There is more than one way to calculate an average. One of the ways is to calculate the mean. Let's look at a simple example. Here are 10 numbers:

3	1
2	3
3	5
4	4
3	3

To calculate the mean, you add up all of the numbers and then divide by the number of numbers (10).

In "math speak" then....

MEAN = S x
            n

Therefore, 31/10 = 3.1

But, what would you do if instead of just having 10 numbers to deal with, you had 10,000,000? It would take you forever to add all those numbers up ... and, you'd have no life!

There is ANOTHER way to calculate an average. It is called a WEIGHTED AVERAGE. What you do have to have is the relative percentage of each of the numbers.

Let's calculate the above 10 numbers using a weighted average:

WEIGHTED AVERAGE = S (decimal fraction x individual number)

So, we first have to know the % of each number represented in the table above:

We have:

10%	1
10%	2
50%	3
20%	4
10%	5

You'll notice that the total % = 100% (naturally....)

To find the weighted average, just convert each of the percentages into a decimal fraction (you should remember how to do this from prealgebra) and then multiply that by each number represented and add 'em all up!

WEIGHTED AVERAGE =

  (0.1)    (1)
+ 0.1     (2)
+ 0.5     (3)
+ 0.2     (4)
+ 0.1     (5)
= 3.1

See, we got the same value as above. But, the advantage of this approach is that we don't need to know the actual individual numbers -- just the relative percentages of each.

Let's use this approach for the 6 isotopes of CALCIUM:

NOTE: Isotopes for the same element are typically written as the SYMBOL of the element and then a whole number which is the MASS NUMBER (mass number = # protons + # neutrons). 

    Example:     Ca-40 means that for this isotope of calcium, it has 20 protons and 20 neutrons.
                       Ca-41 means that for this isotope of calcium, it has 20 protons and 21 neutrons.

Isotope	Atomic mass (amu)	Relative abundance
Ca-40	40.008	96.94%
Ca-42	41.885	0.647%
Ca-43	42.991	0.135%
Ca-44	43.599	2.086%
Ca-46	45.988	0.004%
Ca-48	47.999	0.187%

WEIGHTED AVERAGE =

      0.9694     (40.008) =  38.78376
+    0.00647   (41.885) =      .27099
+    0.00135   (42.991) =      .05804
+    0.02086   (43.599) =      .90948
+    0.00004   (45.988) =      .00184
+    0.00187   (47.999) =      .08976
=                          40.114 amu

That's a whole lot easier to do than to individually add up each and every calcium atom in the world (which you couldn't do anyway!).

SOLVE THESE PROBLEMS:

1. Calculate the atomic mass of Gallium (Ga).  Gallium has two isotopes: ⁶⁹Ga and ⁷¹Ga.  ⁶⁹Ga has a relative abundance of 60.12% and an atomic mass of 68.9257 amu.  ⁷¹Ga has a relative abundance of 39.88% and an atomic mass of 70.9249 amu.  Show all your work!

2. Calculate the atomic mass of element X.  Then, use the periodic table to identify the element.  Show all your work!

3. Calculate the atomic mass of element X.  Then, use the periodic table to identify the element.  Show all your work!

Click here to go to the solutions to these problems.