ch09centreanddispersion.Rmd

# Centre and dispersion {#ch-centre-and-dispersion}

## Introduction

In the preceding chapter, we learnt to count and classify observations.
These allow us to summarise a variable's observations, for example
in a table, a frequency distribution, or in a histogram. We can often 
summarise the observations even further, in characteristics which 
indicate the manner in which the observations are distributed. In 
this chapter we will acquaint ourselves with a number of such characteristics.
Some of these characteristics are applicable to variables of all levels of
measurement (e.g. mode), others only to variables of interval or ratio 
level (e.g. mean). After an introduction on using symbols,
we will firstly discuss how we can describe the centre of a distribution,
and how we can describe the dispersion. 

## Symbols

In descriptive statistics, much work is done with symbols. The
symbols are abbreviated indications for a series of actions.
You already know some of these symbols: the exponent ${}^2$ in the expression
$x^2$ is a symbol which means "multiply $x$
with itself", or $x^2 = x \times x$ (where $\times$ is also again 
a symbol).

Often a capital letter is used to indicate a variable ($X$),
and a lower case letter is used to indicate an individual score
of that variable. If we want to distinguish the individual
scores, we do so with a subscript index: $x_1$ is the
first observation, $x_2$ is the second observation, etc. As such, 
$x_i$ indicates the score of participant number
$i$, of variable $X$. If we want to generalise 
over all the scores, we can omit the index but 
we can also use a dot as an "empty" index: in
the expression $x_.$ the dot-index stands
for any arbitrary index.

We indicate the number of observations in a certain group with a
lower case $n$, and the total number of observations of a variable
with the capital letter $N$. If there is only one group, like in
the examples in this chapter, then it holds that $n=N$.

In descriptive statistics, we use many addition operations, and for these
there is a separate symbol, $\sum$, the Greek capital letter Sigma,
with which an addition operation is indicated. We could say "add all the observed
values of the variable $X$ to each other", but we usually do this
more briefly:
$$\sum\limits_{i=1}^n x_i, \textrm{or even shorter} \sum x
%   \Sigma_i^N x_i, \textrm{or even shorter} \Sigma X$$
This is how 
we indicate that all $x_i$ scores have to be added to each other,
for all values from $i$ (from $i=1$, unless indicated
otherwise) to $i=n$. All $n$ scores of the variable $x$
therefore have to be added up. 

When brackets are used then pay good attention: actions described
within a pair of brackets have priority, so you have to execute
them first. Also when it is not strictly necessary, we will often use
brackets for clarity, like in $(2\times3)+4=10$.

## Central tendencies

### mean {#sec:mean}

The best known measure for the centre of a distribution is the 
mean. The mean can be calculated straightforwardly by adding
all scores to each other, and then dividing the sum by the 
number of observations. In symbols:
\begin{equation} 
  \overline{x} = \frac{\sum x}{n} = \frac{1}{n} \sum\limits_{i}^n x_i
  (\#eq:average)
\end{equation} 

Here we immediately encounter a new symbol, $\overline{x}$, often
named "x-bar", which indicates the mean of $x$. The mean
is also often indicated with the symbol $M$ (mean), amongst others 
in articles in the APA-style.

---

> *Example 9.1*: In 
a shop, it is noted how long customers have to wait
at the checkout before their turn comes. For $N=10$ customers,
the following waiting times are observed, in minutes:\
1, 2, 5, 2, 2, 2, 3, 1, 1, 3.\
The mean waiting time is $(\sum X)/N = 22/10 = 2.2$ minutes.

---

The mean of $X$ is usually expressed with one decimal figure more than
the scores of $X$ (see also §\@ref(sec:significantfigures-means) below about the number of significant
figures with which we represent the mean). 

The mean can be understood as the "balance point" of a distribution:
the observations on both sides hold each other "in equilibrium", as
illustrated in Figure \@ref(fig:waittime-hist), where the "blocks"
of the histogram are precisely "in equilibrium" at the "balance point" 
of the mean of 2.2. The mean is also the value relative to which the
$N$ observations together differ the least, and therefore forms a good
characteristic for the centre of a probability distribution. 

The mean can only be used with variables of the interval or ratio
level of measurement.

```{r waittime-hist, echo=FALSE, fig.cap="Histogram of N=10 waiting times, with the mean marked.", fig.width=4}
# waiting times example, to illustrate the mean
# HQ 20160228 en 20160302
x <- c( 1, 2, 5, 2, 2, 2, 3, 1, 1, 3 ) # see example 9.1
hist(x, breaks=seq(min(x),max(x)+1,1)-0.5, col="grey80", 
     main="", xlab="Waiting time (minutes)", ylab="Frequency" )
# axis(side=1, at=mean(x), labels=expression(symbol("\220")), 
#      cex=5, lwd.ticks=3, tcl=1, col="maroon")
axis(side=1, at=mean(x), labels="", lwd.ticks=6, tcl=1.3, col="maroon")
```

### median {#sec:median}

The median (symbol $Md$ or $\tilde{x}$) is the observation in the middle 
of the sequence of sorted observations [^fn09-1]. When we sort the observed values of variable $X$
from smallest to largest, the median is the midpoint of the sorted
sequence. Half of the observations are smaller than the median, 
and the other half is larger than the median. 

