## π¬ The Idea in 3 Minutes ::: {style="text-align:center"} [Watch this short intuition video before (or after) the slides. Captions: CC button.]{style="font-size:22px"} ::: --- ## π― Learning Objectives ::: {style="font-size: 34px"} By the end of this lecture, you will be able to: - **Distinguish** a population from a sample, and a parameter from a statistic, in a financial setting - **State** the objective of statistics as making an inference together with a measure of its goodness - **Construct** a frequency distribution and a relative frequency histogram from raw return data - **Choose** a sensible number of classes and explain how that choice changes what the picture shows - **Interpret** the shape of a distribution β symmetry, skewness, tail weight β in economic terms ::: --- ## πΊοΈ Overview ::: {style="font-size: 34px"} **Wackerly §§1.1 β 1.3** 1. What statistics is, and what it is for 2. Populations, samples, parameters, statistics 3. The inferential problem and its measure of goodness 4. Describing a data set graphically 5. Frequency distributions and relative frequency histograms 6. Relative frequency as the bridge to probability ::: --- ## π‘ Motivation: One Number Is Never Enough ::: {style="font-size: 32px"} A risk desk in Baku holds a portfolio of bank shares. It has **250 daily returns** from last year. ::: {.fragment} The desk does not care about those 250 days. It cares about **tomorrow**, and about the *process* that produced them. ::: ::: {.fragment} So it must reason from what it saw to what it did not see β and say **how much confidence** that reasoning deserves. ::: ::: {.fragment} That reasoning is the whole subject of this course. ::: ::: --- ## π The Objective of Statistics ::: {style="font-size: 34px"} ::: {.callout-note} ## Definition **Statistics** is the science of making an **inference** about a population, based on information contained in a **sample** from that population, together with a **measure of the goodness** of that inference. ::: ::: {.fragment} Three words carry the weight: *inference*, *sample*, *goodness*. Drop the third and you are not doing statistics β you are guessing with arithmetic. ::: ::: --- ## π₯ Population and Sample ::: {style="font-size: 32px"} ::: {.columns} ::: {.column width="50%"} ::: {.callout-note} ## Population The **whole set** of measurements of interest to the investigator. *All daily returns the AZ banking index will ever generate.* ::: ::: ::: {.column width="50%"} ::: {.callout-tip} ## Sample A **subset** of measurements selected from the population. *The 250 daily returns actually observed in 2025.* ::: ::: ::: ::: {.fragment} The population is usually **conceptual and infinite**. The sample is finite and in your spreadsheet. ::: ::: --- ## π’ Parameter and Statistic ::: {style="font-size: 32px"} | | Describes | Symbol | Known? | |---|---|---|---| | **Parameter** | Population | $\mu,\ \sigma^2,\ p$ | Never | | **Statistic** | Sample | $\bar{y},\ s^2,\ \hat{p}$ | Always | ::: {.fragment} A statistic is a **function of the sample**, so it is computable β and it is a **random variable**, because a different sample would give a different value. That randomness is what we will spend the semester quantifying. ::: ::: --- ## π€ Why Sample At All? ::: {style="font-size: 32px"} - **Cost.** Auditing every loan in a portfolio of 400,000 is not a budget line anyone approves. - **Time.** A credit decision made in six months is not a credit decision. - **Destruction.** Some tests consume the item β stress-testing a system until it fails. - **Impossibility.** The population may not exist yet: next year's returns have not happened. ::: {.fragment} The last reason is the deep one. In finance, the population is almost always **hypothetical**. ::: ::: --- ## π― The Inferential Problem ::: {style="font-size: 32px"} Four questions, and this course answers them in order: 1. What **population** is of interest? 2. What **parameter** of it do we want to know? 3. What **sample information** is available? 4. How **good** is the resulting inference? ::: {.fragment} Chapters 1β5 build the machinery of question 4. Everything before that is bookkeeping. ::: ::: --- ## π Describing a Data Set ::: {style="font-size: 32px"} Before inference, **look at the data**. A list of 250 numbers tells you nothing; the same 250 numbers arranged by size tell you a great deal. ::: {.fragment} Two routes: - **Graphical** β frequency distributions, histograms (Β§1.2, today) - **Numerical** β mean, variance, standard deviation (§§1.4β1.5, Saturday) ::: ::: --- ## 𧱠Building a Frequency Distribution ::: {style="font-size: 30px"} 1. Find the **range**: $\text{range} = \max - \min$ 2. Choose the number of **classes** $k$: between 5 and 20; more data, more classes 3. Class **width** $\approx \text{range} / k$, rounded to something a human reads easily 4. Set **boundaries** so that no observation falls on