12 It started with a line

Last modified on 08. January 2026 at 13:50:54

“It started with a ~~kiss~~ line; […] How could I resist?” — It Started With A Kiss by Hot Chocolate

In statistics, everything is about a line. This is also a lie. More precisely, in parametric statistics, everything is about a line. In a later chapter, we will also explore nonparametric statistical analysis, which is different but somewhat outdated. We have better tools on your shelves to solve problems that were previously solved using nonparametric approaches.

12.1 Geometry in statistics

Before we go any deeper into statistics, we need to either learn or review some geometry. We need to understand what a line is and how this concept differs in statistics. I will then explain the geometry of a square and the square root. You should be familiar with what both look like in a visualisation. We will revisit such geometries repeatedly. Ultimately, we will bring them together. A line can be expressed as a square. This is a key concept in statistics.

12.1.1 The square and the root

“What you see, well, you might not know; You get the feelin’, comin’ after the glow” — The joker & the thief by Wolfmother

If you have never studied statistics before, or have only covered the basics, you may be surprised to learn that everything in statistics revolves around squares. However, this is only true for parametric statistics, which is where we want to start. In the following figure on the left side, you can see a square. A square has the same side length $a$ on all sides, as well as a square area $a^2$. If we want to find the side length of a square and only know its area, we can take the square root $\sqrt{\phantom{a}}$ shown on the right side. This is a simple geometry, but an important one to remember. We will need to refer to squares repeatedly throughout this book.

Figure 12.1: Visualisation of the relationship between a square and its square root. **(A)** The square consists of two sides of equal length. Therefore, the area of a square is the square of its side. **(B)** The square root of the area of a square is the length of its side.

Now, let’s assume that there are lots of squares. Where these squares originate is not currently important. We have these squares and want to summarise them. A standard statistical procedure is to calculate the area of each square and then add up the areas. We call this new square the sum of squares. The sum of the squares is used so often that it is abbreviated to SS.

The following figure shows four squares labelled $a_1, ..., a_4$ on the left. Each square has its own area, labelled $a_1^2, ..., a_4^2$. In order to work with these squares, we will now add up their areas. The sum of the squares is shown on the right. The square root describes the length of the side of the sum of squares. We will use the sum of squares and the square root of the sum of squares repeatedly in our further analysis. They may not appear under the same names, but we will see the squares again.

Figure 12.2: Visualisation of the calculation of the sum of squares. **(A)** The area of each square is determined. **(B)** All the areas are then combined to form one large square and then called sum of squares. The square root can then be used for further statistical analysis.

So far, this has not been very complicated. You might already remember the geometry from seeing a square. The tricky part is that we will work with mathematical expressions that are essentially squares, but don’t look like them. Now let’s combine a line with a square. Or answer the question of why a line can be expressed by a square.

12.1.2 The line as a square

Figure 12.3: Visualisation of the dependency between a line and a square. **(A)** A line through two points, also known as a Euclidean line. Most people think of such a line first when they think about lines through points. This is mathematically correct, but not in statistics. **(B)** The distance between two points, also known as a line, is represented as a square.

12.1.3 The hunter and the rabbit

“Statistics is: When the hunter misses the rabbit once to the left and once to the right, on average the rabbit is dead.” — Mike Krüger, German comedian

I first came across the joke on Tweedback, a tool for anonymous feedback in lectures. One student had written the joke, which intrigued me. At first, I could not understand why. Then I understood the deeper meaning. In statistics, it’s all about lines. Or lines through points. A gunshot is a line. A mean is a line. Single shots deviate from the average shot, as do single observations from the mean. If we measure the distance from the mean to each observation, we will get positive and negative values. Ultimately, these values would sum to zero. I don’t believe for a moment that the comedian truly understands the depth of the joke from a statistical perspective.

In the following Figure 12.4, I have illustrated the example with some numbers. The rabbit is sitting at position 5m. The hunter shoots at $3m$, which is $-2m$ to the left. The second shot goes to the right, at $7m$, which is a deviation of $+2m$. We can now calculate the average of the two shots. This is equal to $5m$. The hunter hits. When we calculate the deviation, we get zero: $(-2) + (+2) = 0$. Therefore, the whole hunt of two shots has no deviation at all. This is wrong. Our model of reality tells us something completely different compared to what our hunter experienced. Therefore, we need a better model.