[^fn09-1]: In American English, the area in the middle of a road (separating traffic in opposite directions) is called the 
"median (strip)" (British English: "central reservation"); this strip typically splits the road into two equally large halves.

For an odd number of observations, the middlemost observation is the median.
For an even number of observations, the median is usually formed from
the mean of the two middlemost observations.

---

> *Example 9.2*: The waiting times from Example 9.1
are ordered as follows:\
1, 1, 1, 2, *2, 2*, 2, 3, 3, 5.\
The median is the mean of the two middlemost (italicised)
observations, so 2 minutes.

---

The median is less sensitive than the mean to extreme values 
of $x$. In the above example, the extreme waiting time of 5 minutes
has a considerable influence on the mean. If we remove 
that value, then the mean changes from 2.2 to 1.9 but the median is still 
2. Extreme values thus have less great an influence on the
median then on the mean. Only if
the ordering of the observations changes, may the median also 
change.

The median can be used with variables of ordinal, interval or ratio
level of measurement. 

### mode

The mode (adj. 'modal') is the value or score of $X$ which occurs
the most frequently.

---

> *Example 9.3*: In the waiting times from Example 9.1
the score 2 occurs the most often ($4\times$); this is the mode.

---

> *Example 9.4*: In 2022, the mean income per household in the Netherlands was about €35,000.
The modal income (per household) was between €22,000 and €24,000[^fn09-2]: this income category contained the highest number of households. 

[^fn09-2]: https://www.cbs.nl/nl-nl/visualisaties/inkomensverdeling

---

The mode is even less sensitive than the mean to extreme values of
$x$. In the Example 9.2 above, it does not matter what the value of 
the longest waiting time is: even if that observation has the value $10$ or
$1,000$, the mode remains invariably $2$ (check it for yourself).

The mode can be used with variables of all levels of measurement.

### Harmonic mean {#sec:harmonicmean}

If the dependent variable is a fraction or ratio, 
such as the speed with which a task is conducted, then 
the (arithmetic) mean of formula \@ref(eq:average) does 
not actually provide a good indication for the 
most characteristic or central value. In that case, it is better
to use the harmonic mean:

\begin{equation}
  H = \frac{1}{\frac{1}{n} \sum\limits_{i}^n \frac{1}{x_i} } = \frac{n}{\sum\limits_{i}^n \frac{1}{x_i}}
  (\#eq:harmonicmean)
\end{equation}

---

> *Example 9.5*: 
A student writes $n=3$ texts. For the first text (500 words) (s)he takes
2.5 hours, for the second text (1,000 words) (s)he takes 4 hours, and
for the third text (300 words) (s)he takes 0.6 hours. What is this student's
mean speed of writing?
The speeds of writing are respectively 200, 250 and 500
words per hour, and the "normal" (arithmetic) mean of these is 
317 words per hour, averaging over three texts. 
Nevertheless, it took 7.1 hours to write 1800 words, hence 
the "actual" mean is
 $(500+1000+300)/(2.5+4+0.6)$ $=1800/7.1=254$ words per hour. The high
writing speed of the short text counts for $1/n$ parts in the arithmetic mean, even though the text only contains $300/1,800=1/6$ of the total number of words.

> Since the dependent variable is a fraction (speed, words/hour), 
the harmonic mean is a better measure of central tendency. We firstly convert
the speed (words per time unit) into its inverse 
(see \@ref(eq:harmonicmean), in denominator, within sum sign),
i.e. to *time* per word: 0.005, 0.004, and 0.002 (time units per
word, see footnote[^fn09-3]). We then average these times, to
a mean of 0.00366 hours per word (13.2 seconds per word), and finally we again take the 
inverse of this. The harmonic mean speed of writing is then $1/0.00366=273$
words per hour, closer to the "actual" mean of 254 words per hour.

---

### winsorized mean

The great sensitivity of the normal (arithmetic) mean for 
outliers can be restricted by changing the most extreme observations
into less extreme, more central observations. The
mean of these (partially changed) observations is called the 
*winsorized* mean.

---

> *Example 9.6*: The waiting times from Example 9.1
are ordered as follows:\
1, 1, 1, 2, 2, 2, 2, 3, 3, 5.\
For the 10% winsorized mean, the 10% of smallest observations (by
order) are made to equal the first subsequent larger value, and 
the 10% of largest observations are made to equal the last
preceding smaller value (changed values are italicised here):\
*1*, 1, 1, 2, 2, 2, 2, 3, 3, *3*.\
The winsorized mean over these changed values is
$\overline{x}_w=2$ minutes.

---

### trimmed mean

An even more drastic intervention is to remove the most extreme observations
entirely. The mean of the remaining observations is called 
the *trimmed* mean. For a 10% trim, we remove the lowermost 10% *and* the
uppermost 10% of the observations;
as such, what remains is then only $(1- (2 \times (10/100))\times n$
observations [@Wilcox12].