one 5. **Count** the observations in each class: the class frequency $f_i$ 6. Divide by $n$: the **relative frequency** $f_i / n$ ::: --- ## π A Worked Example: 40 Daily Returns ::: {style="font-size: 30px"} Forty daily percentage returns on a bank share, sorted into classes of width 1%: | Class (%) | Frequency $f_i$ | Relative frequency $f_i/n$ | |---|---|---| | $[-4,-3)$ | 1 | 0.025 | | $[-3,-2)$ | 3 | 0.075 | | $[-2,-1)$ | 6 | 0.150 | | $[-1,0)$ | 10 | 0.250 | | $[0,1)$ | 11 | 0.275 | | $[1,2)$ | 6 | 0.150 | | $[2,3)$ | 2 | 0.050 | | $[3,4)$ | 1 | 0.025 | The relative frequencies sum to 1. That is not an accident β it is the reason this construction matters. ::: --- ## π Relative Frequency Is a Probability in Waiting ::: {style="font-size: 32px"} ::: {.callout-important} ## The bridge to Chapter 2 If a measurement is drawn at random from the sample, the probability that it lands in a class **is** that class's relative frequency. As $n$ grows, the histogram settles down toward the **probability distribution** of the population. ::: ::: {.fragment} Everything in Chapters 3 and 4 is this idea, made exact. ::: ::: --- ## π Reading the Shape ::: {style="font-size: 30px"} ::: {.columns} ::: {.column width="33%"} **Symmetric, mound-shaped** Returns of a broad index over a calm year. The empirical rule will apply. ::: ::: {.column width="33%"} **Skewed right** Loan losses, insurance claims, waiting times. A long right tail, most mass near zero. ::: ::: {.column width="33%"} **Heavy-tailed** Crisis-period returns. The centre looks normal; the tails are where the money is lost. ::: ::: ::: {.fragment} The histogram is the first place a bad model announces itself. ::: ::: --- ## β οΈ The Number of Classes Is a Choice ::: {style="font-size: 32px"} - **Too few classes**: everything looks like one lump. Structure is hidden. - **Too many classes**: every observation gets its own bar. Noise looks like structure. ::: {.fragment} There is no formula that removes the judgement. There is only the discipline of **trying several** and reporting one you can defend. Move the sliders on the next slide and watch a conclusion change shape. ::: ::: --- ## π¬ Interactive: Sample Size and Bin Width {.smaller} ```{ojs} //| echo: false viewof n_obs = { const input = Inputs.range([20, 1000], {value: 200, step: 20, label: "Sample size n:"}); ['pointerdown','touchstart','mousedown','click','wheel','pointermove','touchmove'] .forEach(e => input.addEventListener(e, ev => ev.stopPropagation())); return input; } viewof n_bins = { const input = Inputs.range([3, 40], {value: 12, step: 1, label: "Number of classes k:"}); ['pointerdown','touchstart','mousedown','click','wheel','pointermove','touchmove'] .forEach(e => input.addEventListener(e, ev => ev.stopPropagation())); return input; } viewof shape = { const input = Inputs.select(["mound-shaped", "skewed right", "heavy-tailed"], {value: "mound-shaped", label: "Population shape:"}); ['pointerdown','touchstart','mousedown','click','wheel','pointermove','touchmove'] .forEach(e => input.addEventListener(e, ev => ev.stopPropagation())); return input; } ``` ```{ojs} //| echo: false // deterministic generator, so the class sees the same picture every time rng = { let s = 42; return () => (s = (s * 1103515245 + 12345) % 2147483648) / 2147483648; } normal = () => { const u = Math.max(rng(), 1e-12), v = rng(); return Math.sqrt(-2 * Math.log(u)) * Math.cos(2 * Math.PI * v); } sample = { const out = []; for (let i = 0; i < n_obs; i++) { if (shape === "mound-shaped") out.push(0.05 + 1.2 * normal()); else if (shape === "skewed right") out.push(Math.exp(0.35 * normal()) - 1); else out.push(1.2 * normal() * (rng() < 0.06 ? 4 : 1)); } return out; } Plot.plot({ width: 1150, height: 330, marginLeft: 78, marginBottom: 58, // Plot's default is 10px, which is unreadable on a projector. style: {fontSize: "18px"}, x: {label: "Daily return (%)"}, y: {label: "Relative frequency", tickFormat: ".2f"}, marks: [ Plot.rectY(sample, Plot.binX( {y: "proportion"}, {x: d => d, thresholds: n_bins, fill: "#8b2635", fillOpacity: 0.85} )), Plot.ruleY([0]) ] }) ``` --- ## π§ Think-Pair-Share ```{r} #| label: tps-timer #| echo: false countdown::countdown(minutes = 3, seconds = 0, top = 0, right = 0, font_size = "2em", warn_when = 30) ``` ::: {style="font-size: 32px"} A colleague shows you a histogram of 500 daily returns built with **3 classes**. It looks perfectly symmetric, and he concludes the returns are normally distributed. **Three minutes, in pairs:** 1. What has his choice of classes hidden? 2. What would you ask him to produce before you accept the conclusion? 3. Which feature of returns is *most* likely to be invisible at $k = 3$? ::: --- ## β
Think-Pair-Share: Solution ::: {style="font-size: 30px"} 1. With three classes every observation falls into a wide bin. Symmetry of three bars says almost nothing about symmetry of the underlying distribution β it is an artefact of coarse binning. 