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Name: ___________________________________Section: _________________Date: ________________ Graded Course Assignment 4: Interpreting Data & Statistics. Interpreting Scientific Data Using Mathematics and Statistics Math and statistics are integral to scientific research—from forming a hypothesis, to collecting and analyzing data, to drawing inferences beyond the data. It is rarely practical for scientists to measure every event or individual in a population. Instead, they typically collect data on a sample of a population and use them to draw conclusions (or make inferences) about the entire population. Our population is 14 day old Bean Plants. Table 1. Bean Plant Heights at 14 days Plant # Height (cm) Plant # 1 8.3 10 2 9.4 11 3 7.7 12 4 8.9 13 5 7.1 14 6 8.9 15 7 8.7 16 8 9.8 17 9 10.1 18 19 Height (cm) 6.7 8.1 9.2 10.6 9.6 8.9 7.5 8.2 9.4 9.3 One of the first steps in describing a data set is to graph the data and examine the distribution. The first thing you notice is that the shape of the distribution resembles the shape of a bell. This is what’s referred to as a normal distribution. Figure 1. The distributions of plant heights Measures of Average: Mean, Median, and Mode A description of a group of observations typically includes a value for the mean, median, or mode. These are all measures of central tendency—in other words they represent a number close to the center of the distribution of values or observations in the data set. Mean The mean (also referred to as the average) is the sum of all numbers in a data set divided by the number of values in the data set. The mean is not always the best measure of central tendency because it can be distorted by extreme values, or outliers, which are extremely different from the rest of the sample. Application in Science To determine the mean of the bean plants: I. 8.3 9.4 Find the sum of the heights: 7.7 8.7 7.1 8.9 8.7 9.8 10.1 6.7 8.1 9.2 10.6 9.6 8.7 7.5 8.2 9.4 9.3 = 166.4 cm II. Count the number of height measurements: There are 19 height measurements. III. Divide the sum of the heights by the number of measurements to compute the mean: Mean = 166.4 cm/19 = 8.76 cm The mean for this sample of eight plants is 8.76 cm and may serve as an estimate for the true mean of the population of bean plants growing under these conditions. In other words, if we collected data from hundreds of plants and graphed the data, the center of the distribution might be around 8.76 cm. Median The median shows the mid-point in a distribution and is, therefore, less easily distorted by extreme values than the mean. For this reason, it may be more useful to use the median as the main descriptive statistic for a sample of data in which some of the measurements are extremely large or extremely small. To determine the median of a set of values, first arrange them in numerical order from lowest to highest. The middle value in the list is the median. If there is an even number of values in the list, then the median is the mean of the middle two values. Application in Science To determine the median of our bean plants: I. Arrange the Height values in numerical order from lowest to highest: 6.7 7.1 7.5 7.7 8.1 8.2 8.3 8.7 8.9 1 2 3 4 5 6 7 8 9 II. 8.9 8.9 9.2 9.3 9.4 9.4 9.6 9.8 10.1 10.6 9 8 7 6 5 4 3 2 1 Find the middle value. This value is the median: Median = 8.9cm Note: In this case the median is 8.9cm, but the mean is 8.76cm. The mean is smaller than the median, and the two might end up close in value, but it’s not guaranteed. Mode The mode is another measure that can help describe a set of data. It is the value that appears most often in a sample of data. In the example above in Table 1, the mode is 8.9 seconds (it appears 3 times). The mode is not typically used as a measure of central tendency in scientific research, but can be useful in describing some distributions. For example, Figure 2 shows a distribution with two peaks, or modes— what’s called a bimodal distribution. Describing these data with a measure of central tendency like the mean or median would obscure this fact. Frequencies of Body Lengths of Weaver Ants Figure 2. Graph of body lengths of weaver ant workers. When to Use Which? The mean is the descriptive statistic most often used to describe the central tendency of a sample of measurements; it is the only one of the three measurements of average that takes into account all the information in a data set. This is why other statistical techniques, such as standard deviation, which is a measure of variability, utilize the mean. However, as discussed above, there are occasions when taking account of the value of every measurement in a distribution may give a distorted picture of the data; in such cases, the median provides a more realistic description of the center of the distribution