Figure 12.4: Illustration of the joke about the hunter and the rabbit. If the hunter shoots past the rabbit on the left and on the right, then on average the hunter will hit the rabbit. The deviation is zero. The joke illustrates the problem of measuring a statistical deviation when positive and negative values can occur.

Figure 12.5: Visualisation of the deviation from the hunter and rabbit example. **(A)** Each deviation is squared. Therefore, no sign is left. **(B)** The deviations are added up. The square root of the sum of squares has the same unit as the original measurement. The example was built in such a way that the root is the standard deviation.

12.2 Normal distributed data

“Can’t explain all the feelings that you’re making me feel; I believe in a thing called ~~love~~ normality.” — I Believe in a Thing Called Love by The Darkness

12.3 Glossary

term: what does it mean.

12.4 The meaning of “Models of Reality” in this chapter.

itemize with max. 5-6 words

12.5 Summary

References

```{r echo = FALSE, warning = FALSE, message = FALSE} source("init.R") source("images/part_5/part_5_model.R") ``` # It started with a line {#sec-started-line} *Last modified on `r format(fs::file_info("chapter-52-it-started-with-a-line.qmd")$modification_time, '%d. %B %Y at %H:%M:%S')`* > *"It started with a ~~kiss~~ line; [...] How could I resist?" --- It Started With A Kiss by Hot Chocolate* In statistics, everything is about a line. This is also a lie. More precisely, in parametric statistics, everything is about a line. In a later chapter, we will also explore nonparametric statistical analysis, which is different but somewhat outdated. We have better tools on your shelves to solve problems that were previously solved using nonparametric approaches. ## Geometry in statistics Before we go any deeper into statistics, we need to either learn or review some geometry. We need to understand what a line is and how this concept differs in statistics. I will then explain the geometry of a square and the square root. You should be familiar with what both look like in a visualisation. We will revisit such geometries repeatedly. Ultimately, we will bring them together. A line can be expressed as a square. This is a key concept in statistics. ### The square and the root > *"What you see, well, you might not know; You get the feelin', comin' after the glow" --- The joker & the thief by Wolfmother* If you have never studied statistics before, or have only covered the basics, you may be surprised to learn that everything in statistics revolves around squares. However, this is only true for parametric statistics, which is where we want to start. In the following figure on the left side, you can see a square. A square has the same side length $a$ on all sides, as well as a square area $a^2$. If we want to find the side length of a square and only know its area, we can take the square root $\sqrt{\phantom{a}}$ shown on the right side. This is a simple geometry, but an important one to remember. We will need to refer to squares repeatedly throughout this book. ```{r} #| message: false #| echo: false #| warning: false #| fig-align: center #| fig-height: 3.5 #| fig-width: 7 #| fig-cap: "Visualisation of the relationship between a square and its square root. **(A)** The square consists of two sides of equal length. Therefore, the area of a square is the square of its side. **(B)** The square root of the area of a square is the length of its side." #| label: fig-square-and-root p1_square + p2_square_root + plot_layout(ncol = 2) + plot_annotation(tag_levels = 'A', tag_prefix = '(', tag_suffix = ')') & theme(plot.tag = element_text(size = 16, face = "bold")) ``` Now, let's assume that there are lots of squares. Where these squares originate is not currently important. We have these squares and want to summarise them. A standard statistical procedure is to calculate the area of each square and then add up the areas. We call this new square the sum of squares. The sum of the squares is used so often that it is abbreviated to SS. The following figure shows four squares labelled $a_1, ..., a_4$ on the left. Each square has its own area, labelled $a_1^2, ..., a_4^2$. In order to work with these squares, we will now add up their areas. The sum of the squares is shown on the right. The square root describes the length of the side of the sum of squares. We will use the sum of squares and the square root of the sum of squares repeatedly in our further analysis. They may not appear under the same names, but we will see the squares again. ```{r} #| message: false #| echo: false #| warning: