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What is a Hypothesis?

As you learned in the lesson in this module titled "Types of Investigations," there are many types of investigations that scientists perform. We examined three types.

Look carefully at the hypothesis and variables manipulated columns. Notice that in descriptive investigations there is no hypothesis and no variables manipulated but in comparative and experimental investigations there is both a hypothesis and variables that are manipulated.

A hypothesis is a proposed explanation for an observable phenomenon. For a hypothesis to be used in science, one must be able to test it in the form of an experiment. The hypothesis is not just an “educated guess.” Hypotheses should be based on previous observations that cannot be explained with available information. A scientific investigation is designed to test this hypothesis.

In an experimental investigation, the hypothesis should establish a cause and effect relationship. The cause is the independent variable in the investigation and the effect or result is the dependent variable and is a prediction of what the student thinks will happen.

In a comparative experiment, the hypothesis should identify the relationship between the independent and dependent variable. Often the hypothesis of a comparative experiment identifies which level or condition of the independent variable will perform the best when the dependent variable is applied.

The basic form of a hypothesis is usually similar to one of the following:

How to Write a Hypothesis

The first step in writing a hypothesis is to identify the independent and dependent variables in the experiment. (If you are not sure what the independent and dependent variables are, review the lesson titled “Experimental Design” in the Related Resources.)

How does the amount of fertilizer affect plant growth (height)?

Look at the scientific research question above. See if you can identify the independent and dependent variables.

Now that you have identified the independent and dependent variable, you can easily write the hypothesis. The hypothesis should be written in the form of an “if-then” or “as-the” statement linking a change in the independent variable to a predicted change in the dependent variable.

An easy way to write a hypothesis is to follow this simple formula.

If the (independent variable) (verb or verb phrase describing how the independent variable is changed) , then the (dependent variable) will (verb or verb phrase predicting how the dependent variable will change) .

The picture below shows the setup of the fertilizer and plant growth experiment. 

plants in an experimental investigation

Notice there are different amounts of fertilizer (zero drops, 2 drops, 4 drops and 6 drops) being applied to the plants. These are the levels or conditions of the independent variable. The conditions or levels of the independent variable will determine the verb you add to the first part of the hypothesis. In this investigation, the number of drops of fertilizer is increasing (from zero up to 6 drops) so you could use the word increases in the first part of your hypothesis.

The verb that follows the dependent variable in the second part of the hypothesis should be your prediction of what will happen to the dependent variable. So, a sample hypothesis might be as follows: 

If the amount of fertilizer is increased , then plant growth will increase .

Sometimes you will see hypothesis written as an “as-the” statement. You would write the same hypothesis but use the words “as-the” instead of “if-then.” For example,

If the amount of fertilizer applied to plants is increased , then plant growth (height) will increase .

As the amount of fertilizer increases the plant growth increases .

Writing a hypothesis for a comparative investigation can be a little more difficult than you think. Remember, the hypothesis of a comparative experiment identifies which level or condition of the independent variable will perform the best when the dependent variable is applied.

Writing a Hypothesis: Practice 1

Read the lab scenario below.

Pendulum Lab     Cut a piece of string 10 cm, 20 cm, 30 cm, 40 cm, and 50 cm.     Tie a paperclip to one end of the 10 cm string.     Tape a pencil to the end of the table.     Tie the other end of the string to the pencil.     Tape the string in place with a small piece of masking tape.     Hang three metal washers on the paperclip.     Raise the washers to be level with the edge of the table.     Start the stopwatch and release the pendulum.     Count the number of times the pendulum makes a complete trip (a period) in 10 seconds.     Record your results in your data table.     Repeat this experiment for four more trials.     Remove the pendulum from the pencil. Disassemble the pendulum by removing the washers and paper clip.     Repeat steps 2-10 with the remaining lengths of string.

Writing a Hypothesis: Practice 2

Watch the video and read the scenario to create a hypothesis for this experiment.

Source: The Extreme Diet Coke & Mentos Experiments, Zorro103, YouTube

Mixing chewy mint candies with a carbonated drink such as diet cola will produce a fountain of soda that spews from the bottle. You decided to do an experiment using diet cola and different types of chewy mint candies. You decided to use a diet soda instead of a regular soda so that your clean up is easier; it won’t have the sugary, sticky residue that comes from regular sodas. Below is the procedure and results that you observed.

In your observations about the chewy mint candies, you notice that there are small pits all over the surface of the chewy mint candies, but the fruit-flavored chewy candies have a smooth coating on them. In your observations, you write that the diet cola with the mint chewy candies looked like a geyser as the diet cola was streaming from the bottle. You note that the diet cola with the fruit-flavored chewy candies remained intact and unchanged.

Copy and paste the link code above.

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May 4, 2017

How to Science 2: Making Hypotheses and Testing Them

How we know what we know about the natural world.

In our last How to Science post, we talked about forming good research questions and applying that to Experiment proposals. In this post, we’ll zoom out of the scientific process and enter the realm of hypotheses, statistics, reasoning and assumptions.

Whether you strive to become a scientist or not, it’s important to know how scientists come to the conclusions they do, so you can understand the effects of their products. How does a scientist decide what information is true?

To answer this question, we have to understand hypotheses.

Once you’ve solidified your research question, you should try to develop hypotheses. Hypotheses are potential answers to your research question, or parts of your research question that are supported by prior knowledge . This is why a lot of people refer to hypotheses as ‘educated guesses’, meaning you use information to predict what your study might find.

In science, it’s important to not swap this term with the term ‘theory.’ While hypotheses are based on prior knowledge — ideas that have already been supported with evidence — we don’t know whether they’re true yet. Meaning, we don’t know whether our idea will match what happens in experiments , or in nature.

Theories, on the other hand, are substantiated with evidence, reproducible , fact based, and repeatedly offer reliable explanations of the natural world . Even if you find a lot of evidence in support of some hypothesis, it still doesn’t make it a theory. It’s a very hard club to get into!

Hypotheses are formed before you start gathering data , but they should be supported with data from previous studies or observations . Science is a living thing that builds upon itself, revises, and renews constantly. If your research question is a person, existing data make up it’s family tree. And a family history is crucial to making the best predictions and designing the best experiments!

Perhaps most importantly, hypotheses need to be testable, and so, they can’t be questions of what’s moral or ethical. To form one you need to consider how you’ll approach the research question, and what factors (called variables ) you are comparing or monitoring.