than the mean. The mode is not used very frequently in science as a measure of central tendency but may be useful in describing some types of distributions—for example, ones with more than one peak. Measures of Variability: Range and Standard Deviation Variability describes the extent to which numbers in a data set diverge from the central tendency or average. It is a measure of how “spread out” the data are. Two common measures of variability are range and standard deviation. Range The simplest measure of variability in a sample of normally distributed data is the range, which is the difference between the largest and smallest values in a set of data. Application in Science Students in a biology class measured the width of eight leaves from eight different maple trees and recorded their results in Table 2. Table 2. Width of Maple Tree Leaves Leaf # Width (cm) 1 2 3 4 5 6 7 8 To determine the range of leaf widths: 7.5 10.1 8.3 9.8 5.7 10.3 9.2 8.7 I. Identify the largest and smallest values in the data set: Largest = 10.3 cm, Smallest = 5.7 cm II. To determine the range, subtract the smallest value from the largest value: Range = 10.3 cm – 5.7 cm = 4.6 cm Standard Deviation The standard deviation is the most widely applied measure of variability, there are two types. 1) Sample Standard Deviation: Used when you want to apply your results to other items not measured, like other bean plant heights. The sample mean ( ) provides a measure of the central tendency of the sample; the sample standard deviation (s) measures the average deviation between each measurement in the sample and the mean ( ). Application in Science You are interested in knowing how tall bean plants (Phaseolus vulgaris) will grow two weeks after planting. You plant a sample of 20 seeds in separate pots and give them equal amounts of water and light. After two weeks, 17 of the seeds have germinated and have grown into small seedlings. Each plant is measured from the tips of the roots to the top of the tallest stem. The measurements are recorded in Table 3. Table 3. Plant Measurements and Steps for Calculating Standard Deviation. Plant # Height (mm) Plant # Height (mm) 1 112 10 110 2 102 11 95 3 106 12 98 4 120 13 74 5 98 14 112 6 106 15 115 7 80 16 109 8 105 17 100 9 106 Fig 3 Mean Height with +/- 1 std.dev. Sample mean = 103 Standard Deviation (s) = 11.7 The mean height of the bean plants in this sample is 103 mm 11.7 mm. If we use the normal distribution table on the left. (Fig 3) We can say that for our sample 68.2% of the plants will fall between 114.7mm and 91.3mm. Fig 4 Normal distribution (bell curve) It is up to the scientist to determine how usable the data may be. Would it be prudent for a bean seed vendor to place this data on their packages based on 17 bean plants? The easy answer is no. It’s very possible that we would get similar data if we grew 10,000 seeds, however it’s also just as likely we may not. 2) Population Standard Deviation: Used when you have measured the entire population of items you are interested in, like the metal cylinders below. The population mean (µ) provides a measure of the central tendency of the sample; the population standard deviation (σ) measures the average deviation between each measurement in the sample and the mean (µ). TABLE 4 Densities of steel cylinders measured by water displacement. Sample Density A 7.9 B 8.1 C 7.7 D 7.9 E 7.9 F 7.8 G 8.0 H 8.2 I 7.9 J 7.7 K 8.4 L 7.9 The data in table 4 shows the densities of 12 different steel cylinders. The volume was measured using water displacement. This set of data is very different from the bean plants in that the density values are expected to be close to identical. The density of this steel shouldn’t have a normal distribution. So using a std. dev. Can help us determine if the method(s), tools or skills used in obtaining the data are appropriately accurate and/or precise. For this data the mean is 7.95 g/ml, and the std dev is 0.19g/ml. The accepted/known value for the density steel is 7.812 g/ml with a Std Dev of 0.0157g/ml. Giving us an acceptable range of 7.796g/ml to 7.828g/ml. ( +/- 1 std dev) While we can say that this data wasn’t that far off ( +0.138 g/ml), it doesn’t fall within one or two std deviations. It is actually 9 std dev’s away from the known. Given the distance from this data set’s mean from the accepted mean and the much higher std dev (0.0157 vs 0.2) it would be difficult to label this data as worth using. Their Formulae aren’t that different: Sample Population Interpolation & Extrapolation Sometimes scientists need to make predictions outside the range of measured data, such as in forecasting future climate change. This process in which scientists make predictions between known data points, or within the range of measured data is called interpolation. Fig 5 shows that we could reasonably expect