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Unformatted text preview: 0.12 and 100.10. These readings are all close to one another, but they ought to be
reading closer to 100.00; they are precise, but not very accurate. Sometimes, these errors
can be corrected for by using the same equipment. For example, if we are using mass
difference, and get the mass of a beaker first, it’s mass will also be off by 0.11 g. Thus,
when we subtract the mass of the beaker from the mass of the liquid and the beaker to get Dakota State University page 226 of 232 Basic Laboratory Statistics General Chemistry I and II Lab Manual just the mass of the liquid, this systematic error will automatically cancel itself.
However, it is important to note that this error cancellation will only occur if we are
consistent with the instruments that we use since it is highly unlikely that a second
instrument, in this case a balance, will be off by the same amount as the other. Thus, if
we switch balances in the middle of the experiment, these systematic errors will not
“Accuracy” is how close the mean is to the true value (which may or may not be
known). For example, because of the density of water, 100.00 mL of water should have a
mass of 100.00 g. You can have high accuracy with low precision. For example,
suppose our balance is outside on a gusty day; we might get five readings of 95.02,
107.80, 96.42, 101.33 and 99.46. The mean of this data (see below) is 100.01 g, which is
very close to what it ought to be, but the values themselves are all over the place.
Naturally, we want results that are both accurate and precise. If we cannot have
both, however, which would you rather have yourself, high accuracy, or high precision?
Mean, Median and Mode
The term “average” is thrown around a lot, but has very low precision in its
meaning (see how I did that?). People usually mean “mean” when speaking of average,
but it can also refer to median or mode. From here on out, get in the habit of using the
terms “mean,” “median” or “mode” instead of “average,” as this will tell people precisely
what you mean.
There are three basic ways to discuss the “middle of the value” or “average” for a
set of data. The “mean” is the most common, this is simply determined by adding up the
individual data points, and dividing by the total number of data points:
N mean = x = ∑x
i =1 i N Thus, for our set of data 95.02, 107.80, 96.42, 101.33 and 99.46, we have
x= 95.02 + 107.80 + 96.42 + 101.33 + 99.46
5 as our mean (see “significant figures” below).
The “median” is the middle value of the data. If we re-arrange this data in
increasing (or decreasing) order, we have 95.02, 96.42, 99.46, 101.33, and 107.80. Since
we have five data points, the middle data point will be the third data point, or, in this
case, 99.46. This is our median. If we have an even number of data points, then the
median is the mean of the two center data points. Notice that this is close to, but
distinctly different from our mean.
Finally, the “mode” refe...
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