Post-testing, it doesn’t matter whether your hypothesis is supported by the evidence you gather or not. Even a refuted hypothesis can provide valuable information about a topic and define a reproducible experiment for other scientists to work from.

Types of Hypotheses

Null hypotheses assume that two variables are not related in any significant way .

Ex. There is no difference in the speed of plant growth when plants are grown with or without fertilizer.

It’s customary to make this hypothesis before the alternative hypothesis, because it allows for a lot less ambiguity in the researchers expected results. The researcher seeks to nullify, provide evidence against, or disprove the null before moving forward.

Alternative hypotheses assume that two variables are related in a significant way. They’re setup to establish a conclusion through testing.

Ex. Plants grow more quickly when fertilizer is used than when it’s not used.

These hypotheses assume, after the null hypothesis is rejected, that there is some significant relationship between two variables.

How to test hypotheses

The point of testing is to gather evidence that can help you decide if your hypothesis is true. And we test through experiments!

This step can get a bit tricky, so let’s first look at an example and then come up with some rules for good experiment making. Here’s our sample hypothesis:

Plants grow more quickly when fertilizer is used than when it’s not used.

You can start to test your hypothesis by getting two, near identical plants and growing one with fertilizer and one without fertilizer, but you’ll also have to make sure the test is reproducible and can provide the necessary data.

The first thing to consider is that the conditions the plants are growing in are identical (or at least as close as you can get), and that the plants themselves are of equal health. If one plant has more sun and fertilizer than the other, you won’t know what caused your results. If one plant has a leaf eating parasite, then it might’ve been doomed regardless of fertilizer levels.

Secondly, if you use the same species of plant then your results will be limited to that species. To make the claim that the above hypothesis is correct (to apply your data to all plants), you’d have to show that the fertilizer influences a trait that is ubiquitous in all plants, like the Calvin Cycle.

Thirdly, since not all fertilizers contain the same ingredients in the same ratios, it’ll be best to pinpoint an active ingredient in all plant fertilizers that you can suggest influences growth (as opposed to all the other ingredients). Testing multiple fertilizers would yield an even more significant result!

Then, of course, you’re going to need some way to keep track of changes to your variables. There are two kinds of variables in most experiments, independent and dependent.

Independent variables are factors a scientist modifies during an experiment (fertilizer, in this case), and dependent variables change because you’ve altered the independent variable (plant growth). They literally ‘depend’ on their independent half in order to change.

Decide on the unit you’ll be using to track changes in your variables, and then plot the relationship between them. If your data doesn’t confirm your hypothesis or you make a mistake in testing, you must record it! Research isn’t about getting everything right, it’s about exploring. Only good things can come from a refuted hypothesis.

Based on this example, we can come up with some general features that are crucial to a good experiment .

Most examples you see of experimentation and hypotheses tend to be very centered in life sciences, but physical scientists (like geologists, and physicists) use this method, too!

Physics asks especially big questions about the entire natural world. While biologists might look at a tree and chemical reactions, microbial activity, transpiration, and a species names — things that distinguish the tree from its surroundings, a physicist might notice the things that apply to all matter in the universe — like forces.

Since most things in physics can be described mathematically, the subject relies on a lot of deduction (we’ll learn what that is further down) and can come to conclusions that are logical necessities . Procedurally, the process of testing a physics hypothesis can be complex and often don’t make the best examples, but physicists still rely on making predictions, seeing what those predictions would lead to, and then comparing that result with nature and experiments.

Evaluating your results

The superstar term of evaluating results is statistical significance , which uses mathematics to assess how valid the results of an experiment are. If you perform an experiment multiple times and the data you collect shows roughly the same pattern, your results are more statistically significant than if multiple experiments have completely different patterns.

In addition to all this formal stuff, we can take a few steps back and answer the simple question: does it work? Do your results match what’s happening in the natural world? Is it effective in application? While you can’t coast on this alone, good science should withstand nature’s scrutiny.

As any scientist will eagerly tell you, science relies on evidence (which we’ve collected in the testing phase) and reasoning (how we interpret the results of our test). But, what is reasoning? Loosely, it’s thinking about things logically and rationally. But, those definitions only lead to more questions about what logic and rationality is, and who decides what it is. A tempting answer to these questions is: ‘you’ll know it when you see it’, but a more effective way to explain reasoning is by exploring it’s many forms.

Reasoning method #1: Inductive

A lot of scientific research is based on this method! It’s all about gathering evidence to support a likely conclusion. However, the conclusions a lot of research relies on aren’t logical certainties, just extreme likelihoods. This method is based on overwhelming evidence. Present what you’ve found in your study, and then tell us what it could mean.

This doesn’t mean that it’s impossible for science to come to conclusions about anything (you can do this mathematically, or by gathering observable things in nature), but most research relies on this method in the initial stages. That’s not necessarily a liability, however, as evidence is crucial to arriving at honest answers. The most important thing about this method is that it’s dynamic — constantly changing. New or contradictory evidence always has the potential to come along, and may cause your conclusions to change.

Reasoning method #2: Deductive

Mathematicians rely heavily on this method to find out what is true in their fields, and scientists use it, too! This method of reasoning starts with assertions — statements that say something is true. Then, from those assertions, you can come to logical conclusions that are guaranteed true .

For example, if you assert that x = 10 and y = 20, then x + y = 30. Any other answer is illogical.

Similarly, if you assert that all people named John wear blue pants, and you meet John Legend at one of his concerts, he has to be wearing blue pants. This example shows why it can be hard for some natural sciences to use this method. In nature, John Legend might not want to wear blue pants, so it’s hard to come up with assertions that would make deduction possible.

Reasoning method #3: Abductive

This is a kind of reasoning science doesn’t rely on too heavily. Instead, it’s the process most of us use to come to conclusions in our everyday lives. The problem, and the reason I’m including this method, is that the observations used in abductive reasoning are incomplete, while the conclusions are typically stated definitively. But, this method isn’t all bad!

Medical diagnoses and judicial cases rely on this kind of reasoning, and thinking this way can throw imagination into the picture, which can be useful for scientists, too.

Coming to conclusions

Conclusions usually begin with an equally pretentious and informative introduction, “The data suggest…”

This is because your conclusions need to be based on the data you collect! What relationships did you find within variables? What can then be said about your test subjects?

