Editing 2118: Normal Distribution

{{comic
| number    = 2118
| date      = March 1, 2019
| title     = Normal Distribution
| image     = normal_distribution.png
| titletext = It's the NORMAL distribution, not the TANGENT distribution.
}}

==Explanation==
This is another comic on [[:Category:How to annoy|How to annoy]] people, particularly targeting statisticians in this instance.

[[File:Standard_deviation_diagram.svg|thumb|{{w|Normal distribution}}s and the intervals of the standard deviation are a topic commonly seen in introductory statistics.  Randall's chart is similar, but his lines are perpendicular.]]
In statistics, a {{w|Probability distribution|distribution}} is a representation that can be understood in terms of how much of a sample is expected to fall into either discrete bins or between particular ranges of values.  For example, if you wanted to represent an age distribution using bins of ten years (0-9, 10-19, etc.), you could produce a bar chart, one bar for each bin, where the height of each bar represents a count of the portion of the sample matching that bin. To turn that bar chart into a distribution, you'd get infinitely many people (technically: a number N which tends to infinity), put them into age bins that are infinitely narrow (technically: bins whose size is O(1/sqrt(N))), and then divide each bin count by the total count so that the whole thing added up to 1. It is common to ask how much of the distribution lies between two vertical lines; that would correspond to asking what percent of people are expected to fall between two ages.

Many statistical samplings form a pattern called a "{{w|normal distribution}}".  A theoretically perfect normal distribution would have an infinite sample size and infinitely small bins.  That would produce a bar chart matching the shape of the curve in the comic.

The area between two vertical lines of the distribution represents the probability that a randomly selected X-value is between the X-values of the lines. Randall instead finds the area between two ''horizontal'' lines, which is mathematically meaningless, because the Y-axis of a probability distribution is typically taken to represent {{w|absolute magnitude|magnitude}} as a fraction of unity. In the age-distribution analogy above, two points with the same X-value could be understood to represent two people with the same age; but two points with the same Y-value cannot easily be understood in terms of the analogy. The items "represented" by the magnitude at any given horizontal position are indistinguishable, unordered, and interchangeable; the fact that two items happen to fall at the same position on the Y-axis doesn't mean they have anything in common.

In short, Randall has invented a new probability distribution, which the title text humorously implies should be called the ''tangent distribution''. This distribution is defined as follows: consider the area between the curve in the comic and the horizontal axis, and consider a random point (X, Y) uniformly distributed in that region.  Then X has the normal distribution and Y has the tangent distribution.  Areas between vertical lines in the comic give probabilities concerning X, and areas between horizontal lines in the comic give probabilities concerning Y.  The comic correctly indicates that if we let ''R'' be the interval of Y values that is 52.682% of the range of Y centered at the midpoint of the range, then any randomly selected Y value has probability 1/2 of falling inside interval ''R''.

This distribution has never been discussed before, and has no known application. Moreover, the distribution of Y is not symmetric: while 50% of Y values fall inside interval ''R'', 41% fall below ''R'' and only 9% fall above ''R''. So the single piece of information in the comic is not a good way to describe this distribution! We do use such intervals for the normal distribution because the normal distribution is symmetric, and the center of symmetry is the mean, median, and mode. (However, it would be just about as ridiculous to observe that 50% of the X values in a standard normal distribution fall between the vertical lines X=-0.2 and X=1.41.)

The title text refers to the notion of {{w|Normal (geometry)|normals}} and {{w|tangent}}s in geometry. Given a 2D curve or 3D surface, a line which points perpendicularly outward from a point on the curve or surface (making a 90-degree angle with the curve) is said to be ''normal'' to the curve, while a line which just grazes the curve, being exactly parallel to the curve at the point of contact, is said to be ''tangent'' to the curve at that point. The joke is that this geometrical notion of ''normal'' is completely unrelated to the statistical ''normal distribution''. Randall observes that if you take a geometric normal and rotate it 90 degrees, you produce a tangent; thus, if you take the ''normal'' distribution and rotate it by 90 degrees, you must get something called the "''tangent'' distribution." Saying this to a statistician would only annoy the statistician further.

This is annoying to a statistician not only because the terms ''normal'' and ''tangent'' come from differential geometry and have no established meaning in probability theory.  Even the word ''perpendicular'' has no established meaning in probability theory.  Of course, the x and y coordinates in the comic are perpendicular (orthogonal) coordinates, but X and Y are not "perpendicular" or "orthogonal" random variables.  Even if we give "perpendicular" or "orthogonal" a probabilistic meaning, and the most obvious such meaning is either {{w|Independence (probability theory)|independent}}, which even uses a symbol related to the geometric symbol for perpendicularity, or {{w|Uncorrelatedness (probability theory)|uncorrelated}}, which makes X and Y orthogonal vectors in the Hilbert space of random variables that are square integrable with respect to Lebesgue measure, X and Y are not perpendicular in either of these senses.

So the more probability and statistics you know, the more annoying this comic becomes.  It is not just about confusing novices.

==Transcript==
:[A bell curve of a normal distribution, with the area between two horizontal lines shaded.]

:[The center of the chart is marked between the two lines:]
:Midpoint

:[The distance between the lines is marked to the right of the midpoint, with the label:]
:52.7%

:[A label on the outside of the graph, describing the distance between the two lines:]
:"Remember, 50% of the distribution falls between these two lines!"

:[Caption below the panel:]
:How to annoy a statistician


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