ENV Laboratory 1: Quadrant Sampling

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1 Name: Date: Introduction Quite often when conducting experiments it is impossible to measure every individual, object or species that exists. We will explore some methods to estimate the number of objects/species/individuals in a sample. Discussion Much of environmental science is concerned with estimating different populations, how many organisms (elk, deer, cows, people, birds, bee s, coyotes etc) are in an area and how the population is changing - increasing, decreasing, remaining the same. One can also compare different environments; normal, polluted, encroachment of humans, the effect of climate change, and many other variables. Whether studying elk, tree, insects, people, all of the populations studied share one frustrating characteristic in common, that they are difficult to count. Generally it is impossible for large population or in a large area to count every individual organism. Many different methods have been devised for sampling these populations and estimating the total population. Instead of measuring the total population of a species we instead measure only a small representative sample. We can define the SAMPLE as consisting of a small amount of something which if properly chosen is representative of the whole. Humans are sometimes bad at choosing representative samples, so when possible we want to chose a RANDOM sample by some means (dice, random number generators, computer programs etc.). Quadrant Sampling One method for sampling static (unchanging) samples is the Quadrant (or Quadrat or Grid) method. In this method one divides the total area to be sampled into a number of smaller samples by overlying a grid on the sample. One can then sample a smaller number of grids (usually between 5-10% of the total sample) and extrapolate to find the total number in the sample. DENSITY is defined as the number of individuals per area sampled. Other useful measures are the Frequency defined as the number of quadrants in which the species occurs and is a measure of how often the species occurs. The Relative Frequency is the frequency of a given species in relation to the total frequency of all species. Relative frequency can also tell us if a given species is distributed randomly (0-30%), clumps (31-80%), or evenly (81-100%). Percent Error When performing any sort of estimation, there will be an error associated with your estimate. There are many complicated ways to statistically determine how good your estimate is, but this is NOT a statistics class, so we won t do any of that. If you are lucky enough to know the actual answer one can determine how close you are to the correct answer by computing the PERCENT ERROR using the formula below. Percent Error = Actual Value Experimental Value Actual Value 100 (1) Page 1 of 8

2 Determining the Correct Sample Size Determining sample size is very important. If your sample is too large, you will be wasting time and money. If your sample is too small, your results will not be very accurate. Like Goldilocks, you need the sample size that is just right. When we collect sample data and take the average of those samples, it will usually be different from the actual population average. Yet as we increase the number of samples, their calculated average gets closer and closer to the actual average for the whole population. And if we take so many samples that we have sampled the entire population... then the sample average would be the same as the actual average. A simple way to determine the appropriate sample size is to graph a running average from the samples. As you can see the data after a sufficient number of trials will oscillate around the correct value. Average and Standard Deviation It should be obvious that very few times will you estimate the correct answer by this method. If multiple studies are attempted and or there are multiple sets of data, one can average the results together and also find the standard deviation. The average is given below in two forms, on the left, is the formal mathematical expression, on the right is the more standard form. x = 1 n n i=1 x i x = Average i = individual measurement Average = x 1 + x 2 + x 3... n n = number of measurements Figure 1: (a) Formal mathematical definition of an Average. (b) Standard definition of Average. The sample standard deviation (s) is used to measure how close each value is to the average value, and quantifies the amount of variation in set of data values. A low sample standard deviation indicates that the data points tend to be very close to the average value, while a high sample standard deviation indicates that the data points are far from the average value. Thus, a small sample standard deviation means that the experimental result is reproducible, while a large sample standard deviation means the experimental results are not very reproducible. The equation for the sample standard deviation is given below: Page 2 of 8

3 s = 1 n (x i x) 2 2 n 1 i=1 Figure 2: Standard Deviation The following example shows the calculation of the average and standard deviation for an example set of data. A student measures the change in temperature of a sample 5 times with the following results: Grid # Balls x = = = 24.3 ( ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 + ( ) 2 ) 1 2 σ = = Figure 3: Example Calculation for Average and Standard Deviation Based on the example above, one would report the measured value as 24.3 ± 1.5 balls. Page 3 of 8

4 Exercise 1 - Estimating the Number of Dandelions in a Field 1. While it is possible to count all the Dandelions in the field (if you are bored enough), just estimate the number of Dandelions in the field (there is no wrong answer!): 2. Write down your assigned group number. 3. Write down the Grids assigned to you in column Using the Grids assigned to you in class by your instructor count the number of Dandelions in each square and complete column Next compute the cumulative number of Dandelions counted and record it in column Compute the cumulative Density and record it in column Estimate the total number of Dandelions in the field: 8. Ask your instructor for the correct number of Dandelions in the field (of if you are bored count them yourself!): 9. Calculate the percent error in your estimate: Page 4 of 8

5 Exercise 2 - Calculating Average and Standard Deviation for the Class You will now collect data from the entire class and calculate the average and standard deviation of the data. 1. Collect data from your class mates and enter it into the table below. 2. Calculate the average. Show your work below the table. 3. Fill in the remaining columns and calculate the standard deviation. 4. Compare your answer to the instructors. Group Avg. # Dandelions Figure 4: Class Data for 12 groups Calculate the standard deviation by completing the following equation. ( (A T1 ) 2 + (A T 2 ) 2 + (A T 3 ) (A T 11 ) 2 + (A T 12 ) 2 ) 1 2 σ = n 1 You may find it useful to complete the following table to solve for the standard deviation using the following steps. Show any work required in the space to the right 1. In Row 1 calculate the difference between the measured value and the average. 2. In Row 2 square the value from Row 1. Group Avg-Trial # (Avg-Trial #) 2 Figure 5: Class Data for 12 groups 3. Sum the data in Row 2 above and place it in the table below. 4. Finish by taking the sum divided by (n-1) where (n-1) is the number of trials -1 (hint, its 11!). 5. Take the final answer and square it. Calculation Sum Sum (n-1) Standard Deviation Result Wow that was A LOT of work! We can finally answer the question of how many Dandelions are in the field AND how certain we are of our answer. Record your answer in the space provided. The number of Dandelions in the field is: and the Standard Deviation is: Page 5 of 8

6 Exercise 3 - Estimating the Number Spilled Atoms We will now scale up the exercise and perform it in a situation where you would NOT (unless you are really bored or my work study) want to count all the objects. Using what you have learned before estimate the number of balls Jay spilled in lab. questions before you begin. 1. Which color of balls were you assigned? Answer the following 2. If you need to sample 10% of the total area, how many squares should you sample? 3. How will you randomly determine which squares to sample? 4. Sketch a data table in the space below to use when collecting your data. 5. How many balls did Jay spill? (Show how you calculated your answer.) Page 6 of 8

7 Group Black Red White Avg: Stnd Dev: Using the Data taken in class, calculate the average and standard deviation for your set of colored balls. Show work for your calculation in the space below. The number of Balls spilled in the field is: and the Standard Deviation is: Exercise 4 - Plotting a Running Average Using Excel One last thing! Using Excel calculate and plot the running average for your data and attach it to the back of the laboratory. Be sure to label the axis properly. Page 7 of 8

8 Blank Page, Have a Nice Day! Page 8 of 8