Exponential Relationships and Models

Exponential Relationships and Models

Exponential Relationships

Like linear relationships, exponential relationships describe a specific pattern of change between a dependent and independent variable
Recognizing and modeling this pattern allows us to make predictions about future values based on an exponential rate of change.

Exponential growth: summary

While in linear relationships, y changes by a fixed absolute amount for each change in x (such as adding 3 each time),
in exponential relationships, y changes by a fixed relative amount for each change in x (such as adding 3% each time).
Quantities related exponentially increase at an increasing rate, producing rapid growth

Exponential relationships

When noticing and describing patterns between variables, we look at the value of the independent variable (x), the dependent variable (y), and the rate of change in the dependent variable relative to the independent variable.

Exponential relationships: example

In Boomtown, a small but increasingly popular destination, the number of new people moving there doubles every year.

Independent variable (x)= time, in years

Dependent variable (y) = total number of people in the town (new arrivals + residents)

Rate of change: doubling the # new people arriving each year

Calculating the growth & the total

Exponential growth: graph

Exponential growth: when a quantity changes by a percentage

The exponential notation can describe percentage change as well:
Say quantity ‘A’ is growing by r% each year. After one year, A will become A + Ar. Algebraically, this can also be written as A(1+r)
If A(1+r) is then increased again the next year by r%, then we can say A(1+r) * (1+r), which is A(1+r)²

Exponential modeling

It is possible to use these equations to make predictions, or to use an exponential trendline on a scatterplot, but making predictions with this type involves solving these equations of an exponential trendline (y=b^x) with logarithmic calculations (calculus).
For the purposes of our data analysis in this course, it is appropriate to model growth with cell-reference equations in an Excel table.

Example: A savings account that grows at a rate of 8% per year

Assume you start with $100 in the fund
At the end of the first year, you will have earned 8% interest. Since 8% of $100 is $8, you have $100 + $8 = $108 in the account.
In the second year, you will be earning 8% of a higher amount, $108. So the interest accrued is .08*108 = $8.64. Your total at the end of the second year is $108 + $8.64 = $116.64.

Example: 8% growth fund

The fact that you are earning a percentage of a successively higher amount each year (compound interest) means that the fund is increasing at an increasing rate, the definition of an exponential relationship.
This can be modeled by using excel table and cell reference formulas.

Example: Fund that grows at 8% per year –Excel formulas to model

How to use Excel for exponential models (summary)

To make predictions with Excel about future values based on exponential growth:
Create a table with the years (or independent variable units) in column A, and extend the table to the year you’re trying to make a prediction for.

Do this with the series fill command (highlight three cells in the column, place the the "+" cursor on the bottom right box, and drag down until desired year is reached)

In column B, use cell references to calculate the successive percentage increase (or exponential increase) in a starting value. Drag formula to end year.

Limits of exponential models

There is no equivalent to the linear model’s r-squared value for exponential models – nothing to tell you how accurate the model is
Exponential models share the limits of linear models in that they cannot predict real-world factors that may intervene
Exponential models are best used when variables are seen to be changing by a certain percentage, sometimes more realistic than linear models in this respect.