no additional information.

5. Compare the Chi Squared value against a table of critical values.a. Refer to the degrees of freedomb. Look up the chi-square value you calculated on the table below. c. Use the table to determine the p value.

6. Decide whether to accept or reject the Null Hypothesis.a. If the p value is greater than .05 then you would accept (fail to reject) your null hypothesis .b. If the p value is .05 or less then you would reject your null hypothesis.

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7. Draw a conclusion.a. The Null hypothesis states that there is no difference between your observed and expected results. b. If p ≤ .05 then Null Hypothesis is Rejected.

i. This means that if the difference between your observed data and your expected data would occur due to chance alone fewer than 1 time in 20 (p=.05) then there is a 95% chance that the differences between your observed and your expected data are due to some other factor beyond chance.

ii. This means there is a statistically significant difference between what you expected and what you observed and you would reject your null hypothesis.

c. If p>.05 then the Null Hypothesis is Accepted.i. This means that there is no statistical difference between the observed and expected

results. ii. Any difference between the observed and expected results are due to random events and

doesn’t represent any sort of biological phenomenon.

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Chi Square PracticeNaked mole rats are a burrowing rodent native to parts of East Africa. They have a complex social structure in which only one female (the queen) and one to three males reproduce, while the rest of the members of the colony function as workers. Mammal ecologists suspected that they had an unusual male to female ratio. They counted the numbers of each sex in one colony.

Sex Number of AnimalsFemale 52Male 34


What is the sex ratio you would EXPECT in the population?

Calculate the Chi Squared Value below:Sex Observed Expected o-e (o-e)2 (o-e)2/eFemale 52Male 34

Total

X2 ___________

What are the degrees of freedom? _________

Based on the degrees of freedom and calculated Chi Squared value what is the p value? (LOOK IT UP ON THE TABLE) ___________

Properly state your conclusion below based on the p value.

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Chi Squared PracticeYou have been wandering about on a seashore and you have noticed that a small snail (the flat periwinkle) seems to live only on seaweeds of various kinds. You decide to investigate whether the animals prefer certain kinds of seaweed by counting numbers of animals on different species. You end up with the following data:

Type of Seaweed Number of animals on each kind of seaweed

Serrated wrack 45Bladder wrack 38Egg wrack 10Spiral wrack 5Other algae 2

TOTAL 100


What proportions would you EXPECT to see of each seaweed?

Calculate the Chi Squared Value below:Seaweed Observed Expected o-e (o-e)2 (o-e)2/eSerrated wrack 45Bladder wrack 38Egg wrack 10Spiral wrack 5Other algae 2

Total 100

X2 ___________

What are the degrees of freedom? _________

Based on the degrees of freedom and calculated Chi Squared value what is the p value? (LOOK IT UP ON THE TABLE) ___________

Properly state your conclusion below based on the p value.

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Practice!Analyze each experiment’s strengths and weaknesses by answering the questions that follow.Experiment 1:Five tomato plants of the same height were placed in the same size pots, in the same type of soil and each was given the same amount of water. Each plant was under a light bulb of the same intensity as the others but each light was of a different color. Each day, the plants were given light (each its own color) for 12 hours and left in darkness for 12 hours. The height of each plant was measured in centimeters at the end of each week for 10 weeks.

Week #→ 1 2 3 4 5 6 7 8 9 10Light Color ↓ Height of plants (cm)Yellow 4 5 6 7 8 9 10 11 12 13Green 4 4 4 3 3 2 2 1 0 0Blue 4 4 4 5 5 5 5 6 6 6Purple 4 4 5 5 6 6 7 7 8 8Red 4 5 6 7 8 9 10 11 12 13

a. What question is being tested in this experiment?

b. Write one null hypothesis and two alternative hypotheses for this experiment. A null hypothesis is one that says that the treatment will have no effect while an alternative hypothesis is one that makes a prediction of a specific effect.

