*Section author: Danielle J. Navarro and David R. Foxcroft*

# Measures of variability¶

The statistics that we’ve discussed so far all relate to *central
tendency*. That is, they all talk about which values are “in the middle”
or “popular” in the data. However, central tendency is not the only type
of summary statistic that we want to calculate. The second thing that we
really want is a measure of the **variability** of the data. That is,
how “spread out” are the data? How “far” away from the mean or median do
the observed values tend to be? For now, let’s assume that the data are
interval or ratio scale, and we’ll continue to use the `afl.margins`

variable. We’ll use this data to discuss several different measures of
spread, each with different strengths and weaknesses.

## Range¶

The **range** of a variable is very simple. It’s the biggest value minus
the smallest value. For the AFL winning margins data the maximum value
is 116 and the minimum value is 0. Although the range is the simplest
way to quantify the notion of “variability”, it’s one of the worst.
Recall from our discussion of the mean that we want our summary measure
to be robust. If the data set has one or two extremely bad values in it
we’d like our statistics to not be unduly influenced by these cases. For
example, in a variable containing very extreme outliers

```
-100, 2, 3, 4, 5, 6, 7, 8, 9, 10
```

it is clear that the range is not robust. This variable has a range of 110 but if the outlier were removed we would have a range of only 8.

## Interquartile range¶

The **interquartile range** (IQR) is like the range, but instead of the
difference between the biggest and smallest value the difference between
the 25th percentile and the 75th percentile is taken. If you don’t
already know what a **percentile** is, the 10th percentile of a data set
is the smallest number *x* such that 10% of the data is less than
*x*. In fact, we’ve already come across the idea. The median of a
data set is its 50th percentile! In jamovi you can easily specify the
25th, 50th and 75th percentiles by clicking the checkbox `Quartiles`

in
the `Exploration`

→ `Descriptives`

→ `Statistics`

screen.

And not surprisingly, in Fig. 12 the 50th percentile is the same as the median value. And, by noting that 50.50 - 12.75 = 37.75, we can see that the interquartile range for the 2010 AFL winning margins data is 37.75. While it’s obvious how to interpret the range it’s a little less obvious how to interpret the IQR. The simplest way to think about it is like this: the interquartile range is the range spanned by the “middle half” of the data. That is, one quarter of the data falls below the 25th percentile and one quarter of the data is above the 75th percentile, leaving the “middle half” of the data lying in between the two. And the IQR is the range covered by that middle half.

## Mean absolute deviation¶

The two measures we’ve looked at so far, the range and the interquartile range, both rely on the idea that we can measure the spread of the data by looking at the percentiles of the data. However, this isn’t the only way to think about the problem. A different approach is to select a meaningful reference point (usually the mean or the median) and then report the “typical” deviations from that reference point. What do we mean by “typical” deviation? Usually, this is the mean or median value of these deviations. In practice, this leads to two different measures: the “mean absolute deviation” (from the mean) and the “median absolute deviation” (from the median). From what I’ve read, the measure based on the median seems to be used in statistics and does seem to be the better of the two. But to be honest I don’t think I’ve seen it used much in psychology. The measure based on the mean does occasionally show up in psychology though. In this section I’ll talk about the first one, and I’ll come back to talk about the second one later.

Since the previous paragraph might sound a little abstract, let’s go through
the **mean absolute deviation** from the mean a little more slowly. One useful
thing about this measure is that the name actually tells you exactly how to
calculate it. Let’s think about our AFL winning margins data, and once again
we’ll start by pretending that there are only 5 games in total, with winning
margins of 56, 31, 56, 8 and 32. Since our calculations rely on an examination
of the deviation from some reference point (in this case the mean), the first
thing we need to calculate is the mean, *X̄*. For these five observations, our
mean is *X̄* = 36.6. The next step is to convert each of our observations
*X*_{i} into a deviation score. We do this by calculating the difference
between the observation *X*_{i} and the mean *X̄*. That is, the deviation
score is defined to be *X*_{i} - *X̄*. For the first observation in our
sample, this is equal to 56 - 36.6 = 19.4. Okay, that’s simple enough. The next
step in the process is to convert these deviations to absolute deviations, and
we do this by converting any negative values to positive ones. Mathematically,
we would denote the absolute value of -3 as |-3|, and so we say that |-3|
= 3`. We use the absolute value here because we don’t really care whether the
value is higher than the mean or lower than the mean, we’re just interested in
how *close* it is to the mean. To help make this process as obvious as
possible, the table below shows these calculations for all five observations:

Description: | which game | value | deviation from mean | absolute deviation |
---|---|---|---|---|

Notation: | i | X_{i} |
X_{i} - X̄ |
| X_{i} - X̄ | |

1 | 56 | 19.4 | 19.4 | |

2 | 31 | -5.6 | 5.6 | |

3 | 56 | 19.4 | 19.4 | |

4 | 8 | -28.6 | 28.6 | |

5 | 32 | -4.6 | 4.6 |

Now that we have calculated the absolute deviation score for every observation in the data set, all that we have to do to calculate the mean of these scores. Let’s do that:

And we’re done. The mean absolute deviation for these five scores is 15.52.

