Mathematical Statistics

What Is Statistics? Populations, Samples and Frequency Distributions

Samir Orujov, PhD

ADA University, School of Business

Information Communication Technologies Agency, Statistics Unit

2026-09-27

🎬 The Idea in 3 Minutes

Watch this short intuition video before (or after) the slides. Captions (English / Azərbaycanca): CC button. ▶ Watch on YouTube

🎯 Learning Objectives

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 — symmetry, skewness, tail weight — in economic terms

🗺️ Overview

Wackerly §§1.1 – 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

💡 Motivation: One Number Is Never Enough

A risk desk in Baku holds a portfolio of bank shares. It has 250 daily returns from last year.

The desk does not care about those 250 days. It cares about tomorrow, and about the process that produced them.

So it must reason from what it saw to what it did not see — and say how much confidence that reasoning deserves.

That reasoning is the whole subject of this course.

📝 The Objective of Statistics

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.

Three words carry the weight: inference, sample, goodness.

Drop the third and you are not doing statistics — you are guessing with arithmetic.

👥 Population and Sample

Population

The whole set of measurements of interest to the investigator.

All daily returns the AZ banking index will ever generate.

Sample

A subset of measurements selected from the population.

The 250 daily returns actually observed in 2025.

The population is usually conceptual and infinite. The sample is finite and in your spreadsheet.

🔢 Parameter and Statistic

	Describes	Symbol	Known?
Parameter	Population	\(\mu,\ \sigma^2,\ p\)	Never
Statistic	Sample	\(\bar{y},\ s^2,\ \hat{p}\)	Always

A statistic is a function of the sample, so it is computable — 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.

🤔 Why Sample At All?

• 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 — stress-testing a system until it fails.

• Impossibility. The population may not exist yet: next year’s returns have not happened.

The last reason is the deep one. In finance, the population is almost always hypothetical.

🎯 The Inferential Problem

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?

Chapters 1–5 build the machinery of question 4. Everything before that is bookkeeping.

📊 Describing a Data Set

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.

Two routes:

• Graphical — frequency distributions, histograms (§1.2, today)

• Numerical — mean, variance, standard deviation (§§1.4–1.5, Saturday)

🧱 Building a Frequency Distribution

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 \(f_i / n\)

📈 A Worked Example: 40 Daily Returns

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 — it is the reason this construction matters.

🔑 Relative Frequency Is a Probability in Waiting

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.

Everything in Chapters 3 and 4 is this idea, made exact.

👀 Reading the Shape

Symmetric, mound-shaped

Returns of a broad index over a calm year. The empirical rule will apply.

Skewed right

Loan losses, insurance claims, waiting times. A long right tail, most mass near zero.

Heavy-tailed

Crisis-period returns. The centre looks normal; the tails are where the money is lost.

The histogram is the first place a bad model announces itself.

⚠️ The Number of Classes Is a Choice

• Too few classes: everything looks like one lump. Structure is hidden.

• Too many classes: every observation gets its own bar. Noise looks like structure.

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.

🔬 Interactive: Sample Size and Bin Width

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;
}

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])
  ]
})

🧠 Think-Pair-Share

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 \(k = 3\)?

✅ Think-Pair-Share: Solution

1. With three classes every observation falls into a wide bin. Symmetry of three bars says almost nothing about symmetry of the underlying distribution — 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 1–2% 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.

The moral: a histogram is a claim, and the number of classes is part of the claim.

💰 Case Study: Getting the Data

📈 The question

1. Is the shape of daily S&P 500 returns symmetric, or does it lean?
2. How much mass sits beyond \(\pm 2\%\) in a single day?
3. Would a coarser or finer set of classes change the story?

📊 The data

Series: S&P 500 daily close (FRED SP500)

Period: Sep 2016 – Sep 2026, about 2,500 trading days

Transformation: daily log return, \(100 \times \log(P_t / P_{t-1})\)

💰 The Frequency Distribution

library(ggplot2)
sp <- read.csv("data/sp500.csv", col.names = c("date", "close"))
sp$close <- suppressWarnings(as.numeric(sp$close))   # FRED marks holidays "."
sp <- na.omit(sp)
returns <- 100 * diff(log(sp$close))                 # daily log return, %

k        <- 20                                       # 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), ]

     class frequency rel_freq
11 (-3,-2]        53    0.021
12 (-2,-1]       189    0.075
13  (-1,0]       862    0.343
14   (0,1]      1043    0.415
15   (1,2]       270    0.107
16   (2,3]        42    0.017
17   (3,4]         9    0.004

Mean return \(\approx 0.05\%\), standard deviation \(\approx 1.14\%\); about \(5.9\%\) of days move more than \(\pm 2\%\).

💰 The Relative Frequency Histogram

What the data say: the centre and shoulders look close to a normal curve of the same mean and spread. But real returns have a taller peak and more days far out in the tails than that curve predicts — the heavy tails that make March 2020 and similar crises more common than a normal model would say.

📝 Quiz #1: Population or Sample?

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
• a sample
• a statistic
• a parameter

📝 Quiz #2: Parameters and Statistics

Which of the following is a statistic?

• The average return of the 250 days in your data set
• The true mean daily return of the index
• The population variance \(\sigma^2\)
• The probability of default of a AAA borrower

📝 Quiz #3: Relative Frequency

In a sample of \(n = 40\) returns, 11 fall in the class \([0,1)\). The relative frequency of that class is:

• 0.275
• 11
• 0.11
• 2.75

📋 Summary

• 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

📚 Practice Problems

Wackerly, 7th edition

• Exercises at the end of §1.3 — construct relative frequency histograms and read them

• Chapter 1 supplementary exercises 1.1 – 1.10

Before Saturday

Read §§1.4 – 1.6. We will replace the picture with three numbers, and then ask how much of the picture those numbers actually keep.

🙏 Thank You

Dr. Samir Orujov

📧 sorujov@ada.edu.az 🏢 Building D, Room D325 🕓 Office hours: Wednesday, 16:00 – 18:00

Next class: Numerical descriptive measures — the mean, the variance, and Tchebysheff’s theorem.

❓ Questions

• 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?