2. Ask for the same data at $k \approx \sqrt{500} \approx 22$ classes, plus the numerical summaries of Saturday's lecture and, ideally, a normal reference curve on the same axes. 3. **Tail behaviour.** Crisis returns live in the outer 1β2% of the data. At $k=3$ they are absorbed into the outermost bars and vanish. In risk management that is precisely the part you cannot afford to lose. ::: {.fragment} The moral: a histogram is a *claim*, and the number of classes is part of the claim. ::: ::: --- ## π° Case Study: Getting the Data ::: {style="font-size: 28px"} On your own machine, this is how you would pull five years of real prices: ```{.r .display-only} library(quantmod) getSymbols("^GSPC", src = "yahoo", from = "2021-01-01", auto.assign = TRUE) returns <- 100 * diff(log(as.numeric(Cl(GSPC)))) # daily log returns, % ``` We will not run it here: it needs a live connection, and the numbers would change between one lecture and the next. Instead we use a **simulated sample of 1,250 daily returns**, drawn to match the S&P 500's centre and spread. ::: --- ## π° The Frequency Distribution ```{r} #| label: case-study-freq set.seed(2026) # Five years of daily returns, in per cent. The t distribution gives the # heavy tails that real returns have and a normal sample would not. returns <- 0.05 + 0.75 * rt(1250, df = 4) k <- ceiling(sqrt(length(returns))) # a defensible starting choice breaks <- pretty(range(returns), n = k) counts <- table(cut(returns, breaks)) rel_freq <- counts / length(returns) # The outermost classes hold a handful of observations each, so print the # part of the table that carries the mass: the classes around the mode. centre <- which.max(counts) data.frame(class = names(counts), frequency = as.integer(counts), rel_freq = round(as.numeric(rel_freq), 3))[(centre - 3):(centre + 3), ] ``` --- ## π° The Relative Frequency Histogram ```{r} #| label: case-study-hist #| fig-height: 4 #| fig-width: 9 #| echo: false hist(returns, breaks = breaks, freq = FALSE, main = NULL, xlab = "Daily return (%)", ylab = "Density", col = "grey85", border = "white") ``` ::: {style="font-size: 28px"} **Ask of the output:** where is the centre, how wide is the spread, and how much mass sits beyond Β±3%? ::: --- ## π Quiz #1: Population or Sample? {.quiz-question} A regulator collects the download speeds of **all 1.2 million** fixed broadband subscribers in the country during March. For the question *"what is the mean speed of subscribers in March?"*, this collection is: - [the population]{.correct data-explanation="β
Every measurement of interest is present, so the mean computed from it is a parameter, not a statistic."} - a sample - a statistic - a parameter --- ## π Quiz #2: Parameters and Statistics {.quiz-question} Which of the following is a **statistic**? - [The average return of the 250 days in your data set]{.correct data-explanation="β
It is a function of the sample, so it is computable and it varies from sample to sample."} - The true mean daily return of the index - The population variance $\sigma^2$ - The probability of default of a AAA borrower --- ## π Quiz #3: Relative Frequency {.quiz-question} In a sample of $n = 40$ returns, 11 fall in the class $[0,1)$. The relative frequency of that class is: - [0.275]{.correct data-explanation="β
11/40 = 0.275. Sum all such values and you get 1."} - 11 - 0.11 - 2.75 --- ## π Summary ::: {style="font-size: 32px"} - Statistics makes an **inference about a population from a sample**, and attaches a **measure of goodness** to it - **Parameters** describe populations and are unknown; **statistics** describe samples and are random variables - A **frequency distribution** groups data into classes; dividing by $n$ gives **relative frequencies** that sum to 1 - The relative frequency histogram is the empirical ancestor of the **probability distribution** - The number of classes is a judgement, and it can hide exactly the tail behaviour that matters in finance ::: --- ## π Practice Problems ::: {style="font-size: 32px"} **Wackerly, 7th edition** - Exercises at the end of Β§1.3 β construct relative frequency histograms and read them - Chapter 1 supplementary exercises 1.1 β 1.10 **Before Saturday** Read §§1.4 β 1.6. We will replace the picture with three numbers, and then ask how much of the picture those numbers actually keep. ::: --- ## π Thank You ::: {style="font-size: 34px"} **Dr. Samir Orujov** π§ sorujov@ada.edu.az π’ Building D, Room D325 π Office hours: Wednesday, 16:00 β 18:00 **Next class:** Numerical descriptive measures β the mean, the variance, and Tchebysheff's theorem. ::: --- ## β Questions ::: {style="font-size: 34px"} - Which populations in your own work are hypothetical rather than physical? - If a statistic is a random variable, what would its distribution look like? - Can a histogram ever be *wrong*, or only misleading? ::: 