to earn ~$480 if the temperature was 21°C. Fig 5 Ice cream sales vs. Temperature In contrast extrapolation (fig 6) involves making predictions outside of known data points. Scientists can extrapolate by using a formula, data arranged on a graph, or programmed into a computer model. In general, extrapolation comes with a higher degree of uncertainty. images: www.mathisfun.com Fig 6 Ice cream sales vs. Temperature Correlation When two sets of data are strongly linked together we say they have a High Correlation. Correlation is positive when the variables change in the same direction. For example, the length of an iron bar will increase as the temperature increases. Correlation is negative when variables change in opposite directions. For example, the volume of a gas will decrease as the pressure increases. If there is no relationship between the two variables such that they change independently of each other, then there is no or zero correlation. On the graph, written below the equation for the line is the Coefficient of Determination (R2) that Excel provides. R2 is a measure that allows us to determine how certain one can be in making predictions from a certain model/graph. R2 = 0 < +1 To find r just take the square root of R2 and adjust the sign + or – according to the trend line. Figure 4. Pressure vs Temperature Another measure of correlation is r - correlation coefficient. r measures the strength and the direction of a linear relationship between two variables. It can have values from -1 < 0 < +1. Fig 5 Correlation Examples Like r, R2 values close to 1.0 are indicators of high correlation. In fig 4 the R2 = 0.9996, we can say that this experiment represents a good display of Gay-Lussac’s Law which relates temperature and pressure, and that the graph/line made from this data is worthy of using. However, correlation implies that the numbers follow the same trend, and that is all. Causation is for the experimenter to determine. In other words: “Did changing the temperature actually cause the pressure to increase?” Statistical Analysis - The t-test Suppose that a researcher wishes to test if a certain kind of growth hormone will produce faster growth in mice. She injects 10 mice with the hormone and uses another 10 as a control. Three weeks later, she weighs the mice and discovers that the mean weight of mice that have received the injections is 12.05 g and the mean weight of control mice is 9.3 g. These values indicate that the mice receiving the hormone are heavier. Is her value of 12.05 significantly different than 9.3? Is it possible that the hormone has no effect; that the weight difference between the two groups is due to chance? This is like flipping a coin 10 times. You expect 5 heads and 5 tails but you might get 6 heads or 7 heads or perhaps 8 heads. Similarly, if the hormone does not work, you expect the mean for the two groups to be similar but it may not be exactly the same. Table 5 – Tumor masses for 2 groups of mice. Group 1 - Hormone - Weight (grams) 12.5 13 12 12 13 14 13 10.5 9.5 11 Mean = 12.05 Group 2 – No Hormone - Weight (grams) 12 8.5 10 8 8 13.5 9 8.5 6.5 9 Mean = 9.3 What is the chance that the two means would be as different as 12.05g and 9.3g if the hormone really did not work? Statistical tests are used to determine whether differences in the data are real differences or whether they are due to chance. In the example above, we test if the mean of group 1 is significantly different than the mean of group 2. The alternative is that the difference is due to chance or random fluctuations and the hormone did not cause additional weight gain. The test gives the probability that difference could be due to chance. If the probability that the difference is due to chance is less than 1 out of 20 (<0.05), then we conclude that the difference is real. If the probability is greater than 0.05, we conclude that the difference is not significant, it could be due to chance. There are several tests available for testing means. A commonly used test for data that are normally distributed is the t-test. The calculations for the test can be performed by hand but computer software can do them very quickly. To perform the test, the weight data for the two groups of mice above are entered into a t-test program. The software reveals that p = 0.0012. The probability that the difference between the two means (12.05 and 9.3) is due to chance (random effects) is 0.0012 (or 12 out of 10,000). Because p < 0.05, we conclude that the two means are really different and that the difference is not due to chance. The researcher accepts her hypothesis that the hormone produces faster growth. If p had been greater than 0.05, we would reject her hypothesis and conclude that the two means are not significantly different; the hormone did not cause one group to be heavier. The word "significant" has a slightly different meaning in statistics than it does in general usage. In a statistical test of two means, if the difference is not due to chance, we conclude that the two means are significantly different. In the example above, the mean weight of group 1 is significantly heavier than that of group 2. NAME: ____________________________________ Section: Date:______________ 1. Range & Standard Deviation Two Methods for measuring volume were examined in an experiment. The densities of twelve cylinders were calculated using data gathered in two sets, each set using a different method for measuring the volume of the cylinder. Figure 1. Density Measurements Method 1 Figure 2. Density Measurements Method 2 For Method 1 the Mean is ___________g/ml; - the std dev is 0.0346 for Method 2 the Mean is 7.95 g/ml. - the std dev is 0.19 TABLE 1 Method 1 Sample Density A 7.800 B 7.844 C 7.825 D 7.883 E 7.819 F 7.824 G 7.875 H 7.859 I 7.882 J 7.764 K 7.821 L 7.863 TABLE 2 Method 2 Sample Density A 7.9 B 8.1 C 7.7 D 7.9 E 7.9 F 7.8 G 8.0 H 8.2 I 7.9 J 7.7 K 8.4 L 7.9 The accepted value for this steel is density of 7.812 g/ml with a Std Dev of 0.0157g/ml. a. Find the Mean for Method 1. ___________________ b. What are the ranges for each method? M 1 _________________ M 2 ________________ c. Which method is better? Why? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ d. Using your own judgement comparing Method 1’s results and the accepted value, is the data from method 1 worth using? Explain using the statistical information. ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 2. Interpolation, Extrapolation & Correlation Figure 3. Pressure vs Temperature Answer the following questions: a. Does this data show good correlation? Why? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ b. What would you expect the pressure of this system to be at a temperature of 315K? ____________ c. Predict the pressure for a temperature of 335K ______________ d. With a temp range of 290K to 325K would it be prudent to extrapolate out to 1,000K? ___________________________________________________________________________ ___________________________________________________________________________ ___________________________________________________________________________ 3. T-tests Complete the practice scenarios and determine if there is a significant difference between the means using a Student’s t-test (alpha level= 0.05). Answer all questions and attach the t-test spreadsheet with calculations associated with each scenario. A researcher wishes to learn if a certain drug slows the growth of tumors. She obtains mice with tumors and randomly divides them into two groups. She injects one group of mice with the drug and uses the second group of mice as a control. After 2 weeks, she sacrifices the mice and weighs the tumors. The mass of tumors for each group of mice is below. The researcher is interested in learning if the drug reduces the growth of tumors. Her hypothesis is: The mean weight of tumors from mice in Group A will be less than the mean weight of tumors from mice in Group B. Table 3 - Tumor masses for 2 groups of mice. Tumor mass (g) of Group A mice Treated with Drug 0.72 0.68 0.69 0.66 0.70 0.63 0.71 0.73 0.68 0.71 Tumor mass (g) of Group B mice Control- Not Treated 0.71 0.83 0.89 0.78 0.68 0.74 0.75 0.80 0.80 0.78 Answer the following questions: (Attach the Excel t-test calculation worksheet.) a. b. c. d. Is this a one-tailed or two-tailed t-test? Why? Perform a t-test using the programmed Excel worksheet. What is the calculated p-value? Is the researcher’s hypothesis accepted or rejected? Why? Reference: Gregory, M. (2014, August). Statistical analysis- The t-test. Retrieved August 5, 2014 from http://biology.clc.uc.edu/courses/bio303/statanal.htm A group of ecology students was studying and comparing two different forest areas by taking 20 sample plots in each area. The researcher formulated the following hypothesis: The mean difference of sugar plus red maple trees between Woods A and B will be different. In each wooded area the combined number of sugar plus red maple trees in each of the 20 plots was found to be: Table 4 Plot # * Woods A Woods B Plot Data Plot Data 1 0 9 2 6 6 3 7 2 4 0 7 5 1 6 6 3 8 7 2 6 8 9 5 9 7 0 10 3 2 11 1 3 12 9 2 13 8 4 14 2 7 15 1 9 16 5 8 17 7 7 18 5 1 19 7 7 20 3 8 *Do not use the plot # in your calculations. Answer the following questions: (Attach the Excel t-test calculation worksheet.) e. f. g. h. Is this a one-tailed or two-tailed t-test? Why? Perform a t-test using the programmed Excel worksheet. What is the calculated p-value? Is the researcher’s hypothesis accepted or rejected? Why? Reference: Carter, S.J., (2005, March 29). Statistical analysis of data. Retrieved August 4, 2014 from http://biology.clc.uc.edu/courses/bio303/statanal.htm