Section author: Danielle J. Navarro and David R. Foxcroft
Mathematical functions and operations¶
In the previous section, I discussed the ideas behind variable transformations and showed that a lot of the transformations that you might want to apply to your data are based on fairly simple mathematical functions and operations. In this section I want to return to that discussion and mention several other mathematical functions and arithmetic operations that are actually quite useful for a lot of real world data analysis. Table 5 gives a brief overview of the various mathematical functions I want to talk about here, or later.[1] Obviously this doesn’t even come close to cataloguing the range of possibilities available, but it does cover a range of functions that are used regularly in data analysis and that are available in jamovi.
| Function | Example input | result | |
|---|---|---|---|
| Square root | SQRT(x) |
SQRT(25) |
5 |
| Absolute value | ABS(x) |
ABS(-23) |
23 |
| Logarithm (base 10) | LOG10(x) |
LOG10(1000) |
3 |
| Logarithm (base e) | LN(x) |
LN(1000) |
6.908 |
| Exponentiation | EXP(x) |
EXP(6.908) |
1000.245 |
| Box-Cox | BOXCOX(x, lamda) |
BOXCOX(6.908, 3) |
109.551 |
| Rounding to nearest | ROUND() |
ROUND(1.32) |
1 |
| Rounding down | FLOOR() |
FLOOR(1.32) |
1 |
| Rounding up | CEILING() |
CEILING(1.32) |
2 |
Logarithms and exponentials¶
As I’ve mentioned earlier, jamovi has an useful range of mathematical functions built into it and there really wouldn’t be much point in trying to describe or even list all of them. For the most part, I’ve focused only on those functions that are strictly necessary for this book. However I do want to make an exception for logarithms and exponentials. Although they aren’t needed anywhere else in this book, they are everywhere in statistics more broadly. And not only that, there are a lot of situations in which it is convenient to analyse the logarithm of a variable (i.e., to take a “log-transform” of the variable). I suspect that many (maybe most) readers of this book will have encountered logarithms and exponentials before, but from past experience I know that there’s a substantial proportion of students who take a social science statistics class who haven’t touched logarithms since high school, and would appreciate a bit of a refresher.
In order to understand logarithms and exponentials, the easiest thing to
do is to actually calculate them and see how they relate to other simple
calculations. There are three jamovi functions in particular that I want
to talk about, namely LN(), LOG10() and EXP(). To start
with, let’s consider LOG10(), which is known as the “logarithm in
base 10”. The trick to understanding a logarithm is to understand
that it’s basically the “opposite” of taking a power. Specifically, the
logarithm in base 10 is closely related to the powers of 10. So let’s
start by noting that 10-cubed is 1000. Mathematically, we would write
this:
The trick to understanding a logarithm is to recognise that the statement that “10 to the power of 3 is equal to 1000” is equivalent to the statement that “the logarithm (in base 10) of 1000 is equal to 3”. Mathematically, we write this as follows,
Okay, since the LOG10() function is related to the powers of 10, you
might expect that there are other logarithms (in bases other than 10)
that are related to other powers too. And of course that’s true: there’s
not really anything mathematically special about the number 10. You and
I happen to find it useful because decimal numbers are built around the
number 10, but the big bad world of mathematics scoffs at our decimal
numbers. Sadly, the universe doesn’t actually care how we write down
numbers. Anyway, the consequence of this cosmic indifference is that
there’s nothing particularly special about calculating logarithms in
base 10. You could, for instance, calculate your logarithms in base 2.
Alternatively, a third type of logarithm, and one we see a lot more of
in statistics than either base 10 or base 2, is called the natural
logarithm, and corresponds to the logarithm in base e. Since you might one
day run into it, I’d better explain what e is. The number e, known as
Euler’s number, is one of those annoying “irrational” numbers whose decimal
expansion is infinitely long, and is considered one of the most important
numbers in mathematics. The first few digits of e are:
e = 2.718282
There are quite a few situation in statistics that require us to
calculate powers of e, though none of them appear in this book.
Raising e to the power x is called the exponential
of x, and so it’s very common to see ex written as
exp(x). And so it’s no surprise that jamovi has a function that
calculates exponentials, called EXP(). Because the number e
crops up so often in statistics, the natural logarithm (i.e., logarithm
in base e) also tends to turn up. Mathematicians often write it
as loge(x) or ln(x). In fact, jamovi works the same
way: the LN() function corresponds to the natural logarithm.
And with that, I think we’ve had quite enough exponentials and logarithms for this book!
| [1] | We’ll leave the box-cox function until later on. |