It’s important to not make conclusions that are so broad that they’re no longer supported by your evidence. For example, if you’ve found that your plants grow better with fertilizer, it might be best to hold off on saying that fertilizer could help all living things grow. Keep your statements honest. There’s no pressure to cure diseases or solve the world’s issues — every little bit of reproducible research helps!

There we have it! A summary of how scientists find truth in the world. There are certainly many ways people decide what to accept as true, and they’re usually based on some information that person has stored in their brain or some method that they’ve internalized. Although I am slightly biased, I like to think science is special because of its vigor. Regardless of the results, science strives to provide as complete information as possible, report data honestly, verify their results with nature, all the while remaining flexible enough to adapt to new information. And I think that idea can be far more valuable than memorizing a list of science’s products.

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Dr. Jacqueline McLaughlin, PhD. Biology 110 section 001 September 6, 1996

We have made the observation that not all coffee bean plants mature identically. We have come up with the following hypothesis: Nutrient resources in fertilizers are essential to coffee bean growth, lack of fertilizer retards growth.

20 coffee bean seeds, soil from a constant source, fertilizer with a known amount of nitrogen and phosphorus, pots to plant the seeds, a constant UV light source.

1.There will be two groups of seeds with 10 plants each: a) the seeds which have fertilizer (independent variable); and b) the seeds which do not have fertilizer (control treatment).

2. Plant all seeds in 30 cm diameter pots with soil. The fertilizer treatment will receive 10 grams of fertilizer.

3. At the end of the growing season, the number of beans,dependent variable, of each plant will be recorded.

Control Treatment:

The plants which have natural (no fertilizer) soil conditions.

Data Interpretation:

A histogram will be used to plot the results. The average number of beans for each group of plants will be plotted on the Y axis (ordinate) and the treatment group will be plotted on the X axis (abscissa). A t-test will be performed to determine if the treatment group differs from the control group. If the treatment group produces more seeds than the control, we can then conclude that the treatment of fertilizer had an effect and the resource in question is limited to plants.


Morgan, I.G. and Brown Carter, M.E., Investigating Biology: A LaboratoryManual for Biology. California: Benjamin/Cummings Publishing Co., Inc. 1993.

Main Page | Introduction and Objectives | Scientific Investigation | Experimental Procedures | Writing Procedures | Mendelian Inheritance |  Monohybrid and Dihybrid Exercises | Reference | Miscellaneous | Overview | The Question | The Hypothesis | The Experiment | The Scientific Paper | Variables | Procedures | Predictions | Protocol | Data Evaluations | Conclusions |


good hypothesis about fertilizer

Hypothesis For Fertilizer Lab Essay

Stickleback fish lab report.

If a lake is cut from the ocean and has no large fish as predators (only dragonflies), then the sticklebacks will have a better chance at surviving and reproducing because the sticklebacks will be better adapted to the environment (presence/absence of a pelvic bone) and have little to no predators. The hypothesis was supported.

Ace Fertilizer Case Study

In the case of Ace Fertilizer Company, Assistant Director of Manufacturing , Abby Conroy is faced with an ethical issue that was presented to her by boss George Smilee who is the Director of Manufacturing. Ace Fertilizer’s business is to produce lawn & gardening fertilizer, and the company is known for delivering the highest quality special order products. The fact that they can deliver on time with top quality is what helps to drive their business. They use a consistent mark up on special orders at an 80% mark up over the cost of the orders. Abby prefers to allocate indirect cost using activity-based costing for these orders, but recognizes that not all costs are driven by volume of output. Abby prepares a

Agricultural Runoff

A major problem present today is agricultural runoff. As farmers have to mass produce in order to supply for the growing population today, fertilizer is essential to improve the quality and growth of the

Spearmint Gum Lab Hypothesis

This scientific question is, does the different kind of gum increase or decrease your concentration? This question was chose because many people in the world always are looking for some tips on what helps them concentrate, and chewing gum is a very easy way to help improve concentration. The Independent Variable is the type of gum, the Control Group is Bubble Gum, the Experimental Group is the Spearmint, the Dependent Variable is test scores, the Constant Variables are same timed test, same timer, and same brand of gum. The hypothesis for this science fair project is if spearmint gum is chewed then the person's concentration will be increased. This hypothesis was picked because spearmint gum is minty and the scientist thought that maybe mint impacted the brain the most. The other two gums that were used in this experiment were Bubblegum and Watermelon. The scientist thought that spearmint would be the fastest in the multiplication and he was not really sure what to expect.

Miracle Gro Lab Report

Human error played a factor because it was not made sure that air was trapped in each bag. In the class data the amount of seeds germinated on day three was higher than the amount of seeds germinated on day four. This is not possible as the amount of seeds germinated cannot decrease, rather they should stay the same or increase. A seed cannot go from being germinated to not being germinated. This could have occurred from miscounting or not accurately counting the number of seeds germinated. If the experiment was completed again, more higher concentrations could be applied to the seeds to see if the data is further supported. Either way the more trials the better and more accurate results, this applies to any experiment. Practical applications of the principles of this investigation may be using the correct concentration for a specific plant or seed. This lab shows that it is necessary to follow the directions and specifications when using the fertilizer miracle

Radish Plant Experiment

Abstract: The purpose of this experiment was to determine whether the amount of topsoil would influence the growth of radish plants. It was hypothesized that if the amount of topsoil increased by 50% would increase because topsoil contains the essential nutrients which are required for proper plant growth. The principle findings indicated that a medium amount of topsoil is ideal for plant growth as the radish plant potted in 50 ml of topsoil experienced the most growth in comparison to the radish plants potted either in 25 ml or 200 ml of topsoil.

Effect Of Surface Area To Volume Ratio On The Rate Of Diffusion

There are independent variables, dependent variables, and controls. the one that is being tested and the one that is the inconsistent variable in the Independent variable like the volume and surface area of the agar cubes. The variable that is kept consistent is the dependent variable such as the percentage of diffusion of pigment in the agar cubes. The constant variable is the features of the experiment that is kept the same throughout the entire experiment such as, the amount of time the agar cubes are left in the beaker and the amount of acid in the beaker.

Biology Quiz

1. Free ears in dogs are controlled by dominant allele (F), and attached ears are controlled by the recessive allele (f). In addition, Short dogs is due to a dominant allele(S), and long hair is due to a recessive allele (s). Which of the following is the genotype of the dogs with free ears and short hair?