---

> *Example 9.7*: The waiting times from Example 9.1
are again ordered as follows:\
1, 1, 1, 2, 2, 2, 2, 3, 3, 5.\
For the 10% trimmed mean, the 10% of smallest observations (by order) are removed,  
and likewise the 10% of largest observations 
are removed:\
1, 1, 2, 2, 2, 2, 3, 3.\
The trimmed mean over these $10-(.2)(10)=8$ remaining values here
is $\overline{x}_t=2$ minutes.

---

### comparison of central tendencies 

Figure \@ref(fig:centraltendencies) illustrates the differences between
the various central tendencies, for asymmetrically distributed observations.

```{r centraltendencies, echo=FALSE, fig.cap="Histogram of a variable with positively skewed (asymmetric) frequency distribution, with (1) the median, (2) the 10% trimmed mean, (3) the 10% winsorized mean, and (4) the arithmetic mean, indicated. The observed scores are marked along the horizontal axis."}
# this script was previously named `centrummaten.R`
set.seed(2015) # date of first version
xx <- rlnorm(100, meanlog=3,sdlog=0.8)*5; 
# hist(xx) # xx is indeed lognormally distributed
# require(scales) # for alpha
# require(psych) # for winsor.mean
nbreaks <- 20 # number of bins in histogram
xupper <- 500
maxfreq <- max( hist(xx, breaks=nbreaks*1.5, plot=F)$counts )
hist(xx, col="grey90", xlab="Score", ylab="Frequency", main="", 
     breaks=nbreaks*1.5, xlim=c(0,xupper) )
# rug(xx,side=1,col=scales::alpha("black",.5), quiet=TRUE) # causes error
# 4 arithmetic mean
abline(v=mean(xx),lwd=2,col="red")
axis(side=1,at=mean(xx),labels=F, col="red", lwd=2, line=0, cex=0.75, las=2, padj=1)
text( x=mean(xx), y=maxfreq-3, 
      paste("4 arithmetic mean =",round(mean(xx))), cex=0.75, adj=0 )
# 3 winsorized mean
abline(v=psych::winsor.mean(xx, trim=.10),lwd=2,col="red")
axis(side=1,at=psych::winsor.mean(xx,trim=.10),labels=F, col="red", lwd=2, 
     line=0, cex=0.75, las=2, padj=0.5)
text( x=psych::winsor.mean(xx,trim=.10), y=maxfreq-2, 
      paste("3 winsorized mean =",round(psych::winsor.mean(xx,trim=.10))), cex=0.75, adj=0 )
# 2 trimmed mean
abline(v=mean(xx,trim=.10),lwd=2,col="red")
axis(side=1,at=mean(xx,trim=.10),labels=F, col="red", lwd=2, line=0, cex=0.75, las=2, padj=0.5)
text( x=mean(xx,trim=.10), y=maxfreq-1, 
      paste("2 trimmed mean =",round(mean(xx,trim=.10))), cex=0.75, adj=0 )
# 1 median
abline(v=median(xx),lwd=2,col="red")
axis(side=1,at=median(xx),labels=F, col="red", lwd=2, line=0, cex=0.75, las=2, padj=0)
text( x=median(xx), y=maxfreq, 
      paste("1 median =",round(median(xx))), cex=0.75, adj=0)
```

The arithmetic mean is the most sensitive to extreme values:
the extreme values "pull" very hard at the mean. This influence
of extreme values is tempered in the winsorized mean, and 
tempered even more in the trimmed mean. The higher the
trim factor (the percentage of the observations that have been changed or 
removed), the more the winsorized and trimmed means will look like the median. 
Indeed, with a trim factor of 50%, out of all
the observations, only one (unchanged) observation remains, and that
is the median (check it for yourself). In 
§\@ref(sec:robustefficient) we will look further into the choice
for the appropriate measure for the centre of a distribution. 

## Quartiles and boxplots {#sec:quartiles-and-boxplots}

The distribution of a variable is not only characterised by the 
centre of the distribution but also by the degree of dispersion around
the centre, i.e. how large the difference is between observations and the mean. 
For instance, we not only want to know 
what the mean income is but also how large the *differences* in 
income are.

### Quartiles

Quartiles are a simple and useful measure for this [@Tukey77].
We split the ordered observations into two halves; the dividing line 
between these is the median. We then halve each of these halves again into quarters.
The quartiles are formed by the dividing lines between these
quarters; as such, there are three quartiles. The first quartile $Q_1$ is the lowermost half's median, 
$Q_2$ is the median of all $n$ observations, and the third quartile $Q_3$ is the uppermost half's median. 
Half of the observations (namely the second and third quarters) are
between $Q_1$ and $Q_3$. The distance between $Q_1$ and $Q_3$ is
called the "interquartile range" (IQR). This IQR is a first measure which
can be used for the dispersion of observations with respect to their
central value.

To illustrate, we use the fictive reading test scores shown in Table 
\@ref(tab:cito).

Table: (#tab:cito) The scores of N=10 pupils on three sections of the CITO test,
taken in the final year of primary school in the Netherlands.

