I am unsure about the accuracy and/or precision of measuring instruments. For example, if we have a graduated cylinder, a pipette and a beaker, how and why can we assert them that they can be used with accuracy, precision, or both?
Beakers, Erlenmeyer flasks, dropping funnels and the like are not volumetric instruments. They are not precisely calibrated, and their scales serve only as approximate guides. Therefore, the beaker that is mentioned in the question is right out.
Graduated cylinders, volumetric flasks, bulb pipettes, graduated pipettes, and burettes are volumetric instruments. They are calibrated; however, like any other measuring instrument, they only have a limited accuracy and precision.
The accuracy and precision do not only depend on the type of the instrument (e.g. graduated cylinder or pipette). They also depend on the dimensions, graduating divisions, and accuracy class of the instrument, which are usually internationally standardized.
In principle, volumetric instruments with a small diameter have a better precision because the same difference of the liquid level (i.e. the meniscus) corresponds to a smaller difference in volume. Therefore, typical long and thin pipettes are usually better than short and thick graduated cylinders. For example:
A standardized graduated pipette (Class A or AS) with a nominal capacity of 10 ml and an overall length of about 360 mm has a maximum permissible error of ±0.05 ml.
A standardized graduated measuring cylinder (Class A) with a nominal capacity of 10 ml and a maximum overall height of 140 mm (tall form) has a maximum permissible error of ±0.1 ml.
If you use an even larger graduated measuring cylinder (Class A) with a nominal capacity of 100 ml and fill it just to the 10 ml mark, the maximum permissible error is ±0.5 ml.
These errors represent the maximum permissible error at any point on the scale, and also the maximum permissible difference between the errors at any two points.
Since the same sources of error are, naturally, inherent both in calibration and use, a volumetric instrument should have a good precision in order to be able to be calibrated with a good accuracy.
However, note that graduated pipettes are usually calibrated ‘to deliver’ (TD, Ex), whereas graduated measuring cylinders are calibrated ‘to contain’ (TC, In). Hence, graduatted pipettes and graduated measuring cylinders may not be interchangeable at will.
There are too many measurement devices to give an answer for all types. However, assuming the scope of your question is limited to measuring fluid volume the way this can be done in a professional laboratory is simple.
You will need:
(1) high purity water
(2) a high precision balance which has been recently calibrated
(3) a type of container called a weigh boat
Tare the weigh boat. This means place the weigh boat on the balance and (if the balance is electronic) press the tare button. This will cause the balance to read zero. (With the old pivot beam style balance another identical weigh boat could have been placed on the other end of the balance to level it. These pivot beam balances are rarely used in modern lab practice as most balances are electronic.)
Using the measuring tool in question, measure a certain volume of high purity water. Pour the measured quantity of high purity water into the weigh boat on the electronic balance. The balance should read the mass of the water. Assuming the temperature of the lab is close to 20°C divide the mass of the water by 0.9982 g/ml. This is an accurate way to tell how many ml were actually measured using the tool.
Repeat the measurement three times. Three true measurements can be used to determine an average and standard deviation. The difference between what was measured and the average of the three true measurements is related to the accuracy of the measurement. The standard deviation is related to the precision of the measurement.
If a pipet is supposed to dispense 10 mL and it actually dispenses 15, 16, and 15 mL then it has high precision (because the numbers are close.) Its accuracy is way off because the average of 15.3 mL is half of an order of magnitude different than 10. It could be used, though it would need to be either calibrated so the measurement is correct or the end numbers adjusted for the inaccuracy.
If a pipet is supposed to dispense 10 mL and it actually dispenses 5, 9, and 16 mL then despite being accurate, its precision is off. Because the precision is off, this pipet is too unreliable to be used in scientific practice.
Realistically, if a pipet were supposed to dispense 10 mL and it dispenses 10.0010, 10.0000, and 9.9998 mL then it would be suitable for lab work. Because both the accuracy and precision is high, this measurement is said to have high trueness. Trueness -- which is both accuracy and precision -- is desirable in a good measurement.
It is always good practice to verify the calibration of any measurement before using the measurement to generate publishable data. Recently published data in high-quality publications is a good source of information to judge if a certain set of data has acceptable trueness.
The term precision means two different things. The number 1.23456789 has more "precision" than the number 1.23. That is, there is a mathematical (and computer science) meaning that is different than the one used by physical scientists.
In physical science, precision means the property of reproducibility of a value or result. If I am precise, I get the same answer every time (or nearly so). If I am accurate, I get the correct answer. Here is one example: I have a machine which counts people going through the door. Whenever someone walks through, it adds 1.1 to its total. This machine is precise but not very accurate. I modify it and now it adds between 0.95 and 1.05 to the count each time someone walks thru. It is definitely not more precise, but may or may not be more accurate.