false #| fig-align: center #| fig-height: 3.5 #| fig-width: 7 #| fig-cap: "Visualisation of the calculation of the sum of squares. **(A)** The area of each square is determined. **(B)** All the areas are then combined to form one large square and then called sum of squares. The square root can then be used for further statistical analysis." #| label: fig-square-sum p1_square_many + p2_square_sum + plot_layout(ncol = 2) + plot_annotation(tag_levels = 'A', tag_prefix = '(', tag_suffix = ')') & theme(plot.tag = element_text(size = 16, face = "bold")) ``` So far, this has not been very complicated. You might already remember the geometry from seeing a square. The tricky part is that we will work with mathematical expressions that are essentially squares, but don't look like them. Now let's combine a line with a square. Or answer the question of why a line can be expressed by a square. ### The line as a square ```{r} #| message: false #| echo: false #| warning: false #| fig-align: center #| fig-height: 3.5 #| fig-width: 7 #| fig-cap: "Visualisation of the dependency between a line and a square. **(A)** A line through two points, also known as a Euclidean line. Most people think of such a line first when they think about lines through points. This is mathematically correct, but not in statistics. **(B)** The distance between two points, also known as a line, is represented as a square." #| label: fig-euclidiean-line p1_euclidean_line + p2_line_squared + plot_layout(ncol = 2) + plot_annotation(tag_levels = 'A', tag_prefix = '(', tag_suffix = ')') & theme(plot.tag = element_text(size = 16, face = "bold")) ``` ### The hunter and the rabbit > *"Statistics is: When the hunter misses the rabbit once to the left and once to the right, on average the rabbit is dead." --- Mike Krüger, German comedian* I first came across the joke on Tweedback, a tool for anonymous feedback in lectures. One student had written the joke, which intrigued me. At first, I could not understand why. Then I understood the deeper meaning. In statistics, it's all about lines. Or lines through points. A gunshot is a line. A mean is a line. Single shots deviate from the average shot, as do single observations from the mean. If we measure the distance from the mean to each observation, we will get positive and negative values. Ultimately, these values would sum to zero. I don't believe for a moment that the comedian truly understands the depth of the joke from a statistical perspective. In the following @fig-rabbit-shot, I have illustrated the example with some numbers. The rabbit is sitting at position 5m. The hunter shoots at $3m$, which is $-2m$ to the left. The second shot goes to the right, at $7m$, which is a deviation of $+2m$. We can now calculate the average of the two shots. This is equal to $5m$. The hunter hits. When we calculate the deviation, we get zero: $(-2) + (+2) = 0$. Therefore, the whole hunt of two shots has no deviation at all. This is wrong. Our model of reality tells us something completely different compared to what our hunter experienced. Therefore, we need a better model. ```{r} #| message: false #| echo: false #| warning: false #| fig-align: center #| fig-height: 2.25 #| fig-width: 7 #| fig-cap: "Illustration of the joke about the hunter and the rabbit. If the hunter shoots past the rabbit on the left and on the right, then on average the hunter will hit the rabbit. The deviation is zero. The joke illustrates the problem of measuring a statistical deviation when positive and negative values can occur." #| label: fig-rabbit-shot p_rabbit_joke ``` ```{r} #| message: false #| echo: false #| warning: false #| fig-align: center #| fig-height: 3.5 #| fig-width: 7 #| fig-cap: "Visualisation of the deviation from the hunter and rabbit example. **(A)** Each deviation is squared. Therefore, no sign is left. **(B)** The deviations are added up. The square root of the sum of squares has the same unit as the original measurement. The example was built in such a way that the root is the standard deviation." #| label: fig-rabbit-deviation p1_square_many_rabbit + p2_square_sum_rabbit + plot_layout(ncol = 2) + plot_annotation(tag_levels = 'A', tag_prefix = '(', tag_suffix = ')') & theme(plot.tag = element_text(size = 16, face = "bold")) ``` ## Normal distributed data > *"Can't explain all the feelings that you're making me feel; I believe in a thing called ~~love~~ normality." --- I Believe in a Thing Called Love by The Darkness* ## Glossary term : what does it mean. ## The meaning of "Models of Reality" in this chapter. - itemize with max. 5-6 words ## Summary ## References {.unnumbered}