Turnip Peroxidase Lab Report

Measuring the independent variable: The pH (the independent variable) is being tested on the turnip peroxidase to observe the reaction rates. 5 levels of pH are required for these series of reactions so pH buffers of 3, 5, 7, 9, and 11 are to be placed in each of the waters that will be put into the cuvettes for the experiment.

Salt Creek And Barker Lake Lab Report

In this lab we were trying to figure out if Salt Creek and Barker Lake had the correct chemical balances to sustain catfish for the years coming. In order to find this out, we tested the water using a Hach Water Testing Kit. Inside were dissolved oxygen reagent powder pillows 1, 2 and 3 which we added and mixed into our sample water to prepare it for testing. Then we added droplets of Sodium Thiosulphate Solution into the prepared water too see how much dissolved oxygen parts per million were in the water. Our independent variable in this experiment was the 5 different testing sites that we went to for water samples. Our dependant variable in this experiment was the dissolved oxygen parts per million in each sample. We recorded this data on the white board back in our classroom, writing down the specific parts per million that we found in the water. The variables that were the same for all

Independent Variable Pre-Lab

An Independent Variable would be health conditions. For example, if one has asthma or not, this will affect the number of breaths that one makes. Thus, those health conditions cause the number of breaths to change. A Dependent Variable would the size of the bag. The Controlled Variables would be time intervals, number of people in a group, etc. The Constant Variable would be environment and temperature. The Materials needed for the experiment are pencil, paper bag, and stopwatch. There are two components of the procedures section, as there was a pre-lab activity, and the actual experiment. For the pre-lab activity, there was one group member who would breathe into the bag for three minutes, while another group member times thirty-second intervals. The other two group members would observe the

Muffin Paper Cup Experiment Essay

The dependent variable, which is the time taken for the empty muffin paper cup to touch the ground (immediately after it was dropped). This will be measured using a stopwatch, which will start when the paper cup is dropped, and stopped when the paper cup touches the ground. This will be done on table, preferably inside a science laboratory. (The table will be parallel to the muffin paper cup). Thus the measurements will repeat for each experiment, where the paper cup’s height from the ground varies in each

Rye Grass Lab

The purpose of this lab was to test one type of grass and four different types of soil in order to figure out which grass will grow tallest in 21 days. The hypothesis states that The Perennial Rye Grass will grow taller in the Miracle-Gro Organic Potting Mix than the other three grasses when given water over 21 days. The independent variable is the soil, the dependent variable is growth measured in height, and the control variables are as follows: 21 days of data collection, 1.5 mL of water a day, sunlight from 8 a.m. to 3 p.m. in greenhouse, temperature is about 70o F, 200ml of soil in each planter, five drainage holes made with a probe, .25 grams of seed, 1 cm into the soil, watered with 1.5ml twice a day and measured after the

Inorganic Fertilizer Literature Review

Fertilizers remove the nutrients of the soil damaging the soil and the local environment and after being mixed with the soil, gradually reduce the fertility of the soil. In the study of (Southland) using fertilizers consists of substances and chemicals like methane, carbon dioxide, ammonia, and nitrogen, the emission of which has contributed to a great extent in the quantity of greenhouse gases present in the environment. These facts are alarming and a serious step needs to be taken as soon as possible to avoid more severe consequences. This in turn is leading to global warming and weather changes. The use of fertilizers for growth and cultivation is keeping our stomach filled for now, but then if things keep on progressing the way they are, it won’t take long to see the times where there is lack of food, water, and health. In fact, nitrous oxide, which is a by product of nitrogen, is the third most significant greenhouse gas, after carbon dioxide and methane. Consequently, you can well imagine as to how harmful is the use of fertilizers for our environment and the ongoing use of fertilizers across the world

Gm Foods Advantages And Disadvantages Essay

World population is increasing day by day which implies scarcity of food will be the major challenge that the world will be facing in the future. Genetically modified foods can meet this rising need.

More about Hypothesis For Fertilizer Lab Essay

The first step in writing a hypothesis is to identify the independent and dependent variables in the experiment. (If you are not sure what the independent and dependent variables are, review the lesson titled “Experimental Design” in the Related Resources.)

Look at the scientific research question above. See if you can identify the independent and dependent variables.

Interactive exercise. Assistance may be required.

Now that you have identified the independent and dependent variable, you can easily write the hypothesis. The hypothesis should be written in the form of an “if-then” or “as-the” statement linking a change in the independent variable to a predicted change in the dependent variable.

An easy way to write a hypothesis is to follow this simple formula.

If the (independent variable) (verb or verb phrase describing how the independent variable is changed) , then the (dependent variable) will (verb or verb phrase predicting how the dependent variable will change) .

The picture below shows the setup of the fertilizer and plant growth experiment.

image is of 4 plants with water and fertilizer being added

Notice there are different amounts of fertilizer (zero drops, 2 drops, 4 drops and 6 drops) being applied to the plants. These are the levels or conditions of the independent variable. The conditions or levels of the independent variable will determine the verb you add to the first part of the hypothesis. In this investigation, the number of drops of fertilizer is increasing (from zero up to 6 drops) so you could use the word increases in the first part of your hypothesis.

The verb that follows the dependent variable in the second part of the hypothesis should be your prediction of what will happen to the dependent variable. So, a sample hypothesis might be as follows: If the amount of fertilizer is increased , then plant growth will increase .

Sometimes you will see hypothesis written as an “as-the” statement. You would write the same hypothesis but use the words “as-the” instead of “if-then.” For example,

Writing a hypothesis for a comparative investigation can be a little more difficult than you think. Remember, the hypothesis of a comparative experiment identifies which level or condition of the independent variable will perform the best when the dependent variable is applied.

University of Florida Institute of Food and Agricultural Sciences

Fertilizer Experimentation, Data Analyses, and Interpretation for Developing Fertilization Recommendations—Examples with Vegetable Crop Research


Fertilizer recommendations contain several important factors, including fertilizer form, source, application timing, placement, and irrigation management. Another important part of a fertilizer recommendation is the amount of a particular nutrient to apply. The optimum fertilizer amount is determined from extensive field experimentation conducted for several years, at multiple locations, with several varieties, etc. Although rate is important, rate should be considered as a part of the overall fertilization management program. The important components of a fertilizer recommendation are discussed in Hochmuth and Hanlon (2010a) Principles of Sound Fertilizer Recommendations for Vegetables , available online at https://edis.ifas.ufl.edu/ss527 . This EDIS publication focuses on the research principles behind determining the optimum rate of fertilizer, including experimentation and interpreting research results for optimum crop production and quality in conjunction with minimal environmental consequences. We use examples from research with vegetable crops in Florida. How we interpret the results is as important as how we conducted the research.