However, whilst our calculations for this little example are at an end,
we do have a couple of things left to talk about. First, we should
really try to write down a proper mathematical formula. But in order do
to this I need some mathematical notation to refer to the mean absolute
deviation. Irritatingly, “mean absolute deviation” and “median absolute
deviation” have the same acronym (MAD), which leads to a certain amount
of ambiguity so I’d better come up with something different for the mean
absolute deviation. Sigh. What I’ll do is use AAD instead, short for
*average* absolute deviation. Now that we have some unambiguous
notation, here’s the formula that describes what we just calculated:

## Variance¶

Although the average absolute deviation measure has its uses, it’s not
the best measure of variability to use. From a purely mathematical
perspective there are some solid reasons to prefer squared deviations
rather than absolute deviations. If we do that we obtain a measure
called the **variance**, which has a lot of really nice statistical
properties that I’m going to ignore,[1] and one massive psychological
flaw that I’m going to make a big deal out of in a moment. The variance
of a data set *X* is sometimes written as Var(X), but it’s more commonly
denoted *s*² (the reason for this will become clearer shortly).

The formula that we use to calculate the variance of a set of observations is as follows:

As you can see, it’s basically the same formula that we used to calculate the average absolute deviation, except that instead of using “absolute deviations” we use “squared deviations”. It is for this reason that the variance is sometimes referred to as the “mean square deviation”.

Now that we’ve got the basic idea, let’s have a look at a concrete example. Once again, let’s use the first five AFL games as our data. If we follow the same approach that we took last time, we end up with the following table:

Description: | which game | value | deviation from mean | squared deviation |
---|---|---|---|---|

Notation: | i | X_{i} |
X_{i} - X̄ |
(X_{i} - X̄)² |

1 | 56 | 19.4 | 376.36 | |

2 | 31 | -5.6 | 31.36 | |

3 | 56 | 19.4 | 376.36 | |

4 | 8 | -28.6 | 817.96 | |

5 | 32 | -4.6 | 21.16 |

That last column contains all of our squared deviations, so all we have to do
is average them. If we do that by hand, i.e. using a calculator, we end up with
a variance of 324.64. Exciting, isn’t it? For the moment, let’s ignore the
burning question that you’re all probably thinking (i.e., what the heck does a
variance of 324.64 actually mean?) and instead talk a bit more about how to do
the calculations in jamovi, because this will reveal something very weird.
Start a new jamovi session by clicking on the main menu button (`☰`

; top
left hand corner) and selecting `New`

. Now type in the first five values
from the `aflsmall_margins`

data set in column A (56, 31, 56, 8, 32). Change
the variable type to `Continuous`

and under `Descriptives`

click the
`Variance`

check box, and you get the same values for variance as the one we
calculated by hand (**324.64**). No, wait, you get a completely *different*
answer (**405.80**) - see Fig. 13.
That’s just weird. Is jamovi broken? Is this a typo? Am I an idiot?

As it happens, the answer is no.[2] It’s not a typo, and jamovi is not
making a mistake. In fact, it’s very simple to explain what jamovi is doing
here, but slightly trickier to explain *why* jamovi is doing it. So let’s start
with the “what”. What jamovi is doing is evaluating a slightly different
formula to the one I showed you above. Instead of averaging the squared
deviations, which requires you to divide by the number of data points *N*,
jamovi has chosen to divide by *N - 1*.

In other words, the formula that jamovi is using is this one:

So that’s the *what*. The real question is *why* jamovi is dividing by *N - 1*
and not by *N*. After all, the variance is supposed to be the *mean* squared
deviation, right? So shouldn’t we be dividing by *N*, the actual number of
observations in the sample? Well, yes, we should. However, as we’ll discuss in
chapter Estimating unknown quantities from a sample,
there’s a subtle distinction between “describing a sample” and “making guesses
about the population from which the sample came”. Up to this point, it’s been
a distinction without a difference. Regardless of whether you’re describing
a sample or drawing inferences about the population, the mean is calculated
exactly the same way. Not so for the variance, or the standard deviation,
or for many other measures besides. What I outlined to you initially (i.e.,
take the actual average, and thus divide by *N*) assumes that you literally
intend to calculate the variance of the sample. Most of the time, however,
you’re not terribly interested in the sample *in and of itself*. Rather,
the sample exists to tell you something about the world. If so, you’re
actually starting to move away from calculating a “sample statistic” and
towards the idea of estimating a “population parameter”. However, I’m
getting ahead of myself. For now, let’s just take it on faith that jamovi
knows what it’s doing, and we’ll revisit the question later on when we
talk about estimation.