      Pupil         Reading   Arithmetic    Geography
  ---------------- --------- ------------- ------------------
         1            18       22               55
         2            32       36               55
         3            45       34               38
         4            25       25               40
         5            27       29               48
         6            23       20               44
         7            29       27               49
         8            26       25               42
         9            20       25               57
         10           25       27               47
      $\sum x$      270       270              475
   $\overline{x}$    27.0      27.0             47.5

---

> Example 9.8:
> The scores for the reading section in
Table \@ref(tab:cito) are
ordered as follows:\
18, 20, 23, 25, 25, 26, 27, 29, 32, 45.\
The median is $Q_2=25.5$ (between the 5th and 6th observation in this 
ranked list). The median of the lowermost half is $Q_1=23$ and that of the 
uppermost half is $Q_3=29$. The interquartile range is 
$\textrm{IQR}=29-23=6$.

---

### Outliers {#sec:outliers}

In the reading scores in Table \@ref(tab:cito), we encounter one extreme value, namely the score 45, which differs markedly from the mean. A marked value like this is referred to as an"outlier". The limit for what we consider to be an outlier
generally lies at $1.5 \times \textrm{IQR}$. If a value is more than 
$1.5 \times \textrm{IQR}$ below $Q_1$ or above $Q_3$, we consider that
observation to be an outlier. Check these observations again (recall the principle of diligence, see §\@ref(sec:integrity-introduction)).

---

> Example 9.9:
> For the aforementioned reading scores in 
Table \@ref(tab:cito), we found
$Q_1=23$, $Q_3=29$, and $\textrm{IQR}=Q_3-Q_1=29-23=6$. The uppermost 
limit value for outliers is 
$Q_3 + 1.5 \times \textrm{IQR} = 29 + 1.5 \times 6 = 29+9 = 38$. The
observation with the score 45 is above this limit value, and is therefore
considered to be an outlier. 

---

### Boxplots {#sec:boxplot}

We can now show the frequency distribution of a variable with five
characteristics, the so-called "five-number summary", namely the minimum value, $Q_1$,
median, $Q_3$, and maximum value. These five characteristics are represented graphically
in a so-called "boxplot", see 
Figure \@ref(fig:cito-boxplot) 
for an example [@Tukey77 §2C].

```{r cito-boxplot, echo=FALSE, fig.cap="Boxplots of the scores of $N=10$ pupils on the Reading and Arithmetic sections of the CITO test (see Table 9.1), with outliers marked as open circles. The observed scores are marked along the vertical axes.", fig.width=4}
require(foreign)
cito <- read.spss("data/cito.sav")
# Columns in `cito.sav` have Dutch names:
# in Dutch: Leerling Lezen Rekenen Wereldorientatie stadplat rek.f
# in English: Pupil Reading Arithmetic World UrbRural Arith.factor
# from script cito.R:
op <- par(mar=c(4,4,1,2)+0.1) # smaller margins
with(cito, 
  boxplot(Lezen, Rekenen, col="grey80", lwd=2, lty=1, ylab="Score", ylim=c(17,45) )
)
axis(side=1, at=c(1,2), labels=c("Reading","Arithmetic") )
# require(plotrix)
plotrix::axis.break(axis=2)
rug(cito$Reading, side=2)
rug(cito$Arithmetic, side=4)
# saved as cito_boxplot.pdf
```

The box spans (approximately) the area from $Q_1$ to $Q_3$, and 
thus spans the central half of the observations. The thicker line in the 
box marks the median. The lines extend to the smallest and largest
values *which are not outliers* [^fn09-4]. The separate outliers
are indicated here with a distinct symbol. 

## Measures of dispersion {#sec:measures-of-dispersion}

### Variance {#sec:variance}

Another way to show the dispersion of observations would be to look at how 
each observation deviates from the mean, thus $(x_i-\overline{x})$. However, if we 
add up all the deviations, they always total zero! After all, the positive and negative
deviations cancel each other out (check that out for yourself in 
Table \@ref(tab:cito)).
Instead of calculating the mean of the deviations themselves, we thus calculate the mean
of the squares of those deviations. Both the positive and negative deviations
result in positive squared deviations. We then calculate the mean of all 
those squared deviations, i.e. we add them up and divide them by 
$(n-1)$, see Footnote[^fn09-5]. We call the result the 
*variance*, indicated by the symbol $s^2$: 
\begin{equation}
  s^2 = \frac{ \sum (x_i - \overline{x})^2 } {n-1}
  (\#eq:variance)
\end{equation}

The numerator of 
this fraction is referred to as the "sum of squared deviations" or
"sum of squares" (SS) and the denominator is referred to as the number of 
"degrees of freedom" of the numerator (d.f.; see
§\@ref(sec:ttest-freedomdegrees)).

Nowadays, we always calculate the variance with a calculator
or computer. 

### standard deviation {#sec:standarddeviation}

To calculate the above variance, we squared the deviations of the
observations. As such, the variance is a quantity which is not expressed
in the original units (e.g. seconds, cm, score),
but in squared units (e.g. $\textrm{s}^2$,
$\textrm{cm}^2$, $\textrm{score}^2$). In order to return to the
original units, we take the square root of the variance. We call the result the
*standard deviation*, indicated by the symbol $s$:

\begin{equation}
  s = \sqrt{s^2} = \sqrt{ \frac{ \sum (x_i - \overline{x})^2 } {n-1} }
  (\#eq:standarddeviation)
\end{equation}

---

> Example 9.10:
> The mean of the previously stated reading scores in 
Table \@ref(tab:cito) is
$27.0$, and the deviations are as follows:\
-9, 5, 18, -2, 0, -4, 2, -1, -7, -2.\
The squared deviations are 81, 25, 324, 4, 0, 16, 4, 1, 49, 4.\
The sum of these squared deviations is 508, and the variance is
$s^2=508/9=56.44$. The standard deviation is the root of the 
variance, thus $s=\sqrt{508/9}=7.5$.

