Introduction to Advanced Statistical Methods

This course is a first session in advanced statistical methods, built for a class of about 445 students spread over 16 contact sessions of 32 hours in total. The session opens with four real-world case studies showing how statistics can be misused, then builds the descriptive foundation you need for everything later: the types of data, measures of central tendency, variance, and a preview of the normal distribution. The recurring promise of the course is simple: what is taught is what will be examined, nothing more. Everything in the notes below follows the order of the session itself — the cases first, then the descriptive building blocks, then the preview of what inference will need.

1.1 Statistical Literacy: How Statistics Can Mislead

1.1.1 Case Study 1 — The COVID-19 Bar Chart That Faked a Decline

Any scientific information can be communicated in many ways. One way is a research paper: you publish an article and it becomes publicly available. From a mathematics background, another way is a mathematical model — people build differential equations, publish them, and convey the information through equations and symbols. But for a layman, equations are a very difficult way to communicate. That is why, on top of theoretical discussions and mathematical models, it is always recommended to convey the same thing with pictorial or diagrammatic representations: pie charts, bar graphs, and heat maps. Not all people know differential equations or mathematical symbols, so it is better to give them the information in their own way. Real-world: data visualization is one way to stay informed about developments such as the spread of a virus.

The first case is a bar graph posted around May 2020 — about five months after COVID-19 started spreading around the world — by the Georgia Department of Public Health in the United States. The chart aimed to show the top five counties that had the highest COVID-19 cases in the past 15 days, and the number of cases over time. The class was asked to inspect the chart and find the two major problems in it.

To see the chart in your mind: imagine a bar graph with the date running along the horizontal axis (the x-axis), and the number of reported cases running up the vertical axis (the y-axis). Five bars stand on each date, one per county, and the counties were ordered from the most cases to the fewest. The trap is in that ordering — the bars line the counties up by case count, not by time, and the dates themselves are not in sequence. The class was asked to inspect the chart and find the two major problems in it.

The visual takeaway: when the dates hop backward and forward — 28 April followed by 26 April, with other dates interleaved — the eye reads a smooth descending staircase, but the staircase exists only in the layout, not in the data. Take the same numbers, put the dates in real chronological order, and the "decline" is gone.

The point of this case: without knowing the background — statistical illiteracy — the public gets misled. For every problem you can draw a diagram, and for every problem you can run a statistical test, but this course is about giving those diagrams and tests some sense: how to minimize that error and how to improve statistical literacy. The problem here is with the lack of knowledge — not knowing how to portray data or which diagram fits the situation — and the fix is to understand statistics well enough to represent results accurately and reach the audience in a proper way.

1.1.2 Case Study 2 — The Truncated Axis That Exaggerates

The misuse of statistics is present everywhere, and news outlets are not the exception. The American news channel Fox News has been under scrutiny several times for its charts. In one recent case, a graph claimed that the number of Americans identifying as Christians had collapsed over the last decade. The trick was in the y-axis: it did not start from zero. The axis ran from 58 to 78, so a drop from 77% of Americans identifying as Christians in 2009 to 65% in 2019 looked dramatic — a 12-point fall in the plotted frame. Draw the same two bars from zero, and the difference between the two decades looks minor.

Picture both versions side by side. In the truncated version, the y-axis (percentage of Americans) runs from 58 to 78, so the two bars — 77 and 65 — reach nearly the top and bottom of the frame, and the 12-point gap eats most of the plot height. In the honest version, the y-axis runs from 0 to 80, so the same two bars sit near the top of a tall frame, nearly equal in length, and the gap looks like what it is: a small change, not a collapse.

The lesson here is broader than axis scales: the way you choose the statistical test can also give you blunders. It is always our responsibility to take care of all those things — the balancing act between accurate representation and honest framing is what the entire course is about.

1.1.3 Case Study 3 — The Pie Chart That Totals 193%

During the 2020 presidential run, a news network showed a pie chart for the race in which the percentages totaled 193%. A pie chart must always total 100% if you represent parts in percentage terms, or 360 degrees if you represent them in degrees.

To see why this is absurd: a pie is a circle, and a circle is a whole. Each slice claims a share of that whole, so the slices must account for exactly the whole — no more, no less. A pie with 193% of the circle is like a delivery that weighs nearly twice what was ordered: someone is either careless with the measuring or deliberately inflating the portions.

Such errors come from one of two places. Either it is a lack of knowledge — not knowing how to portray the data or which diagram fits the situation — or it is deliberate: purposefully showing misleading information to grab the attention of the audience or to instigate controversy. We cannot do anything about deliberate misuse, but when the misuse is unknowing, we can minimize it with the help of sound statistical procedures, which is the business of this course. The same two-way split — knowingly versus unknowingly misleading the public — is the session's standing classification of statistical misuse.

