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It has lessons should require five days in total, possibly six if students have never worked with a spreadsheet program in the past. The sample spreadsheet included was prepared on Excel 5.0, and the functions described herein pertain to that program. However, if the instructor prefers or only has access to a different spreadsheet, only minor adjustments should be required to make use of this unit plan. Day 1: Accuracy vs. Precision. To become familiar with the difference between accuracy and precision, students will simulate measurements of some quantity, such as cylinder compression, by dropping pennies on a target on the floor. Each penny will represent a compression measurement and the bull’s eye will represent the actual compression. Day 2: Variation: The Penny Lab. Students will calculate the densities of pre-1983 (copper) pennies and post-1983 (copper/zinc) pennies. After measuring masses and volumes, students will enter their data in tables or on a spreadsheet, prepare a stem-and-leaf diagram, and use the tables and diagram to analyze the variation in the data. Days 3 and 4: Variation: An automotive lab. Students will use a spreadsheet to analyze automotive data and discover how both individual measurements and team averages vary. The instructor will have some options when it comes to deciding which automotive quantity will be measured. During this lab students will be introduced to several important spreadsheet features including data entry and analysis. Day 5: Variation: Variance and Standard Deviation. Without actually exposing them to the rather foreboding formulae for variance and standard deviation, students will be introduced to these concepts and use them to describe sets of data simulated on a spreadsheet. NCTM STANDARDS: This unit was designed to meet several of the goals put forth in the NCTM Statistics Standard (Standard 10). In particular, students will summarize data from real-world situations, analyze data in terms of central tendency and variability, and investigate the role of sampling size in making decisions pertaining to accuracy. Day 1: Let’s begin with an explanation of the difference between accuracy and precision. Accuracy refers to how close a measurement is to the true value of what is being measured. Precision refers to how close measurements of the same quantity are to each other, even if they are not close to the true value. For example, the darts on the dart boards below represent sets of measurements. A bull’s eye represents a perfect measurement--a measurement exactly the same as the true value.  EMBED Word.Picture.6  NEITHER PRECISE NOR ACCURATE Since none of the darts are close to the bull’s eye, the measurements they represent are not very accurate. Also, since the darts are not very close to each other, the set of five measurements here is not very precise either.  EMBED Word.Picture.6  BOTH ACCURATE AND PRECISE The measurements are all close to the true value, so they are accurate. Also, the measurements are all close to each other, so they are precise.  EMBED Word.Picture.6  PRECISE BUT NOT ACCURATE Since all of the measurements are close together, they are precise, but since they are not close to the true value, they are not accurate. Activity 1: See if you can draw and describe (as done in the pictures and captions above) the case where a set of measurements is fairly accurate but not very precise. Activity 2: Draw a dart board on a sheet of paper and set it on the ground. Have each member in a group of about five drop a penny on the target from a height of about one meter, aiming for the bull’s eye. Each penny represents a measurement of some quantity, say, the compression in the cylinder. 1. What does the bull’s eye represent? 2. How many compression measurements were made? 3. Were the measurements accurate? Why or why not? 4. Were the measurements precise? Why or why not? Now simulate a new set of measurements by dropping pennies from a height of 2 meters. 5. Has the accuracy and/or precision of your measurements changed? If so, how. Explain. The equipment, tools, and technology available to you when you make measurements will affect your accuracy and precision. Depending on the design and age of a measuring instrument as well as other factors such as the skill of the people using the instruments, real-life measurements can be accurate or inaccurate, precise or imprecise, or any combination of these. 