In the experimental work are two kinds of numbers: the exact, which are those whose value is accurately known, and the inaccurate, whose value has some uncertainty. Exact numbers have many fixed values defined, for example, in one kilometer there are exactly 1000 m, or in a liter there are 1000 mL. The number 1 in any conversion factor, as in 1 min = 60 s, is also an exact number. The exact numbers can be the result of counting objects such as, for example, the number of sheep from a flock or the number of beads on a string.
The numbers that are obtained by measurements are always inaccurate. The instruments used to measure always have inherent limitations (equipment errors) and there are also differences in how different people make the same measurement (human error). Suppose five students with five scales determine the weight of the same piece, a coin, for example. It is likely that the five measures vary slightly from each other for various reasons. The scales can be calibrated slightly differently and there may be differences in how each student “reads” the weight that determines the balance.
It is very important to remember that when quantities are measured there is always an uncertainty. When speaking of quantities measured the terms precision and accuracy are used. The precision of a measure is defined as the closeness between all individual measures (one to each of the others) and accuracy as proximity of each individual measure with the correct or “real” value. Bias is a systematic error (always happens) that makes all the measures are deflected by a certain amount so that if we measure something several times and values are close to each other, can all be wrong if there is bias.
Typically, in the laboratory several “tests” of the same experiment (replicas) are made, and the results are presented as an average. The accuracy of the measurements is often expressed in terms of so-called standard deviation, reflecting the difference in an individual measurement with the average value of all measurements. If we get values very close to each other our confidence in our measures increases because the value of the standard deviation is small. However, as shown in the figure, we can have very precise measurements but little accurate or, put another way, biased. If a very sensitive balance is wrongly calibrated, the weights we measure will be consistent but higher or lower than they should. Will be inaccurate although precise.
To know in detail a series of data, it is not enough to know the measures, we also need to know the deviation that these data show regarding the average (or arithmetic mean) in order to have a more realistic view of them to describe and interpret them correctly . The standard deviation “s” of a dataset is a measure of how much specific data deviate of the average thereof.
For information only, standard deviation “s” is defined as:
where N is the number of measurements, is the average (also known as mean), and represents each of the individual measurements. That is, is the diference between each specific value and the average (it may be a positive or a negative value); if we squared it we get (always will be positive) and represents the sum of all the squares of the differences between each specific value and the (still is a positive value). Finaly, if we divide taht number by the number of measurements minus one and get thet suqre root, we get the standard deviation value “s”. The smaller the value of s, smaller is the dispersion of the data regarding its average or higher is the accuracy, implying that the data are grouped very close to the average.