---

The variance and standard deviation can only be used with variables
of the interval or ratio level of measurement. The variance and 
standard deviation can also be based again on the winsorized or trimmed
collection of observations.

We need the standard deviation
(a) when we want to convert the raw 
observations to standard scores (see §\@ref(sec:standardscores) below),
(b) when we want to describe a variable
which is normally distributed (see §\@ref(sec:normaldistribution), and
(c) when we want to test hypotheses with the help of a normally distributed variable (see
§\@ref(sec:ttest-onesample) et seq.). 


### MAD

Besides standard deviation, there is also a robust counterpart
which does not use the mean. This measure is therefore less
sensitive for outliers (more robust), which is sometimes useful. 

For this, we look for the deviation of every observation from
the median (not the mean). We then take the absolute value
of these deviations[^fn09-6] (not the square). Finally, we 
determine again the median of these absolute deviations (not the mean). 
We call the result the "median absolute deviation" (MAD):
\begin{equation}
  \textrm{MAD} = k ~~ Md ( |x_i - Md(x) |)
  (\#eq:MAD)
\end{equation}

We normally use $k=1.4826$ as a constant here; with this scale factor the MAD
usually roughly matches the standard deviation $s$, if $x$ is
normally distributed (§\@ref(sec:normaldistribution)).

---

> Example 9.11:
> The median of the previously mentioned reading scores in 
Table \@ref(tab:cito) is
25.5, and the deviations from the median are as follows:\
-7.5, 6.5, 19.5, -0.5, 1.5, -2.5, 3.5, 0.5, -5.5, -0.5.\
The ordered absolute deviations are\
0.5, 0.5, 0.5, 1.5, *2.5, 3.5*, 5.5, 6.5, 7.5, 19.5.\
The median of these 10 absolute deviations is 3, and
$\textrm{MAD} = 1.4826 \times 3 = 4.4478$. Notice that the MAD
is smaller than the standard deviation, amongst others because the MAD is less sensitive
for the extreme value $x_3=45$.

---

## On significant figures {#sec:significantfigures} 

### Mean and standard deviation {#sec:significantfigures-means}

A mean result is shown in a limited number of significant figures, i.e. 
a limited number of figures, counted from left to right, ignoring the decimal place. 
The mean result's number of significant figures must be equal to the
number of significant figures of the *number of observations* from which the 
mean is calculated. (Other figures in the mean result are not precisely
determined.) The mean result must firstly be rounded to the 
appropriate number of significant figures, before the result is interpreted
further, see 
Table \@ref(tab:signiffiguresmean).

Table: (#tab:signiffiguresmean) The number of significant figures in the reported mean is
equal to the number of significant figures of the number of observations.

       Num.obs.     Num.signif.figures        example mean           reported as
  ----------------- --------------------- -----------------------  ----------------
      $1\dots9$              1                  21/8 = 2.625              3
     $10\dots99$             2                 57/21 = 2.714286          2.7
    $100\dots999$            3               317/120 = 2.641667          2.64
   $1000\dots9999$           4             3179/1234 = 2.576175         2.576

The number of significant figures in the reported standard deviation is
the same as in the mean, in accordance with 
Table \@ref(tab:signiffiguresmean).


#### Background

Let us assume that I have measured the distance from my house to my work 
along a fixed route a number of times. The mean of those measurements
supposedly amounts to $2.954321$ km. By reporting the mean with 7 figures,
I am suggesting here that I know precisely that the distance is $2954321$ millimetres,
and at most $1$ mm more or less: 
the last figure is estimated or rounded off. The number of significant figures
(in this example 7) indicates the degree of precision. In this example, the 
suggested precision of 1 mm is clearly
wrong, amongst other reasons because the start point and end point cannot be determined
within a millimetre. It is thus usual to report the mean of the measured
distance with a number of significant figures which indicates the precision
of those measurements and of the mean, e.g. 
$3.0$ km (by car or bike) of $2.95$ km (by foot).

The same line of thought is applicable when measuring a characteristic
by means of a survey question. With $n=15$ respondents, the average score
might be $43/15 \approx 2.86667$. However, the precision
in this example is not as good as this decimal number suggests. In fact,
here one deviant answer already brings about a deviation of 
 $\pm0.06667$ in the mean. Besides, a mean score 
 is always the result of a division operation, and 
"[for] quantities created from measured quantities by multiplication and division, the calculated result should have as many significant figures as the measured number with the least number of significant figures" [^fn09-7]. 
 In this example, the mean's numerator ($43$) and its denominator ($15$) both consist
 of 2 significant figures. The mean score should be reported as $2.9$ points, with only
 one figure after the decimal point. 

