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Unformatted text preview: rs to the number that is most frequently seen. We have no mode in this data set, since none of the data values repeat. Suppose, however, that we actually have six data values, where two of them were both 101.33; then the mode would Dakota State University page 227 of 232 Basic Laboratory Statistics General Chemistry I and II Lab Manual be 101.33, and the median would be (99.46+101.33)/2=100.40. Notice that this would also change the mean to 100.27; why? The closer the mean, the median and the mode are to one another, the more “normalized” the data is (that is, the clo ser the data would fit to a curve created with no systematic errors at all). Variance and Standard Deviation We would like to have a measure of how close our data points are to one another, or, better still, how close they are to the mean. For any given data point, we can simply do a subtraction, x − x i , but the problem is, if we try to take the average distance from N ∑ (x − x ) the mean, or i i =1 , we will find that this gives us a value of zero, because of the N way that the mean is defined. In other words, we will have the same error above the mean value as we have below the mean value. How can we get around this problem? Well, instead of taking the absolute difference between each data point and the mean, 2 let’s square this value instead. That is, let’s find ( x − xi ) . Since the square value of any negative number is positive, NOW we can add these points together, and get a legitimate non-zero value. We call this the “Variance”: N Variance = V = ∑ (x − x ) 2 i i =1 N Notice that this does not give us the mean value of each data point from the mean, but rather the square of this value. We are really not interested in the square distance from the mean, so we define the standard deviation as Stddev = σ = V The smaller our standard deviation is, the less spread out our data is. Dakota State University page 228 of 232 Significant Figures General Chemistry I and II Lab Manual Significant Figures Possibly one of the most confusing subject for students is the concept of “significant figures.” One reason that this is so might be that students tend to not understand why significant figures are, well, significant. The reason is simply this; it is the quickest and easiest way for a scientist to show to the reader how reliable their data is. An older method used to be using the “±” symbol. For example, suppose I want to tell you the temperature is 106.234456756316846541684±0.01o C. Two things immediately come to mind; first, why did I bother to write out all of those digits if I can only trust the result to ±0.01? The second is, why bother writing all of this out, including “±0.01”, when 106.23 means exactly the same thing. THIS is what significant figures are. The assumption is an error of ±1 in the last significant decimal point. This is why your instructor will, much to your dismay, insist that you write down “0” from time to time to show the significant figures; if we write 101.2, we are implying an accurac...
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