6. What variable in your experiment corresponds to the reliability of your measurements (in terms of both accuracy and precision)? Activity 3: Suppose your team is challenged to determine the mass of an object using a balance. Each member of the team measures the mass on the same balance. Here are you data: Team memberMass in grams139.97240.06339.98439.97540.02 Your team decides to report your mean (average) mass, which is 40.00g. Your instructor then informs you, however, that the actual mass is 45.00g. 7. Who (or what) would your instructor blame for the fact that your reported mass was too low by 5/45 = 1/9 or about 11%? 8. What can you conclude about the precision of your balance? 9. What can you conclude about the accuracy of your balance? 10. How might this type of error (called a systematic error) have been avoided? Day 2: In this lab you will calculate the densities of pre-1983 and post-1983 pennies. You will find different densities since the older pennies contain only copper, while the newer ones contain both copper and zinc, making them less dense. With a partner, use a triple-beam balance to find the mass of the pennies a graduated cylinder to measure their volume (via water displacement). Then make use of the density formula D = M/V, recording your answers in grams per cubic milliliter (g/mL). It is recommended that about ten to fifteen pennies be used (of each type) for better accuracy. (Instructor: Students who are very concerned about accuracy may wish to find the mass and volume of the exact same pennies, finding the mass first while they are still dry. A short discussion before the lab regarding why it should not matter how many pennies are used and why the answer does not need to be divided by the number of pennies may be in order.) With the other pairs of students in the class, there should be enough density data to make a nice stem-and-leaf diagram. After you and your partner calculate the density of your pennies, display your calculations on the board in a table. An example is shown below. Density Table (in g/mL) Pre-1983 Post-1983 8.958.618.048.098.828.977.848.179.328.937.958.078.748.917.678.059.199.028.107.998.599.108.088.188.909.568.068.168.969.088.037.898.858.957.837.859.268.817.908.008.739.027.667.969.018.778.197.75 You can then use the data to make a separate stem-and-leaf diagram for the each type of penny as shown below. Grams and tenths of grams together should make up the stems (left side of the table); hundredths of grams will comprise the leaves. Stem-and-Leaf-Diagram for Pre-1983 Pennies 859861873 4 7881 2 5890 1 3 5 5 6 7 901 2 2 8910 992693294956 1. By looking at the stem-and-leaf diagram, describe how the density measurements vary. What do you think is the best approximate value for the density of older pennies to the nearest tenth of a g/mL? Why? (Here “best” means the density that is most likely closest to the true density of the pennies.) 2. Using the spreadsheet software or a calculator, find the mean (average) densities for the older and the newer pennies. 3. How close were your measurement to the values found in numbers 1 and 2 above? What does this probably say about the accuracy of your measurement? 4. What is the range of density data gathered by your class for the newer pennies? 5. What are some possible explanations for this range in data? 6. What does the range suggest about the precision of your measurements? Activity 2: Copper has a density of 8.96 g/mL, and zinc has a density of 7.14 g/mL. Use this information to answer the following questions. 7. Using your stem-and-leaf-diagram, estimate what proportion of the newer pennies is zinc and what proportion is copper. 8. If you were to ask several people at random to calculate the densities of the pennies that they happened to have in their pockets, would you expect them to report nearly identical answers? Why or why not? How could you determined who had the highest proportion of older pennies? 9. Suppose that in the future the value of copper rises relative to that of zinc, prompting the government to increase the zinc content of pennies to 90%. About what would expect to find for the density of these pennies? Explain. Days 3 and 4: In this lab you will take another look at the variation in a set of measurements, this time in an automotive setting. Each team of students will measure the thickness of a brake pad (or make some other accessible measurement such as oil pressure, spark plug gaps, etc., as determined by your instructor). In order to analyze the variation in the measurements made, it is essential that the same brake pad be measured by each team. Otherwise, we would be looking at the variation in thicknesses among a set of different brake pads. The teams should record their measurements using spreadsheet software like Excel 5.0. Day 3 will be devoted primarily to making the necessary measurements and recording the individual and team data in a