H0:

Ha:

Ha:

c. What is the independent variable?________________________________________________________________

d. What is (are) the dependent variable(s)?___________________________________________________________e. What factors are being held constant in this experiment?

f. Is there a control or control group? If not, what control would you suggest?

g. What was the sample size in this experiment?_______h. Describe any replicates included within this experiment.

i. What are two conclusions that could be drawn from this experiment?(1)

(2)

J. Describe the strengths of this experiment.

K. Describe the weaknesses of this experiment.

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Experiment 2Five soup cans were painted black and five cans were painted white. A quarter liter of 24°C water was added to each can each morning at 8am and then the temperature of the water in each can was recorded in degrees Celsius at noon each day for seven days.

Temperature in °CMon Tues Wed Thurs Fri Sat Sun

Black cans

1 45 37 40 41 32 35 402 45 37 40 41 32 35 403 45 37 40 41 32 35 404 45 37 40 41 32 35 405 45 37 40 41 32 35 40

White cans

1 41 33 36 37 28 31 362 41 33 36 37 28 31 363 41 33 36 37 28 31 364 41 33 36 37 28 31 365 41 33 36 37 28 31 36

a. What question is being tested in this experiment?

b. Write one null hypothesis and two alternative hypotheses for this experiment. H0:

Ha:

Ha:

c. What is the independent variable?___________________________________________________________________

d. What is (are) the dependent variable(s)?______________________________________________________________

e. What factors are being held constant in this experiment?_________________________________________________

f. Is there a control or control group? If not, what control would you suggest?

g. What was the sample size in this experiment?_________________________h. Describe any replicates included within this experiment.

i. What are two conclusions that could be drawn from this experiment?(1)

(2)

J. Describe the strengths of this experiment.

K. Describe the weaknesses of this experiment.

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A group of students decided to investigate whether there is really an average body temperature for all humans. Their research question was: Is 98.6°F actually the average body temperature for humans? Imagine that you have designed a study to answer this question and have randomly selected 130 healthy 18- to 40-year-old adults—65 males and 65 females. You have measured their body temperature, orally, at the same time of day to a 10th of a degree precision. Table 4 shows the data for body temperature collected, in degrees Fahrenheit.

You will be trying to answer the following question: How well does this sample represent the entire population of humans?

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1. What should you construct first to determine how data is distributed? ___________________a. Is the data parametric or nonparametric?__________________________

2. Based on the distribution of the data, calculate the appropriate descriptive statistic(s) for this set of data.

a. IF data is normally distributed what should you calculate? _______________b. IF data is nonparametric what should you calculate? ___________________

3. The next question to answer is: Does the mean value of the data set above represent an abnormal mean temperature (representing a true difference between the accepted mean and the one we calculated), or is the difference an acceptable average that just happened by chance, since we tested only 130 individuals and not the entire human race?

a. How would you test the variability in the data?_________________________b. What value did you calculate for this test? ____________________________c. Explain what this value means________________________________________________

________________________________________________________________________________________________________________________________________________

4. For this study, the students randomly chose 130 people to have their body temperature measured. Discuss what would happen if the students took another random sample of 130 from the true population. Would that sample mean equal the first sample mean? Explain!!_______________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

5. What if would you expect if students repeated samplings like the one explained in #4 10 more times or even 100 more times? _________________________________________________________________________________________________________________________________________________________________________________________________________________

6. What would you calculate to determine how confident you are that the sample mean is close to the true mean? ___________________________ What formula would you use? ________________Calculate this value for this data set_______________________

7. Explain in words what your calculation in #6 means. _____________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________

8. If you did everything correctly prior to this you would know that we can be 95% confident that the true population mean lies between 98.13 F and 98.37 F. Because 98.6 F lies outside those boundaries, is there sufficient evidence to claim that 98.6 F is wrong and 98.25 F is a more accurate representation of the true population mean? You could use a T test to determine this and would get at t-value = (98.25-98.6)/.06 = -5.45. If df = 129 then is much less than .0003. Explain how you would interpret these results.

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