### Percentages

A percentage is a fraction, multiplied by $100$.
Use and report a rounded off percentage (i.e. two significant 
figures) only if the fraction's numerator is larger
than 100 (observations, instances). If the numerator is smaller than 100 
(observations, instances), then percentages are misleading,
see Table \@ref(tab:signiffigurepercentage).

Table: (#tab:signiffigurepercentage) The number of significant figures in the reported
proportion (or percentage) is related to the number of significant figures of the number
of observations in the denominator of the fraction. 

   num.obs.(denominator)   num.signif.figures    example fraction       report as
  ---------------------- --------------------- --------------------  ----------------
      $1\dots9$              1                     3/8 = 0.4            3/8
     $10\dots99$             2                   21/57 = 0.36          21/57
    $100\dots999$            3                 120/317 = 0.378          38\%
   $1000\dots9999$           4               1234/3179 = 0.3882         38.8\%


#### Background

The rules for percentages arise from those in 
§\@ref(sec:significantfigures-means) applied to division operations.
If the denominator is larger than 100, the percentage (with two significant
figures) is the result of a scaling "down" (from a denominator
larger than 100 to a denominator of precisely 100 percentage points).
The percentage scale is less precise than the original 
ratio; the percentages are rounded off to two significant figures;
the percentage's last significant figure is thus secured.

However, if the denominator is smaller than 100, then the percentage (with
two significant figures) is the result of a "scaling upwards" (from a denominator
smaller than 100 to a denominator of exactly 100 percentage points). The percentage
scale then suggests a
pseudo-precision which was not present in the original fraction,
and the precision of the percentage scale is false. As such, if the denominator is 
smaller than 100, percentages are misleading. 

---

> Example 9.12:
> In a course of 29 students, 23 students passed. In this case, we often
speak of a course return of $23/29=$ 79%. However, a rendering as a 
percentage is misleading in this case. To see this, let us look
at the 6 students who failed. You can reason that the number of 6 failed
students has a rounding error of $1/2$ student(s); when converted to the
percentage scale this rounding error is also thereby increased so that
the percentages are less precise than the whole percentages (2
significant figures) suggest. Or put otherwise: the number of 6 
failed students (i.e. a number with one significant figure) means we have
to render the proportion with only one significant figure, and thus not 
as a percentage. It is preferable to report the proportion itself ($23/29$), or
the "odds" ($23/6=4$) rounded off to the correct number of significant
figures[^fn09-8].

---

On the basis of the same considerations, a percentage with one decimal 
place (i.e. with three significant figures, e.g. "36.1%") is only
meaningful if the ratio or fraction's denominator is larger than 1000.

---

> Example 9.13:
> In 2013, 154 students began a two-year research master's degree. After 
2 years, 69 of them had graduated. The nominal return for this cohort is
thus $69/154=$ 0.448052, which should be rounded off and reported as 45%
(not as 44.81%).

---

## Making choices {#sec:robustefficient} 

You can describe the distribution of a variable in various 
manners. 
If variable $X$ is measured on the interval or ratio level of measurement,
always begin with a histogram (§\@ref(sec:histograms))
and a boxplot (§\@ref(sec:boxplot)). 

The centre measures and dispersion measures can be arranged
as in Table \@ref(tab:centredispersionmeasures). 

Table: (#tab:centredispersionmeasures) Overview of discussed centre measures and dispersion measures. For assumptions abbreviated to *(a & b & c)*, see text below table. 

  Distribution           Centre measure               Dispersion measure
  ------------------- ---------------------------  ----------------------
  all                 median                       quartiles, IQR, MAD
  ...                 trimmed or wins. mean        trimmed or wins.std.dev.
  (a & b & c)         mean                         standard deviation                     

The most **robust** measures are
at the top (median, quartiles, IQR, MAD). These measures are robust:
they are less sensitive for outliers or for potential *a*symmetry in the
frequency distribution, as the examples in this chapter show.

The most **efficient** measures are at the bottom of 
Table \@ref(tab:centredispersionmeasures): mean and standard deviation.
These measures are efficient: they represent the centre and the dispersion
the best, they have themselves the smallest standard deviation, and they need 
the (relatively) smallest number of observations for this. The other measures 
occupy a between position: the trimmed measures are somewhat more robust,
and the winsorized measures somewhat more efficient. 

However, the most efficient measures also demand the furthest reaching
assumptions (and the most robust measures demand the fewest assumptions).
These efficient measures are only meaningful if the distribution of $X$
satisfies three assumptions: (a) the distribution is more or less
symmetrical, i.e. the left and right halves of the histogram and
the uppermost and lowermost halves of the boxplot look like each other's
mirror image, (b) the distribution is unimodal, i.e. the distribution has 
a unique mode, and (c) the distribution contains no or hardly any outliers.
Inspect these assumptions in the histogram and the boxplot of $X$. If one
of these assumptions is not satisfied, then it is better to use
more robust measures to describe the distribution.