spreadsheet. The spreadsheet can be set up in a fashion similar to sample spreadsheet provided. (This spreadsheet has been designed under the assumption that all the students will be measuring the same quantity on the same car but, with only a little modification it could be used to analyze data from many cars.) The students should work in teams of about five, but each student will make and record his or her own measurement. In the sample, these are located in column C. Notice that the students are grouped according to teams. In column D the students of each team will calculate their team average. This can be done by making use of Excel’s AVERAGE function. In the sample Team 1 entered the following formula in cell D4: =AVERAGE(C2:C6) This is asking Excel to enter the average, or mean, of the numbers in cell C2 down through cell C6, which contain the individual measurements for Team 1. Do not forget to start with an equal sign when entering a formula or Excel will just enter the text “AVERAGE(C2:C6)” in cell D4. Day four will be spent learning to use Excel to analyze the data and drawing conclusions from the analysis. This portion of the lab will run more efficiently if each team has access to a computer. Each team should open a copy of the spreadsheet containing the previous day’s data. Off to the side of the spreadsheet in cell I4 (see the sample), use the average function to find the mean for the entire class. In the cell next to it enter “class mean” (without an equal sign) so that you will know to what quantity the number in the adjacent cell refers. Note: Not only are the built-in spreadsheet functions like AVERAGE faster to use than a calculator, but with them the spreadsheet is very versatile. For example, if you made a mistake in recording any of you original data, a new average will be computed automatically as soon as the correction is made! Column E will contain the individual deviations from the class mean that you just computed. These deviations tell how far away each measurement was from the class average. For example, Connie from Team 4 measured the brake disc thickness to be 0.428 inches, which was 0.0015 inches below the mean for the whole class. Therefore, her measurement deviated from the mean by 0.0015 inches. Here we are just interested in how far away the measurements are from the mean and not whether they were above or below it. For this reason we will use Excel’s absolute value function. In cell E2 enter the following formula: =ABS(C2-I$4) This will find the deviation of Joe’s measurement from the class mean since Joe’s individual measurement is located in cell C2 and the class mean was computed in cell I4. The dollar sign in the formula will be explained shortly. Notice that functions in Excel operate on whatever is inside the parentheses. In this case, we’re taking the absolute value of the difference of the values found in two different cells. So far, we have just found the deviation for the first person in Team 1. We could enter a similar formula for each person, but that would be time consuming. Excel can copy that formula into all the cells in column E, automatically adjusting them so that in the next row the formula would read =ABS(C3-I$4), and in the row after that, =ABS(C4-I$4), and so on. This will allow us to find the deviations for each student without typing in a separate formula for each. The dollar sign between the “I” and the “4” will keep that cell the same in each formula, which is what we want since that is the only cell that contains the class mean. To “fill down” like this, select cell E2 (click on it once) and then position the pointer to the lower right hand corner of the cell. Here the pointer should change from a fat, white plus sign to a thin, black plus sign. Now click the left mouse button, hold it, and drag it down column E to the bottom of the data table. Then release the button. The new individual deviations should then appear a_927481633џџџџџџџџ РF€aiЙŸЌМ€aiЙŸЌМPIC џџџџџџџџџџџџLMETA џџџџшPICT џџџџџџџџџџџџ4ѕњ-№ќџџџ-№ќџџџ-њ-ЛЋRD-№-№ќџџџ-њ-whŸ‡-№-№ќџџџ-њ-2ша-№-№ќ-њ-w%j-№-№ќ-њ-K>|-№-№ќ-њ-УEЖ2-№-№ќ-њ-•НˆЊ-№-№ќ-њ-cP-№-№-џџџџџџџџџџџџџџџџџџџџџџаЯрЁБс4{ƒџ џџџџƒ{Ёd PPNTЁdPPNTTimes New Romanџџџџџџ {ƒ T{xXџџџџџџTjfXџџџџџџT& ZVXџџџџџџT82IBX џџџџџџџџTW~ZƒXT[_XT, /XT^)a-XTXЁd PPNTџаЯрЁБсўџ џџџџ РFMicrosoft Word 6.0 Document MSWordDocWord.Document.6аЯрЁБс;џўаЯрЁБс;џў CompObj џџџџbObjInfoџџџџџџџџџџџџObjectPool џџџџ€aiЙŸЌМ€aiЙŸЌМWordDocumentџџџџ џџџџ › мЅe3Р e › j jj j j j j ~ ~ ~ ~ ~ ~ ˆ ~ ђ 1’ ’ ’ ’ ’ № № № № ђ ђ ђ ђ ђ ђ # Tw $ђ j № ’ ^№ № № ђ № j j ’ ’ № № № № j ’ j ’ № ~ ~ j j j j № № № №  ЄР!