1. The mass of a metal block was measured three times.The three measurements are 100.2 grams, 100.3 grams, 100.1 grams.  What is the average? 

How 'good' is the precision of these measurements?  Is the average good to 1 part in 10, or 10%; good to 1/100 or 1%;  good to 1/1000 or 0.1%;  good to 1/10,000 or 0.01%. 

The true (correct) mass is 110.2 grams. Is the accuracy as good as the precision?  Why could they differ?

2. Today there are 118 known elements. click here to see a list of them. There are 86 metals among these. What percentage of the known elements are metals?

3.  A loaf of bread weighs 675 grams. It made using 500 grams of flour, 65 grams of eggs, 8 grams of salt, 35 grams of milk 67 grams of water and 1 gram of yeast. Why might the final mass be less than the starting mass? What is the percent salt as NaCl in the original mixture? click for sources of salt in food

1.  To figure the average we add the values for the measurements and divide by the count for the measurements.. 

Total of measurements = 100.2 grams  + 100.3 grams + 100.1 grams = 300.6 grams

average  =   total / number of measurements

average =  300.6 grams   / 3  =  100.2 grams  with four sf  

Remember the "3" is a count so it is an exact number.

The values all agree in the first three digits and disagree in the fourth digit. The average (100.2 g ) has four significant figures.

The precision is good to 0.1 grams out of 100.2 grams.

A different way of expressing this is by the ratio 0.1 grams / 100.2 grams. 

This is basically the same as  1 part out of 1 thousand  (1/ 1000). The precision is good. The measurements agree well with one another.

The accepted value is 110.2 grams. This is substantially different from the average of 100.2 grams for the measurements. The accuracy is not as good as the precision. The accuracy is only good to 10 g out of 110.2 grams. This means they differ from the correct value by about 10% or 10 parts out of 100 or  1/10. 

When the tool used for measuring is not calibrated properly the precision can be good while the accuracy is poor.  Repeated measurements can be consistent with one another but they will not match the actual correct value.

3.  The total mass of the loaf of bread will probably be les than the mass of the ingredients used at the start because some water will be 'boiled' off during the baking also the reactions in the baking process will release carbon dioxide from the yeast activity and the baking powder.  To calculate the percent NaCl first add all the masses to get a total.

Percent   = 100 ( part / total )

Total mass = 500 g + 65 g + 8 g + 67 g + 1 g = 676 g perce

Percent salt, NaCl = 100 [ 8 g salt / 676 g total ] = 1.18 %  rounded to 1 % by mass

NOTE: When the amount of sodium is listed for a serving the number of milligrams refer to Na and not NaCl.  Since sodium chloride is always has the formula NaCl it has a definite percent composition.  The proportions for each element in table salt, NaCl, are 23 g Na for every 35.5 g Cl. 

The percent compositon by mass for NaCl is 

percent Na  = 100 x (23 g / 58 g) = 39.6 % Na sodium ions

percent Cl = 100 x ( 35.5 g / 58 g) = 61 % Cl chloride ions

The milligram listings for a serving on food labels ignore the chloride and refer to the sodium ion from all sources such as sodium bicarbonate (NaHCO₃ ,baking soda), preservatives and artificial sweeteners. click to see more about NaCl.

4. The average is  

Average = total / number measurements 

Average = [ 23.4 cm + 23.0 cm + 23.3 cm + 23.7 cm + 22.7 cm ] [ 1 / 5 ]

Average = 116.1 cm [ 1/ 5]  =  23.2 cm 

which should be rounded to 23. cm with two significant figures since the second digit is not the same for all measurements .  The uncertainty in the average begins with the second digit.