The target audience for this article includes Extension state specialists, county Extension faculty members, and professionals conducting or working with research in nutrients, agrochemicals, and crop production. The authors assume that the reader has an understanding of basic probability and statistics. Statistical information presented in this publication is intended to demonstrate the process involved in fertilizer experimentation. Explanation of the statistics and their calculations is beyond the scope of this document.


The goal of research on fertilizer rate is to determine the amount of fertilizer needed to achieve a commercial crop yield with sufficient quality that is economically acceptable for the grower. In Florida, these types of studies take a slightly different approach depending on whether soil testing for the nutrient in question is involved. For example, rate studies with nitrogen (N) on sandy soils would not involve soil testing, but rate studies with phosphorus (P) or potassium (K) would. In the case of N on sandy soils, the researcher assumes there is minimal N supplied from the soil that would satisfy the crop nutrient requirement. In the case of P or K, a properly calibrated soil test will reveal if a response (yield and fruit quality) to the nutrient is likely or not. Rate studies are best conducted on soils low in the particular nutrient so that maximum crop response is likely and that response can be modeled.

Proper experimental design and statistical data analyses are critical to interpretation of the results. Research begins with a hypothesis or a set of hypotheses. One possible hypothesis may be that there will be no effect on yield associated with N fertilization. This hypothesis, called the null hypothesis, is evaluated with an experiment to test crop yield response against a range of N rates in a field likely to produce a large response to the addition of N fertilizer.

The researcher applies a range of fertilizer rates thought to capture the likely extent of possible crop yield responses. A zero-fertilizer treatment is always included. Crop response without an actual fertilizer application demonstrates and measures the soil-supplied effects, if any. In some cases, sufficient nutrients, or at least a low portion of the crop nutrient requirement, may come from the soil, while in other cases, nutrients may come from the irrigation water.

The researcher may decide to divide the total seasonal amount of fertilizer into split-applications, following what would likely be a recommended practice for the crop being studied. Multiple applications avoid potential large losses of fertilizer because of rainfall events, especially for nutrients that are mobile in the soil. Typically, all treatment rates are handled similarly for timing and placement of the fertilizer to minimize any confounding effects with rate.

During the growing season, the researcher may sample the plant for nutrient concentrations, using whole dried leaves and/or fresh petiole sap. These samples will help the researcher prove the response in yield was related to the plant's nutrient status. Typically, soil samples are not used because there is a chance of including a fertilizer particle in the sample, or there may be questions of where to sample if the fertilizer is applied by banding or through a drip tape. Photographs taken during the season are useful for documenting both growth and potential plant deficiency symptoms.

The crop response of interest, typically marketable yield, is measured at the appropriate harvest time(s). For vegetables, the fruits are evaluated according to USDA grade standards to detect any effects of fertilization on fruit quality (size, color, sugar content, etc.). Yields are expressed in the prevailing commercial units per area of production (e.g., 28-lb boxes/acre, 42-lb crates/acre, bushels/acre, tons/acre, etc.). The raw data should be plotted in a scatter diagram (Figure 1) to gain insight into the type and magnitude of response. Plotting the raw data allows the researcher to inspect for apparent atypical data points that may illustrate errors somewhere in the data entry process.

Figure 1. Theoretical (not actually measured data) crop response to nitrogen fertilization. There were 5 replicates of each N rate (some data points are hidden behind others). Note the rapid increase in yield with the first few increments of N fertilization, then a leveling off, and possibly an indication of yield reduction with excessive fertilization. Also, note there is some yield with zero fertilization, in this case approximately 20% of the maximum yield. We will use the term percent relative yield to express this percentage of the maximum yield. There appears to be no further increase in yield after 150–200 lbs/acre N. This example is used for illustration purposes; typically in field experimentation, there is more variation among the replications, especially at the lower rates of fertilization.

Once the data have been collected and inspected, they are analyzed statistically with analysis of variance (ANOVA). Did fertilization have a significant effect on yield? ANOVA is particularly useful in cases where a researcher might be evaluating the effect of fertilizer rate across several varieties of crop. Here, the researcher is interested in whether varieties differed in response to fertilizer, which will be exposed through a significant interaction term in the ANOVA source table. If the fertilizer treatment effect was significant, then the researcher will want to graphically present the results with a mathematical equation sometimes called a "model."

In fertilizer rate experiments, the rate of fertilizer is referred to as a continuous variable because there are many possible rates in addition to the ones the researcher selected to use in the experiment. Using ANOVA, especially if the experiment had the treatments arranged in a factorial arrangement, is a good approach to test for treatment effects and interactions. Fertilizer rate main effects can be subjected to polynomial contrasts, a statistical method to determine if there are linear or quadratic components in the overall response. Then regression methods can be applied to the continuous variable to develop an equation that explains the significant trend in response (see the section below about models).

The ANOVA statistics for a randomized complete-block N experimental design (data in Figure 1) with five replications and nine N rates indicate that one or more N rate treatments were statistically different from the others (Table 1). In this case, our null hypothesis would have been rejected. Since ANOVA tables contain estimates of several variance components, these tables should be included in research manuscripts but are seldom included. For example, other researchers may be able to use this information when summarizing numerous, similar studies. While simply reporting means and treatment effects is good for a simple research report or presentation, this method does not contain measures of variance, and the ANOVA table does.

Treatment Significance

Researchers cannot study every possible experimental treatment (rate) or combinations of treatments. In addition, there is natural variation in the field where the research will be conducted. The field may have variations in organic matter, soil pH, or moisture, all of which may lead to variations in yield response having nothing to do with the N treatment(s). Therefore, the notion of probability comes into play. What are the chances that the observed differences in yield are because of natural variation from plot to plot? This inherent variability is where statistical analysis of the data helps to sort out the differences most likely caused by treatment (N fertilization) from the so-called "noise" or random error in the production system. If we repeat the application of treatments, called replication, we can estimate the relative amount of natural variation. Experiments should always include replication as part of a properly designed experiment, one that would pass a peer-review process. Analysis of variance is the mathematical tool we use for this analysis, and with this statistical tool we can test the relative proportion of the variation due to treatment effects against the variation due to chance.