## Standard scores {#sec:standardscores}

It can sometimes be useful to compare scores which are measured on 
different scales. Example: Jan got an 8 as his final grade for       
maths at Dutch secondary school, and his IQ is 136. Is the deviation of Jan with
respect to the mean as large on both of the scales? To answer a question
like this, we have to express the scores of the two variables on the 
same measurement scale. We do so by converting the raw scores to 
standard scores, or z-scores. For this, we take the deviation of every
score with respect to the mean, and we divide the deviation by the 
standard deviation: 

\begin{equation}
    z_i = \frac{(x_i-\overline{x})}{s_x}
   (\#eq:zscores)
\end{equation}

The standard score or z-score
thus represents the distance of the $i$'the observation to the mean
of $x$, expressed in units of standard deviation. For a 
standard score of $z=-1$, the observed score is precisely $1 \times s$
below the average $\overline{x}$. For a standard score of $z=+2$,
then the observed score is precisely $2 \times s$ above the 
mean[^fn09-9].

Z-scores are also useful for comparing two variables which
are in fact measured on the same scale (for example, a scale of 
$1 \dots 100$), but which nevertheless have different means and/or
standard deviations, like the scores in 
Tabel \@ref(tab:cito).
In Chapter \@ref(ch-probability-distributions), we will work more with z-scores.

The standard score or z-score has two useful characteristics which you
should remember. Firstly, the mean is always equal to zero:
$\overline{z}=0$, and, secondly, the standard deviation is equal to 1:
$s_z = 1$. (These characteristics follow from the definition in 
formula \@ref(eq:zscores); we omit the mathematical proof here.) Thus,
transformation from a collection of observations to 
standard scores or z-scores always yields a distribution with 
a mean of zero and a standard deviation of one. Do remember that
this transformation to standard scores is only meaningful, provided that and 
to the extent that the mean and the standard deviation are also meaningful measures
to describe the distribution of $x$ (see §\@ref(sec:robustefficient)).

## SPSS

For **histogram, percentiles and boxplot**:
```
Analyze > Descriptive Statistics > Explore...
```

Select variable (drag to Variable(s) panel)\
Choose `Plots`, tick: `Histogram`, and confirm with `Continue`\
Choose `Options`, tick: `Percentiles`, and confirm with `Continue` and
afterwards with `OK`.\
The output comprises descriptive statistics and histogram and
boxplot.

For **descriptive characteristic values**:\ 
```
Analyze > Descriptive Statistics > Descriptives...
```

Select variable (drag to Variable(s) panel)\
Choose `Options`; tick:
`Mean, Sum, Std.deviation, Variance, Minimum, Maximum`, and confirm with
`Continue` and afterwards with `OK`.\
The output comprises the requested statistical characteristics of the variable's distribution.\

For **median**:\
```
Analyze > Compare Means > Means...
```

Select variable (drag to Variable(s) panel)\
Choose `Options`; tick:
`Mean, Number of cases, Standard deviation, Variance, Minimum, Maximum`
and also `Median`, and confirm with `Continue` and afterwards with `OK`.\
The output comprises the requested statistical characteristics of the variable's distribution.\

Calculate and save **Standard scores** in a new column:\
```
Analyze > Descriptive Statistics > Descriptives...
```

Select variables (drag to Variable(s) panel)\
Tick: `Save standardized values as variables` and confirm with `OK`.\
The new variable(s) with z-scores are added as new 
column(s) to the data file.\

## JASP

For **histogram and boxplot**:\
From the top bar, choose 
```
Descriptives
```
Select the variable(s) to summarize and place them into the field "Variables". Open the "Plots" field and check the option `Distribution plots` (under the heading "Basic plots") to obtain a histogram or bar chart (depending on the measurement level). 
If desired, check the option `Boxplots` (with `Boxplot element`) under the heading "Customizable plots".

For **summary numbers and quantiles**:\
From the top bar, choose 
```
Descriptives
```
Select the variable(s) to summarize and place them into the field "Variables". Open the "Statistics" field and check the option `Quartiles` (under the heading "Percentile values") to obtain quartiles. 
Other quantiles are also possible: check the option `Cut points for:` and enter a number, e.g. `6` will produce sextiles. If you want to know a specific percentile, e.g. the 17th percentile, check the option `Percentiles:` and enter the desired percentile value (here `17`). 

For summaries of central tendency and dispersion, check the options (under the heading "Central Tendency") for `Mean`, `Median`, `Mode` en `Sum` , as well as those (under the heading "Dispersion") for `Variance`, `Std.deviation`, `MAD Robust` (with constant value fixed at 1; see Eq.\@ref(eq:MAD) above), `IQR`, `Minimum` and `Maximum` . This will produce a summary table showing the requested descriptive statistics of the variable(s). 

In JASP, a column of **standard scores** can be created by first creating a new variable (column) and subsequently filling that column with standard scores.  

To create a new variable, click on the **+** button to the right of the last column name in the data tab. A "Create Computed Column" panel appears, where you can enter a name for the new variable, e.g. `Lezen_Z`. You can also choose between `R` and a hand-shaped pointer. These are the two options in JASP to define formulas with which the new (empty) variable is filled: using `R` code, or manually using JASP. The paragraphs below explain how standard scores can be computed using these two options. Finally, you can check which measurement level the new variable should be (see Chapter \@ref(ch-levelsofmeasurement)). For standard scores, you can leave this at `Scale`. Next, click on `Create Column` to create the new variable. The new variable (empty column) appears as the rightmost variable in the data set. 

