Lab 1: Introduction to Measurement

Transcription

1 Lab 1: Introduction to Measurement Instructor: Professor Dr. K. H. Chu Measurement is the foundation of gathering data in science. In order to perform successful experiments, it is vitally important to understand the basics of measurement. 1. Measuring with Analog or Digital - Digital scales make rounding easy they do it for you! However, with analog scales, like your car speedometer or a meter stick, the user has to round appropriately. To do this, you must first identify the Least Count, or smallest scale markings on the scale. When you make your measurement with an analog scale, always record one place value MORE than the least count. Let s try it out. Using a meter stick (analog scale) measure the length in cm of your pen or mechanical pencil (don t use a wooden one). What is the least count of the ruler? Using the digital mass scale, measure the mass of your pen or mechanical pencil. 2. Statistics in Measurement Lets say that we want to get an idea of the industry standards for pens and how similar different styles are to each other in size and mass. We can find the typical size of pens by calculating the average of the values for the entire class. To see how much variation there is between different pen types, we can calculate the standard deviation which gives an estimate of how much a typical pen s length or mass deviates from the average. Enter the class numbers for length and mass into Excel and calculate the average (=AVERAGE) and the standard deviation (=STDEV). Compare the value of standard deviation to the average and comment on whether pen models have a lot of variation or are fairly standard. 3. Combining measurements In most experiments we will combine multiple measurements to find some new quantity by applying an equation. For instance if we measure the distance traveled by an car and the time it took to travel that distance, then we can take distance/time and find the average speed of the car. Go into the hallway outside the physics lab and take a meter stick with 1

2 you. You will notice that the floor has a grid system built in. Have each lab member measure the length of one of these grids (in m) and take the average value. Starting at the beginning of a grid, have your partner stand 5 grids down. How long is the total distance? Now time yourself while walking at a normal pace across the grids. Repeat this measurement 3 times and take the average value. Using the measured average distance and time, what is your average walking speed (in m/s)? How long would it take you to walk one mile at that rate (in minutes)? How long to walk around the world (in days)? 4. Finding the density of an object - Mass density is a measure of the mass of an object divided by its volume. Measure the length, width, and height of the block of wood in cm. Measure the mass of the block in g. Calculate the volume of the block (recall that volume = L*W*H) Calculate the density of the block in g/cm³. If you found a dead tree limb made of this same wood in your yard that had a mass of 58 kg, what is its volume? 5. Comparing Values - Many times we want to compare our measured quantities to a known value. For instance if we measured the speed of light, we could compare it to the known value of. We do this by using the Percent Difference which is calculated as follows: If we measured the speed of light to be, what is our percent difference? Compare your value for the density of wood to a neighboring lab group by finding the percent difference (use yours as the measured value). 2

3 Appendix: Standard Deviation and Variance The Standard Deviation is a measure of how spread out the data are. Its symbol is σ (the greek letter sigma). It is the square root of the Variance. Example: You and your friends have just measured the heights of your dogs (in millimeters): The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm. Find out the Mean, the Variance, and the Standard Deviation. Your first step is to find the Mean: Answer: Mean = = 5 5 = 394 so the mean (average) height is 394 mm. Let's plot this on the chart: Now, we calculate each dog difference from the Mean: 3

4 To calculate the Variance, take each difference, square it, and then average the result: So, the Variance is 21,704. And the Standard Deviation is just the square root of Variance, so: Standard Deviation: σ = 21,704 = = 147 (to the nearest mm) And the good thing about the Standard Deviation is that it is useful. Now we can show which heights are within one Standard Deviation (147mm) of the Mean: So, using the Standard Deviation we have a "standard" way of knowing what is normal, and what is extra large or extra small. Rottweilers are tall dogs. And Dachshunds are a bit short. but don't tell them! But. there is a small change with Sample Data Our example was for a Population (5 dogs were the entire number of dogs we were interested in). But if the data is a Sample (a selection taken from a bigger Population), then the calculation changes! When you have "N" data values that are: The Population: divide by N when calculating Variance (like we did) A Sample: divide by N-1 when calculating Variance All other calculations stay the same, including how we calculated the mean. Example: if our 5 dogs were just a sample of a bigger population of dogs, we would divide by 4 instead of 5 like this: 4

5 Sample Variance = 108,520 / 4 = 27,130 Sample Standard Deviation = 27,130 = 164 (to the nearest mm) Think of it as a "correction" when your data is only a sample. Standard Deviation Formulas The "Population Standard Deviation": The "Sample Standard Deviation": Looks complicated, but the important change is to divide by N-1 (instead of N) when calculating a Sample Variance. 5