Okay, one last thing. This section so far has read a bit like a mystery
novel. I’ve shown you how to calculate the variance, described the weird
“*N - 1*” thing that jamovi does and hinted at the reason why it’s
there, but I haven’t mentioned the single most important thing. How do
you *interpret* the variance? Descriptive statistics are supposed to
describe things, after all, and right now the variance is really just a
gibberish number. Unfortunately, the reason why I haven’t given you the
human-friendly interpretation of the variance is that there really isn’t
one. This is the most serious problem with the variance. Although it has
some elegant mathematical properties that suggest that it really is a
fundamental quantity for expressing variation, it’s completely useless
if you want to communicate with an actual human. Variances are
completely uninterpretable in terms of the original variable! All the
numbers have been squared and they don’t mean anything anymore. This is
a huge issue. For instance, according to the table I presented earlier,
the margin in game 1 was “376.36 points-squared higher than the average
margin”. This is *exactly* as stupid as it sounds, and so when we
calculate a variance of 324.64 we’re in the same situation. I’ve watched
a lot of footy games, and at no time has anyone ever referred to “points
squared”. It’s *not* a real unit of measurement, and since the variance
is expressed in terms of this gibberish unit, it is totally meaningless
to a human.

## Standard deviation¶

Okay, suppose that you like the idea of using the variance because of
those nice mathematical properties that I haven’t talked about, but
since you’re a human and not a robot you’d like to have a measure that
is expressed in the same units as the data itself (i.e., points, not
points-squared). What should you do? The solution to the problem is
obvious! Take the square root of the variance, known as the **standard
deviation**, also called the “root mean squared deviation”, or RMSD.
This solves our problem fairly neatly. Whilst nobody has a clue what “a
variance of 324.68 points-squared” really means, it’s much easier to
understand “a standard deviation of 18.01 points” since it’s expressed
in the original units. It is traditional to refer to the standard
deviation of a sample of data as *s*, though “sd” and “std dev.”
are also used at times.

Because the standard deviation is equal to the square root of the variance, you probably won’t be surprised to see that the formula is:

and in jamovi there is a check box for `Std. deviation`

right above the
check box for `Variance`

. Selecting this gives a value of **26.07** for
the standard deviation.

However, as you might have guessed from our discussion of the variance,
what jamovi actually calculates is slightly different to the formula
given above. Just like the we saw with the variance, what jamovi
calculates is a version that divides by *N - 1* rather than *N*.

For reasons that will make sense when we return to this topic in chapter Estimating unknown quantities from a sample I’ll refer to this new quantity as \(\hat\sigma\) (read as: “sigma hat”), and the formula for this is:

Interpreting standard deviations is slightly more complex. Because the
standard deviation is derived from the variance, and the variance is a
quantity that has little to no meaning that makes sense to us humans,
the standard deviation doesn’t have a simple interpretation. As a
consequence, most of us just rely on a simple rule of thumb. In general,
you should expect 68% of the data to fall within 1 standard deviation of
the mean, 95% of the data to fall within 2 standard deviation of the
mean, and 99.7% of the data to fall within 3 standard deviations of the
mean. This rule tends to work pretty well most of the time, but it’s not
exact. It’s actually calculated based on an *assumption* that the
histogram is symmetric and “bell shaped”.[3] As you can tell from
looking at the AFL winning margins histogram in Fig. 20,
this isn’t exactly true of our data! Even so, the rule is approximately
correct. As it turns out, 65.3% of the AFL margins data fall within one
standard deviation of the mean. This is shown visually in Fig. 14.

## Which measure to use?¶

We’ve discussed quite a few measures of spread: range, IQR, mean absolute deviation, variance and standard deviation; and hinted at their strengths and weaknesses. Here’s a quick summary:

*Range*. Gives you the full spread of the data. It’s very vulnerable to outliers and as a consequence it isn’t often used unless you have good reasons to care about the extremes in the data.*Interquartile range*. Tells you where the “middle half” of the data sits. It’s pretty robust and complements the median nicely. This is used a lot.*Mean absolute deviation*. Tells you how far “on average” the observations are from the mean. It’s very interpretable but has a few minor issues (not discussed here) that make it less attractive to statisticians than the standard deviation. Used sometimes, but not often.*Variance*. Tells you the average squared deviation from the mean. It’s mathematically elegant and is probably the “right” way to describe variation around the mean, but it’s completely uninterpretable because it doesn’t use the same units as the data. Almost never used except as a mathematical tool, but it’s buried “under the hood” of a very large number of statistical tools.*Standard deviation*. This is the square root of the variance. It’s fairly elegant mathematically and it’s expressed in the same units as the data so it can be interpreted pretty well. In situations where the mean is the measure of central tendency, this is the default. This is by far the most popular measure of variation.

In short, the IQR and the standard deviation are easily the two most common measures used to report the variability of the data. But there are situations in which the others are used. I’ve described all of them in this book because there’s a fair chance you’ll run into most of these somewhere.

[1] | Well, I will very briefly mention the one that I think is coolest, for a
very particular definition of “cool”, that is. Variances are additive.
Here’s what that means. Suppose I have two variables X and Y, whose
variances are Var(X) and Var(Y) respectively. Now imagine I want to define
a new variable Z that is the sum of the two, Z = X + Y. As it turns
out, the variance of Z is equal to Var(X) + Var(Y). This is a very useful
property, but it’s not true of the other measures that I talk about in this
section. |

[2] | With the possible exception of the third question. |

[3] | Strictly, the assumption is that the data are normally distributed, which
is an important concept that we’ll discuss more in chapter Introduction
to probability and will turn up over and over again later
in the book. |