The generally accepted probability level of 0.05 (5%) is used in agricultural research as the probability that there could be a real difference when ANOVA indicates no such difference. This probability level is the level of error that scientists are willing to accept. In other words, a real difference is so rare that it is of minimal practical concern. If the experiment were repeated 20 times, there would be a 1 in 20 chance that our hypothesis would not be rejected. Said another way, if the ANOVA indicates a difference between one or more treatments, we are 95% certain that this difference is a real effect. We call these differences "significant" differences. If ANOVA detects significant differences among treatment means, then we reject our null hypothesis.

In the "real world," finding no significant differences has two major implications. First, it means that farmers should not be interested in spending extra money each year (for "insurance" applications) just to gain the rare possibility of a real crop response. These unjustified expenses would reduce profitability. The second implication is the potential negative impacts on the environment when a rate of fertilization is applied to the crop when not needed.

One common misinterpretation about treatment differences needs clarification. For example, assume an experiment was conducted to test the effect of N rate on tomato yield and the ANOVA found no significant difference between the grower rate and the recommended (lower) rate at the 5% probability level. This finding means that there is such a rare chance of a real treatment difference occurring that we can be confident the grower can reduce the commercial fertilizer rate. The actual means may be 2,950 and 2,920 boxes/acre for the grower and recommended rates, respectively. An argument could be made to someone without knowledge in statistics that the 30 boxes/acre "difference" is "worth" $600 (30 boxes at $20/box) and that amount will more than pay for the added fertilizer with the grower rate. This conclusion is erroneous because the ANOVA indicated no significant difference between the two treatment means. Therefore, the appropriate representation of the response to fertilizer is the average of the two means (i. e., 2,935 boxes per acre). Said another way, other factors on the farm impact yield more than fertilizer rate.

A more complex experiment may be to test the response of two cultivars to N rate. Here, ANOVA is used to test the significance of the main effect of N rate, the main effect of cultivar, and the interaction in the response of cultivar to N rate. There are two outcomes depending on whether or not there is an interaction of N rate and cultivar (i. e., that the cultivars differed in their response to N rate). If there was no interaction, then the response to N can be averaged using both cultivar means. If an interaction is observed, then each cultivar response must be evaluated separately.

Mathematical Descriptions of the Response (Models)

In statistical terms, fertilizer rate research employs various levels of a quantitative variable, the amount of fertilizer. If the ANOVA indicates a significant N treatment effect, as in Table 1, then the researcher will wish to further evaluate the response with the development of the mathematical model. Responses to a quantitative variable can be statistically inspected along the full range of the levels of the variable, and the responses to rates in between those actually applied in the field can be calculated. In most fertilizer experiments, a set of 4 to 5 levels of fertilizer plus a zero-fertilizer control is sufficient for most models. The results can be presented graphically by an equation or model. The model can be used to predict results if a second experiment similar to the first were conducted. Models are typically developed with regression analyses.

Various models can be fit to a set of data to explain the responses. A linear model might explain a response that continues upward or downward in a straight line within the range of tested fertilizer rates. A linear response may mean the chosen range of treatments was insufficient to determine the maximum (or minimum) yield. A quadratic response is typical of crop yield in which the response increases with fertilizer rate to a point where yield approaches a maximum but then might decrease at higher rates. In other words, there is a point at which increased fertilizer does not result in a significant increase in yield. Quadratic models also typically have a linear component, meaning that as fertilizer rates increase from low to medium rates the yield also increases. At a certain point, the rate of yield increase starts to stabilize or decline.

Linear and quadratic models are the simplest equations to use for explaining crop responses to fertilizer, and they have served scientists well as long as the main interest in the research was maximizing yield. However, today there are other goals in fertilizer research, including economics and environmental issues. Several researchers have explored different models for explaining crop responses to fertilizer (see the articles in the list of references at the end of this publication). Studies have found that the quadratic model leads to overestimation of fertilizer recommendations derived from responses to fertilizer (Cerrato and Blackmer 1987; Hochmuth et al. 1993a; 1993b; 1996; Willcutts et al. 1998). If the goal of the research was to select a fertilizer rate to be used as a recommended practice, then the quadratic model will usually predict a greater fertilizer need if the maximum point from the model is taken as the putative recommendation. The maximum yield mean is not always significantly different from one or more means resulting from lesser fertilizer rates. If we inspect the plot of data in Figure 1, we might predict that there is little difference in yields among the fertilizer rates from 150 lb/acre or greater. Other models have been identified that result in a lower, but agronomically acceptable, recommended fertilizer rate, saving fertilizer expense and reducing the risk of excessive fertilizer applications that might endanger the environment.These models include the logistic and the linear-plateau models. Using the data in Figure 1, these three models are illustrated in Figure 2, Figure 3, and Figure 4.

Figure 2. The quadratic model. Yield (tons/acre) = 6.86 + 0.14N - 0.00026N2. Yield max = 25.7 tons/acre at 270 lbs/acre N.

Researchers use statistics and mathematical models as tools to help explain crop response to fertilizer. We should keep in mind that models are tools, and we should exercise care in their use. The three models depicted here have been fit to the same data set first presented in Figure 1. We know from the ANOVA that crop responded to fertilizer in a significant way, but ANOVA does not identify which fertilizer rate was superior. However, each model tells a different story about the response, if we focus only on a model's parameters. The most commonly used model in agronomic and horticultural crop response research is the quadratic model (Figure 2). The quadratic model is easy to derive by computer statistical packages, and most researchers are familiar with it from their graduate training. Also, the quadratic model is easily differentiated to show a peak yield and its associated fertilizer rate.

The problem with relying solely on the quadratic model occurs on inspection of the mean yields versus fertilizer rate. It could be argued and can be shown by orthogonal contrasts that there is a leveling-off of yield. Further, this leveling-off occurs at a fertilizer rate less than the peak yield derived from the quadratic model. In an environmentally aware society, perhaps researchers should not simply interpret the quadratic model maximum as the putative fertilizer recommendation for rate.

An optional model being used by scientists more frequently is the linear-plateau model (Figure 3). This model also yields critical model parameters, the plateau and the shoulder point. The plateau illustrates the notion that there is a leveling-off of crop yield response to fertilizer. However, the linear-plateau model shoulder point could be argued to be too conservative as a putative fertilizer recommendation.

