Math

The proper way of using mathematics is to precisely define what do we mean and how do we measure things.

We use basic statistics to characterize and measure, while discarding all the underlying abstract theories.

An averages are just information-losing aggregate measurement, variances are just ranges, etc.

Archery

The best application of basic statistics and frequency-based probabilities is when one tries to analyze the result of some archery shooting events, either personal or a whole competition.

Just by observing the marks on a target one could form some informed opinions, based on the patterns the marks have been formed.

the average distance from the center.
the “density” of marks around the center
the percentage of centered vs scattered

These are non-abstract, objective metrics and there are no bullshit assumptions about independence, on the contrary, each shot depends on multiple factors, including the technique, the emotional state, the arrows and the bow.

For a competition it is easy to infer that a shooter with the least spread of marks (and no sudden outliers) has better chance to win, and what we measure is his skills, self-control and equipment using a “natural” consistency metrics (less volatility).

Basic math

Non-bullshit statistics is just distances (differences) and their aggregates (averages).

Metrics are mostly ratios, which is counting (measuring) one quantity in terms (in units) of another.

Derivatives are exactly such ratios. Setting \(dx = 1\) makes a \(dy\) to be just a difference (a distance).

This corresponds to superimposing a ruler (or a scale) of equally spaced notches and to counting them. Usually a ruler is an abstract time divided into equal, arbitrary length units.

Or systems should be modeled as a discrete, time invariant sequences so we will not have the notion of an angle, just of a length or a distance.

Percentages is a general technique of dividing a quantity into 100 equal parts and then selecting one or more of these units (percentage points).

Basic statistics

The fundamental principle is that statistics begins with averages and thus loses a lot of information by just discarding it or reducing to a single number. When you talk about an average price you have discarded all the surges and drops through the day, their speed, etc.,

The Central tendency

This is conceptually equivalent to measuring height and spread of a pile. Or characterizing all the marks (the sample space) on a target.

Notice that everything is just an average or a distance from it (difference). Distances are universal so there is no abstract bullshit so far.

Mean (an average)

Reduction to a single number - an average (finding a center). \(\mu = \frac{\sum x}{n}\)

Median

A value that makes a 50/50 split. No information loss. Sort a sample (to make a distribution) and

take the middle index if length is odd
an average of two middle elements if length is even

Mode

The most occurring value (has a highest frequency). No information loss. Do a bucket-sort (same as frequency) and take the fullest bucket.

Variability

Distances from the center - spread or dispersion of values.

Range

This is just max(xs) - min(xs), which is a difference. It shows how spread out the values are (on a number line).

Variance (how much it varies)

An average /distance from the mean (how close to the center). It characterizes the shape of a pile - how it spreads out. This is an average too (of distances) - a single number. \(var = \frac{\sum(x-\mu)^{2}}{n}\)

Standard deviation

The unit of measurement of a variability. How many of such units away (distant). \(\sigma = \sqrt \frac{\sum(x-\mu)^{2}}{n}\)

z-score

This is how far is a particular value is from the mean. In terms of units of variability. \(z = \frac{x - \mu}{\sigma}\)