If the **computed with R code** option is chosen to define the new variable, a field containing the text "Enter your R code here" appears above the data. Here you can enter R code (see below) that produces standardized values. \
This snippet of R code produces standard scores of the variable `Lezen`: 
```
((Lezen - mean(Lezen)) / sd(Lezen))
```
Enter this R code, fill out the other fields, 
and click on the button `Compute column` below the work sheet to fill the empty variable. 

*Note that in JASP, applying mathematics to variable values is only possible if the measurement levels of all variables involved are set to 'Scale'.*

If the **hand pointer** or **drag-and-drop** option is chosen to define the new variable, a work sheet will appear above the data. To the left of the work sheet are the variables, above it are math symbols, and to the right of the work sheet are several functions. From those functions you can pick the ones required to compute standard scores. If something goes wrong, items on the work sheet can be erased by dragging them to the trash bin on the lower right bottom. After you have completed the specification on the work sheet, then click on the button `Compute column` under the work sheet, to fill the new variable with the generated numbers.\
Drag the variable to convert into the empty sheet. Pick function `mean(y)` from the right. 
Drag the variable to convert to "values". Pick $\div$ from the math symbols at the top. 
Pick the function $\sigma$_y from the right and drag it to the denominator part of the fraction (below the fraction bar). For this function too, drag the variable to convert to "values". 
Eventually the definition of standard scores (here for a variable named `Lezen`) should look like this: \[\frac{(Lezen-mean(Lezen))}{\sigma (Lezen)}\] \
Fill out the other fields, and then click on the button `Compute column` below the work sheet, to fill the empty variable with the newly computed standard scores.  

*Note that in JASP, applying mathematics to variable values is only possible if the measurement levels of all variables involved are set to 'Scale'.*

If you have made a mistake or want to adjust the code of the new variable, you can always return to this 'Computed Column' field by clicking on the formula icon $f_x$ next to the variable name, or by clicking on the variable name.

## R

For **quartiles and boxplot** like Figure \@ref(fig:cito-boxplot), we use the commands  `fivenum`, `quantile`, and `boxplot`: 
```{r cito-summary-boxplot}
require(foreign) # for foreign::read.spss
cito <- read.spss("data/cito.sav")
# Columns in `cito.sav` have Dutch names:
# in Dutch: Leerling Lezen Rekenen Wereldorientatie stadplat rek.f
# in English: Pupil Reading Arithmetic Geography UrbRural Arith.factor
fivenum(cito$Lezen) # minimum, Q1, median, Q3, maximum
quantile(cito$Lezen, c( 1/4, 3/4 ) ) # Q1 and Q3, calculated differently 
op <- par(mar=c(4,4,1,2)+0.1) # smaller margins
with(cito, 
  boxplot(Lezen, Rekenen, col="grey80", lwd=2, lty=1, ylab="Score", ylim=c(17,45) )
)
axis(side=1, at=c(1,2), labels=c("Reading","Arithmetic") )
plotrix::axis.break(axis=2) # break in left Y-aixs
rug(cito$Lezen, side=2) # markings on left Y-axis
rug(cito$rekenen, side=4) # markings on right Y-axis
```

Many **central tendencies** are pre-programmed as functions in R:
```{r cito-centrummaten}
mean(cito$Lezen) # mean
psych::winsor.mean(cito$Lezen, trim=.1) # winsorized mean, from psych package
mean(cito$Lezen, trim=.1) # trimmed mean
median(cito$Lezen)   # median
```

Various **dispersion measures** are also pre-programmed:
```{r cito-spreidingsmaten}
var(cito$Lezen)  # variance
sd(cito$Lezen)   # standard deviation, sd(x) = sqrt(var(x))
mad(cito$Lezen)  # MAD
```

In contrast, we have to calculate **standard scores** ourselves, and save them ourselves as a
new variable, called here `zReading` (note the parentheses in the first command line):
```{r cito-zscores}
# standardized (or z) reading scores
zReading <- (cito$Lezen-mean(cito$Lezen)) / sd(cito$Lezen) 
head(zReading) # first few observations of variable zReading
```



[^fn09-3]: This is comparable with sports like rowing, swimming, cycling, ice skating, etc., where 
the time over an agreed distance is measured and compared, rather than the speed over an 
agreed time. 

[^fn09-4]: In a classic boxplot, the lines extend to the minimum and maximum [@Tukey77] and
outliers are not indicated separately.

[^fn09-5]: We divide by $n-1$ and not by $n$, to get a better estimation of the dispersion in the
*population*. In this way, we take into account the fact that we are using a characteristic of the
sample (namely the mean) to determine the dispersion. If you are only interested in the 
dispersion in your *sample* of observations, and not in the population, divide it by $n$.

[^fn09-6]: Positive deviations remain unchanged, negative deviations are reversed.

[^fn09-7]: <https://en.wikipedia.org/wiki/Significant_figures>

[^fn09-8]: These "odds" indicate that there are 23 successful students to 6 failed students, i.e., rounded up, 4 successful students for every failed student.

[^fn09-9]: Check: $z = +2 = \frac{(x_i-\overline{x})}{s_x}$, thus $2 s = (x_i-\overline{x})$, thus $x_i = \overline{x}+2s$.