Several recent research studies with vegetables in Florida have illustrated the challenges with the quadratic and linear-plateau models if used alone (Hochmuth et al. 1993a; 1993b). These researchers proposed using the midpoint between the shoulder point in the linear-plateau model and the peak in the quadratic model as a putative recommended rate. For our data, this midpoint would be 200 lbs/acre of N fertilizer.

A third model (Figure 4), the logistic model, has been proposed by Overman and colleagues in studies with agronomic and vegetable crops (Overman et al. 1990; 1992; 1993; Willcutts et al. 1998). The logistic model is a reasonable compromise between the quadratic and linear-plateau models. First, this model illustrates the law of diminishing returns. As the rate of nutrient is increased, the yield increases until an area of diminishing returns. Second, the slope of this model is not unusually steep. Third, the function does not pass through the origin; therefore, no negative yields would be predicted, nor are zero yields predicted with zero fertilizer added. Thus, this model accounts for native soil fertility. These attributes make the logistic model particularly useful for making fertilizer recommendations that avoid under- or over-fertilization.

In typical agronomic or horticultural crop yield response data, rarely are yields between 90% and 100% of maximum declared significantly (probability = 5%) different. Selecting 95% of maximum yield to derive the putative recommended fertilizer rate would be a conservative approach to ensure a most suitable fertilizer rate that would result in profitable yields with due diligence in considering the risk to the environment.

Using the data set above, the considerations for a fertilizer recommendation would include the following:

Quadratic model: The predicted peak crop response is 25.6 tons/acre with 270 lbs/acre N.

Linear-plateau model: The plateau yield is 25 tons/acre and the shoulder point fertilizer rate is 129 lbs/acre N.

Logistic model: 95% maximum yield (25 tons/acre) occurs at 168 lbs/acre N, and 97% maximum occurs with 190 lbs/acre.

The list above shows that, depending on the level of conservatism applied, the putative fertilizer recommendation could range from 129 to 270 lbs/acre N, a 100% difference. Selecting the midpoint between the shoulder point of the linear-plateau and the peak of the quadratic model or taking a conservative 97% maximum yield with the logistic model yields similar results. This analysis yields a putative fertilizer recommendation of approximately 200 lbs/acre N. Choosing 200 lbs/acre instead of 270 lbs/acre as the recommendation results in no sacrifice in yield but saves 70 lbs/acre of fertilizer. This is both an economic savings as well as a real removal of nutrient load from the environment.

An Example from Actual Research in Florida

The figures above are helpful to illustrate the principles of research and data presentation. What about actual data from Florida? There have been several research studies conducted with vegetables in Florida evaluating yield and fruit quality responses to fertilization with various models. One such study was conducted with watermelon (Figure 5).

Figure 5. Graph of the data for the quadratic and linear-plateau models for describing the response of watermelon to phosphorus fertilizer from the same on-farm study in northeastern Florida. Note the variation among replicates, especially with the zero-P treatment.

In the watermelon study, the shoulder point for the linear-plateau occurred at 26.4 kg ha -1 P or approximately 53 lbs/acre P 2 O 5 . The quadratic model maximum yield occurs with 75 kg ha - 1 P or 150 lbs/acre P 2 O 5 . Statistical analysis (ANOVA and contrasts) of the data showed no significant difference in yield from 50 to 200 lbs/acre P 2 O 5 . The shoulder value is on the verge of steep yield reduction with less than 53 lbs/acre P 2 O 5 , but the quadratic maximum yield occurred with excessive fertilization. The authors of this research paper proposed using the midpoint between the linear-plateau shoulder point and the quadratic maximum point as a reasonable compromise fertilization recommendation. In this case, the recommendation could be about 100 lbs/acre P 2 O 5 . This recommendation would result in considerable savings in P fertilizer compared to the current recommendation of 160 lbs/acre P 2 O 5 for soils with low or very low Mehlich-1 P concentration.

Using the logistic model (Figure 6) yields a conclusion similar to using the midpoint between the quadratic maximum and the shoulder point of the linear-plateau model. Using 97% of the maximum yield would result in a fertilizer recommendation of approximately 55 kg/ha P or 115 lbs/acre P 2 O 5 .

Figure 6. The logistic equation for describing the response of watermelon to phosphorus fertilizer from on-farm studies in northeastern Florida.

There are additional reasons (beyond environmental) for making recommendations closer to the conservative side of the response curve. There are numerous research reports about excessive fertilization, especially N, having a negative impact on yield and fruit quality. The slight depression in yield at excessive fertilizer rates, coupled with the cost of the extra fertilizer, may lead to significant reductions in farm profits. Furthermore, research results have been published in the peer-reviewed literature documenting reductions in fruit and vegetable quality parameters by excessive fertilization (Hochmuth et al. 1996; 1999).

Some Comments about Percent Relative Yield (RY)

Crop responses are an integration of many different aspects of the entire production system to which the crop is exposed. Research completed during one season is affected by the crop integration process during that entire season, as well as some antecedent contributors, such as nitrogen mineralization from crop residue or soil organic matter. The problem with crop responses associated with different experiments conducted by separate research groups, and often for different purposes, is that the observed crop yields in each of the individual experiments will display variation. Plotting all the data from many experiments in the original units yields a scatter-graph that renders a general interpretation very difficult. One method that can be used to get a sense of the crop response to fertilization across numerous studies is the percent relative yield . The highest yield obtained in that particular experiment in that particular season is assigned as 100% relative yield. All other yields are calculated by dividing the observed yield by the highest actual yield and are expressed as a percentage.

Transforming the original data in this manner adds to the flexibility of looking at the relative yields, which have been brought to a common scale. The value of this type of transformation is that researchers get a sense of how that particular crop responded to fertilizer additions throughout many seasons, locations, and production practices. Relative yield should be used with caution to avoid putting too much emphasis on this data transformation and resulting graph alone. For example, using all the RY values from several experiments for subsequent regression can be quite misleading, especially for calculating actual yields. However, noticing that the variability among all responses decreases after fertilizer rate exceeds a certain range becomes quite obvious.

There are a number of assumptions built into this transformation process. The primary assumption is that most or all of the response that we note in an RY graph is due to fertilizer. There have been extensive arguments both for and against making this assumption. In summarizing this debate, Black (1992) indicates that the assumption can be considered valid when using the RY plot to explore variation across the years, seasons, and other production practices. Black cautions the reader to avoid additional statistical evaluations of the RY plot due in part to its statistical characteristics (not normally distributed) and the true shape of the yield response to added fertilizer is site-specific. The RY plot generalizes the site-specific variations in nature of soil, fertilizer, climate, and plant interactions. Problems with this generalization are avoided if the RY plot is not used for subsequent regression analysis involving actual yields and further interpretation.