Ѕ 2ІЙ ЇК Ј€Љ.0$Р ‡X ‘ џџџџџџK2.0$ў ЗСџџџџџџK2.0$?y4 џџџџџџK2.0$жH_џџџџџџK2.0$ЇGX:џџџK2.0$зЇX:џџџK2.0$Ї чX:џџџK2.0$чзX:џџџK2.0$7 ‡X:џџџK2аЯрџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџВ@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@В@@ИAу@@@@$%4@   џџџ€@ ”"j№@@И@@В@@В@@В@@В@@В@@В@@В@@В  Мљїu uDa ўK@ёџNormala "A@ђџЁ"Default Paragraph Font  џџџџМ  LzЈж2`Žџ@1Times New Roman Symbol &Arial"€аhAUFƒ$Louie BeuschleinаЯрЁБс;џў ўџ р…ŸђљOhЋ‘+'Гй0H ˜SummaryInformation(џџџџџџџџџџџџOx_929873172џџџџџџџџ РF€aiЙŸЌМ€aiЙŸЌМPIC џџџџџџџџџџџџYLMETA џџџџ[ШМ р  (Lp ”Им $џџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџџ#?C:\WINWORD\NORMAL.DOTLouie Beuschlein@†уfсuМ@@@Microsoft Word 6.01аЯрЁБсLљo Q шшаЯрЁБс;џў љo&  V&џџџџWordMicrosoft Word  ёу ћD-ћжџTimes New Roman€- ќџџџ-њ-ЏЃF<њ-№ќџџџ-s if by magic. If you select any of the cells that you have just filled, you should see the appropriate formula in the formula bar above. Try it. In column F you will need to find more deviations, but this time the deviation of each team from the class average. Select cell F4 and see if you can enter the correct formula to do this for Team 1. If you are new to spreadsheets, you may want to simply repeat this procedure in the appropriate cell for each team. A quicker option would be, rather than filling down as you did last time, to copy cell F4 (select it and choose Copy from the Edit menu). Then select the other cells that will contain team deviations (cells F9, F14, and F19) simultaneously by holding down the Control button while you’re selecting these cells the mouse. Then from the Edit menu choose paste. Now let’s find the range of the class data. The range is how far the data spans, that is, the difference between the largest and smallest pieces of data. Here you have at least three options, with option 3 being the most efficient if you’re familiar with spreadsheets: Option 1: Visually scan the class data for the largest and smallest measurements. In the example, Mellisa had the largest measurement, while Tom and Tony tied for the smallest. So, in cell I2 you could enter the formula =C3-C9 to compute the range. This method is not much fun, however, and likely to result in errors, especially when data lists are long. Option 2: Select the list of measurements in column C. Copy and paste them somewhere else in spreadsheet. Then, with the new list still selected, choose Sort from the Data menu. This will list them in ascending order. The range of the data, therefore, is the bottom entry minus the top. Option 3: In cell I2 simply enter the following formula =MAX(C2:C21)-MIN(C2:C21) The MAX function finds the largest value in the data contained in cells C2 through C21. The MIN function finds the smallest value among the same cells. Then the two values are subtracted, yielding the range of the data. In cell I3, see if you and your team can use the MEDIAN function to find the median of the individual measurements. In the example, the median corresponds to the brake disc thickness that would be found right in the middle of a list of thicknesses arranged in ascending order. That is, there will be the same number of measurements above the median as below. If there are an even number of measurements, as there are in the example, the median is the average of the two middle values in the sorted list. When sorted, as described in option 2 above, the tenth and eleventh measurements in the example data are 0.429 in. (Jodie and Sarah) and 0.430 in. (Ben and Amy). Use this sorting method to see if you get the same median as you did with the built-in Excel function. Now consult your sorted list to see to determine which value came up the most. This value is called the mode. In the example, 0.431 in. came up three times, more than any other value, so this is the mode. See if Excel returns the same value when you use the MODE function on the measurements in column C. To finish off your spreadsheet, use Excel to compute the mean team average (the average of the team averages), the mean individual deviation, and the mean team deviation. Compare the average of the team averages to the class average. Explain how you could predict how close these values are to each other in terms of how varied the sizes of the teams are. How should they compare if all the teams have exactly the same number of people. Is the mode the largest measurement made? Does it have to be? Can it be? Explain. Which is more, the mean individual deviation or the mean team deviation? Is this what you would expect? Explain. Which do you think is likely to have more accuracy, an individual measurement a team average? Does the variation within a team appear to “smooth out” any individual measurements that seem very high or very low? Why does this happen? If accuracy were very important, would you prefer to be part of a large team or a small one. Explain. Day 5: People in a variety fields--business, medicine, technology, etc.