A: Understanding Measurements
Length, Area, and Volume
Mathematical Symbols There are several common mathematical terms used in expressing measurements and standards. Some common mathematical symbols and their definitions are given in the table below.
Centimeter vs. Millimeter When making measurements of length, area, or volume, the conversion from one unit of measurement to another is affected. Thus, where one centimeter is equal to ten millimeters, one square centimeter is equal to one hundred square millimeters. This fact is illustrated in the following figure: 1cm=10mm
How to Make Measurements Measurements are made constantly in Quality Assurance. These measurements provide a permanent record of the condition of any given sample. The instrumentation used in making measurements may be as simple as a ruler, or as complex as a spectrophotometer. However, regardless of the sophistication of the instrument, there is always a degree of uncertainty in its result. Significant figures are numbers included in a measurement which are believed to be correct by the person making the measurement, and reflect the sophistication and accuracy of the measuring device used (Note: it is assumed that the person making the measurement is competent in the use of the device). The last significant figure, in any given measurement, is the first digit of that measurement which has some uncertainty to it. If the equipment has a digital readout, the last digit of that readout is
usually the first digit that has some uncertainty. For example, a balance that has a
readout to 0.001g may have an accuracy of "0.003g. It is
clear that the digit in the thousandths position (0.001) is the first digit with some
uncertainty (of "0.003g). Therefore, if you were to the
maximize the precision If the equipment does not have a digital readout, the first digit with some uncertainty is the digit that can be estimated by interpolating between the finest divisions on the equipment. AInterpolating@ means to estimate the reading between divisions to the nearest tenth of a division. For example, the illustration to the right depicts a 50ml graduated cylinder which has 1ml divisions, 5ml divisions, and 10ml divisions. The finest divisions found on the cylinder are the 1ml divisions. By interpolating between these divisions, it is possible to make a reading to 0.1ml. Because there are no 0.1ml marks for a guide, any measurement made to 0.1ml is an estimation and clearly contains some uncertainty. Thus, for this 50ml graduated cylinder, the first digit after the decimal place is the last significant figure. To maximize the precision of the cylinder, all readings must be made out to 0.1ml. In our illustration, the observer has determined that the liquid lies between the 43ml mark and the 44ml mark. By interpolation, the reading was given as 43.3ml. This means that the observer perceives the liquid as being approximately one third of the way between the 43ml mark and the 44ml mark. Thus, the last significant figure in this reading is the 3 after the decimal. This is the digit which represents the uncertainty of the measurement, or the interpolation made by the observer. But what if the equipment being used has a capacity for accuracy which is greater than that needed in the procedure? For example, what if the measurement in the figure need only be reported in milliliters? For such situations (which may often be encountered in Quality Assurance testing), one rounds the measurement off to the desired decimal. This may be accomplished according to the following conventions. For any digit in a number which is to be rounded, it shall be rounded off to the nearest whole number. Thus, if the cylinder reading in the illustration need only be reported in whole milliliters, it would be reported as 43ml. We see that 43.3ml is closer to 43ml than it is to 44ml. Had the level of liquid been at 43.7ml (as interpolated by the person making the reading), it would be reported as 44ml. But what of the case when the liquid is interpolated to be exactly at the 43.5ml position? This is neither nearer to 44ml or 43ml, but is directly in-between. It is a generally accepted convention that when readings which are to be rounded are reported as being exactly half way between two distances, one rounds to the nearest even number. Thus, if our reading were 43.5ml, the level of liquid would be reported as 44ml, since this is the nearest even number. Though this convention may at first seem straight forward, it may lead to counterintuitive results. For example, suppose that the liquid in the above cylinder was found to be at the 42.5ml. When reporting this reading in whole milliliters one may be inclined to round up, reporting the level as 43ml. According to the above convention, however, this would be incorrect. The reading should be reported as 42ml, since this is the nearest even number. Reporting to the nearest even number has been proven to be statistically sound and accurate. |
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of the balance, you would record
each weight to the thousandths position.