This chart lists the cost of sending a 1st class letter in the US from 1919 until 1997. What we are interested in here is to discuss ways of using a spreadsheet to analyse this data and make some predictions.

For starters, it will be nice to get a graph of the data. Since our chart reflects all the changes in rate, we can assign each intervening year the same rate. This yields the following graph:

This kind of a graph is called a step function. It behaves very much like other functions with one additional property--values on various intervals are rounded down to some constant value. It frequently looks like a staircase--hence the name "Step Function."

The first step we will take is to try to find a function whose graph somewhat like this data. Then we will turn it into a step function. Finally, we will make some predictions from our model.

One way to look at this graph is to break it into two parts--thus we will construct a piece-wise defined function. We can construct a linear functionto describe the years from 1919 -1968, and then another linear function for 1968 -1997. Our function will then be defined as follows:

Let's start by choosing the linear functions which connect the endpoints of these intervals. i.e.

Then (Note: is the least integer function)

We now let the spreadsheet find the values of this function and compare them with our original graph:

Although there are many other kinds of functions which we could try (exponential, polynomial, etc), it appears that this piece-wise defined function works well.

Let us address several questions. Although our actual values do not go past 1997, we can project what they might be in the future by using our function. Here is a small piece of the spreadsheet we have been using to manipulate our data. If you would like to see the actual Excel document (you must have the Excel program on your computer) click here.

1. How much do we expect the postal rate to be in the year 2000?

From the data, we can estimate that the rate will be 34 cents in the year 2000.

2. When will the rate go up again?

We can see that we expect this next increase to occur in 1999.

3. When will the rate be $1?

Although this portion of the spreadsheet does not give this value, we can go to the spreadsheet and try values until we get the desired amount. We find that . So we would expect the rate to be $1 in the year 2073.

Now, there is at least one major problem with the function that we have created here. We noticed that in 1968 the rate started increasing at a much higher rate and constructed our function accordingly--it is fairly likely that this will happen again and we have no way of predicting when that will be. This fact may lead us to try other kinds of functions like exponential or polynomial.

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