--must deal with data on a regular basis, and one important characteristic of any set of data is the degree to which the data is “spread out.” In other words, it is often very important to know how much variation exists within a set of data. Two common tools, or indicators, that can give us an idea of just how much variation we’re dealing with in our data are variance and standard deviation. They both, of course, have formulae by which they can be calculated, but let’s not worry about that right now. Rather, let’s just try to get a feel for what information these numbers provide by experimenting a little on a spreadsheet and letting it take care of the complicated arithmetic. To simulate some data, we’ll let Excel generate some random numbers for us with the RANDBETWEEN function. (Instructor: If this function is not already available on your copy of Excel, the Analysis ToolPak add-in macro must be installed. You will find the add-ins under the Tools menu.) In cell A1 of a worksheet, enter = RANDBETWEEN(20,30) Then fill down to cell A15 to simulate fifteen measurements in the range of 20 to 30 units. This will give us 15 random numbers between 20 and 30 inclusive. Note: Random numbers in Excel, being random, will change every time the worksheet is updated. To prevent our data from changing as we analyze it, we can “lock them in” by copying the data and then choosing Paste Special from the Edit menu and selecting Values. In cell in A17 enter the formula =VAR(A1:A15) In cell A18 enter the formula =STDEV(A1: A15) These formulae will return, respectively, the variance and standard deviation of simulated data. Now let’s repeat the same steps in columns B, C, and D using progressively smaller ranges for the random numbers: Let the random numbers range from 22 to 28 in column B and from 24 to 26 in column C. Each column should contain 15 pieces of simulated data. Note: You can fill to the right the variance and standard deviation formulae rather than re-entering them each time. 1. Do the variances and standard deviations increase or decrease as you confine your random numbers to a smaller and smaller interval? 2. Find the means of each set of 15 measurements. Are they about the same? Why should this be so? Repeat this experiment in columns D, E, and F, confining the random numbers as follows: Column D: 120-130; Column E: 122-128; Column F: 124-126. 3. About how much bigger were the means in this experiment than in the first experiment (columns A-C)? 4. Did the same pattern among variances and standard deviations hold as in the first experiment (see question 1)? 5. How did the variances and standard deviations differ from the first experiment? Explain. 6. In column A the simulated data was allowed to span from 20 to 30 units, a maximum range of 10 units, larger than the ranges in B and C. But it is possible to have a large range and still have a small variance and standard deviation. For example, consider these 15 measurements that also have a span from 20 to 30 units: {20.0, 24.8, 24.8, 24.9, 24.9, 24.9, 25.0, 25.0, 25.0, 25.1, 25.1, 25.1, 25.2, 25.2, 30.0} 7. The mean is these data is 25.0 units. How does this mean and range compare to your mean in column A? 8. How would you guess the variation and standard deviation compare to that in column A? (If you aren’t sure or want confirmation, use Excel to actually finds these values for you.) 9. If columns A, B, and C all represent separate sets of measurements of the same quantity, which set of measurements is the most precise? Why? 10. Use the square root function in Excel, SQRT to take the square roots of the variances in your spreadsheet. What do you notice? 11. Let’s take another look at the data you gathered on day 3 and try to get a handle on how “spread out” the measurements were. Which team was most precise in the measurements?  This, by the way, is an option if the students have access to a large enough sample of brake pads. Each student could measure one or more of the pads and record their data in a spreadsheet. Then the data could then be analyzed as recommended above. One possible drawback to this option is that there are, in a sense, two variables: the actual thickness of the different brake pads and the accuracy of the measurements being made. If the students are fairly proficient with the necessary measuring techniques, however, the latter error is probably negligible when compared to the scale of variation among the pads, especially if the pads came from cars of different models and ages. 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