For those who are interested in statistics, this type of transformation also has a weighting factor based upon the selection of the maximum yield. Again, this weighting factor makes the assumptions above and is reduced to insignificance by using the RY plot on a visual basis only and not trying to further statistically analyze the regression by other means. Black (1992) states that while these objections are worthy of note, the RY plot can be a useful tool in fertilizer research.

To further illustrate the usefulness of the percent relative yield approach, watermelon yield is plotted in Figure 7. Note that the yields increase in all experiments and then tend to level off somewhere between 100 and 200 lbs/acre N. The current UF/IFAS N recommendation is 150 lbs/acre N. While this graph was not used to set the UF/IFAS recommendation, the graph indicates that the recommendation is reasonable and supported by research.

Figure 7. Percent relative yield for drip-irrigated watermelon response to increasing N rates across several studies in Florida. The number in parentheses after the indicated year of the report is the year in which that experiment was conducted (Hochmuth and Hanlon 2010b).

Crop fertilizer response research should be carefully conducted to account for the economics to the grower and protection of the environment from nutrient losses due to excessive fertilization. There are several mathematical models to describe crop yield response to fertilizer, and these models should be employed with caution. Using a single model to explain crop response may not account for economics and potential environmental impact together. This problem is evident with the quadratic and linear-plateau models. Incorporating both models in the data response interpretation and calculating the midpoint as we have demonstrated above will consider both goals. The logistic model appears to be the best single model at considering both economics and environmental goals. There is increasing accumulation of research documenting the impacts of over-fertilization on yield and quality, thus reducing profits. Added to these reasons is the need to protect the environment from nutrient pollution related to farming activities. It becomes evident that how research is conducted and how the data are analyzed and interpreted are critical to developing an informed fertilizer recommendation.

Black, C. A. 1992. Soil Fertility Evaluation and Control . Boca Raton, FL: Lewis Publishers.

Bullock, D. G., and D. S. Bullock. 1994. "Quadratic and Quadratic-plus-plateau Models for Predicting Optimal Nitrogen Rate of Corn: A Comparison." Agron. J. 86:191-5.

Cerrato, M. E., and A. M. Blackmer. 1987. "Comparison of Models for Describing Corn Yield Response to Nitrogen Fertilizer." Agron. J. 82:138-43.

Dahnke, W. C., and R. A. Olson. 1990. "Soil Test Correlation, Calibration, and Recommendation." In Soil Testing and Plant Analysis , 3 rd edition, edited by R. L. Westerman, 45-71. Madison, WI: Soil Sci. Soc. Amer.

Hochmuth, G. J., E. E. Albregts, C. C. Chandler, J. Cornell, and J. Harrison. 1996. "Nitrogen Fertigation Requirements of Drip-irrigated Strawberries." J. Amer. Soc. Hort. Sci. 121:660-5.

Hochmuth, G. J., J. K. Brecht, and M. J. Bassett. 1999. "N Fertilization to Maximize Carrot Yield and Quality on a Sandy Soil." HortScience 34(4): 641-5.

Hochmuth, G. J., J. Brecht, and M. J. Bassett. 2006. "Fresh-Market Carrot Yield and Quality Responses to K Fertilization of a Sandy Soil Validated by Mehlich-1 Soil Test." HortTechnology 16:270-6.

Hochmuth, G. J., and E. A. Hanlon. 2010a. Principles of Sound Fertilizer Recommendations . SL315. Gainesville: University of Florida Institute of Food and Agricultural Sciences. https://edis.ifas.ufl.edu/ss527 .

Hochmuth, G. J., and E. A. Hanlon. 2010b. Summary of N, P, and K Research with Watermelon in Florida. SL325. Gainesville: University of Florida Institute of Food and Agricultural Sciences. https://edis.ifas.ufl.edu/cv232 .

Hochmuth, G. J., E. A. Hanlon, and J. Cornell. 1993a. "Watermelon Phosphorus Requirements in Soils with Low Mehlich-1 Extractable Phosphorus." HortScience 28:630-2.

Hochmuth, G. J., R. C. Hochmuth, M. E. Donley, and E. A. Hanlon. 1993b. "Eggplant Yield in Response to Potassium Fertilization on Sandy Soil." HortScience 28:1002-5.

Nelson, L. A., and R. L. Anderson. 1977. "Partitioning of Soil-test Response Probability." In Soil Testing: Correlation and Interpreting the Analytical Results, spec. publ. 29, edited by T.R. Peck, J.T. Cope, and D.A. Whitney, 19-38. Madison, WI: Am. Soc. Agron.

Overman, A. R., F. G. Martin, and S. R. Wilkinson. 1990. "A Logistic Equation for Yield Response of Forage Grass to Nitrogen." Commun. Soil. Sci. Plant Anal. 21:595-609.

Overman, A. R., M. A. Sanderson, and R. M. Jones. 1993. "Logistic Response of Bermudagrass and Bunchgrass Cultivars to Applied Nitrogen." Agron. J. 85:541-5.

Overman, A. R., and S. R. Wilkinson. 1992. "Model Evaluation for Perennial Grasses in the Southern United States." Agron. J. 84:523-9.

Willcutts, J. F., A. R. Overman, G. J. Hochmuth, D. J. Cantliffe, and P. Soundy. 1998. "A Comparison of Three Mathematical Models of Response to Applied Nitrogen: A Case Study Using Lettuce." HortScience 33:833-6.

Analysis of variance for the data in Figure 1, testing crop response to rate of N fertilizer. In this case, the experimental design was a randomized, complete-block design with 5 replications.

Publication # SL345

Date: 1/30/2018

Soil and Water Science

Soil and Water Science

Related topics.

Fact Sheet

About this Publication

This document is SL345, one of a series of the Department of Soil and Water Sciences, UF/IFAS Extension. Original publication date March 2011. Visit the EDIS website at  https://edis.ifas.ufl.edu  for the currently supported version of this publication. 

About the Authors

George Hochmuth, professor, Department of Soil and Water Sciences; Ed Hanlon, professor, UF/IFAS Southwest Florida Research and Education Center, Department of Soil and Water Sciences; and Allen Overman, professor, Agricultural and Biological Engineering Department; UF/IFAS Extension, Gainesville, FL 32611.

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