1.1.4 Case Study 4 — Doctors vs Gun Owners: Arithmetic Without Context

This case shows how perfectly valid arithmetic can still produce a disastrous conclusion. Take two sets of data. From health statistics for the United States: the number of physicians is about 700,000 (7 lakh), and accidental deaths caused by physicians per year is somewhere around 120,000 (1,20,000). From the Federal Bureau of Investigation (FBI): accidental gun deaths per year across all age groups is about 1,500.

Now work the per-person rates. For physicians, divide deaths by people:

$$\frac{120,000}{700,000}=\frac{12}{70}\approx 0.1714\approx 0.17$$

So there are about 0.17 accidental deaths per physician each year. (A second figure, 0.197, was also spoken in the session; it does not come from these two counts, since $0.197\times 700,000\approx 138,000$ deaths, not 120,000. The rate that closes with the stated numbers is 0.17.)

For gun owners, the rate quoted was 0.00188. Check the arithmetic: divide 1,500 deaths by a number of gun owners and ask which denominator produces 0.00188.

$$\frac{1,500}{800,000}=0.001875\approx 0.00188$$

The denominator that makes the number work is about 800,000 gun owners. The count spoken in the session was 80,000, but that smaller number would give $1,500/80,000=0.01875\approx 0.019$ — about ten times larger. The honest statement is: the quoted rate of 0.00188 implies roughly 800,000 gun owners in the comparison.

That is the famous quote the session closed this case with: if you torture the data sufficiently, it will confess to almost anything. The honest ratio of the two stated rates is only about 90, not 9000 — and that gap between claim and check is the whole lesson. Every number in the doctor-gun comparison could be verified by the reader; the disaster comes from the context-free reading, not from the sums.

1.1.5 The Statistician's Responsibility

The session paired two contrary statements: it is easy to lie with statistics, but it is hard to tell the truth without statistics. We witnessed the first half with the doctors claim — the statistics gave you that information even though it is not true. The second half is the rescue: for real problems, statistics is the tool that lets you bring out the truth and bring out accurate information to the public.

Real-world: that is where the intervention of the statistician and the statistical analyst comes in — working together with company authorities, they review the numbers so the reports submitted to the public are reliable and, to some extent, accurate. Simply drawing diagrams and submitting reports is not enough; the statistician's job is to check whether the picture the numbers paint is the true picture. In practice this shows up in every industry that publishes figures — health agencies, banks, newsrooms, market researchers — where the person who owns the analysis is the last line of defense between a chart and the public.

1.2 Types of Data

1.2.1 Qualitative Data: Nominal and Ordinal

For any statistical analysis, the first question is what type of data you are holding. Broadly, data falls into two families: qualitative and quantitative. Qualitative data is subcategorized into nominal and ordinal.

A nominal variable (nominal means "in name only") is a way of labeling without any numerical quantification — gender, hair color, and ethnicity are the examples given. A nominal scale has no order and no arithmetic meaning; the labels are just names. Gender is nominal: "male" and "female" are names, and neither comes before the other. Hair color is nominal: "black", "brown", "blonde" are categories, and averaging them makes no sense. The only legitimate arithmetic on nominal data is counting — how many observations fall in each category.

An ordinal variable (ordinal means "ordered") is one where the order matters — first, second, third, or the grades A, B, C. Ordinal data ranks things, but the gaps between ranks are not necessarily equal. The grade A beats B and B beats C, so the order is meaningful; but the step from C to B is not the same distance as the step from B to A — you cannot say the gap is equal, because the labels only promise an order, not a spacing.

1.2.2 Quantitative Data: Discrete and Continuous

The other family is quantitative data, which is the focus once numerical justifications enter the picture. Quantitative data is subdivided into discrete and continuous.

A discrete variable (from the same root as "discreet", meaning separate) takes separate, countable values with gaps between them. The number of students in a class — 445, or 26 — is discrete: you cannot have 445.7 students. A continuous variable can take any value inside a range. Height, weight, and time are continuous: between 5 feet and 6 feet, a height of 5.5 feet, 5.55 feet, or 5.555 feet are all possible. The session also mentioned the interval and ratio scales, but left them for later reading — the idea to hold onto is that quantitative measurements sit on either an interval or a ratio scale, the difference being whether a true zero exists (temperature in Celsius is interval — 0 is not "no heat"; height is ratio — 0 height means nothing).

Real-world: nowadays data is information — you may not just expect numbers; you may expect text, images, or what-not. Whatever form it takes, it is mandatory to understand the nature of the data before making any data analysis. Broadly, you will come across nominal, ordinal, discrete, and continuous data types, and each specific data type has its own suitable statistical technique.

1.2.3 Matching the Data Type to the Technique

The essential rule: for any data you can apply any statistical method, but it need not be reliable — at some points it will be a meaningless way of doing the analysis. The relevant question is always: is this a nominal problem or an ordinal problem? If it is, what is the relevant technique to address it? And once you have the results, you have to infer from them.

Later in the course — especially in bivariate and multivariate techniques such as regression analysis — we will first categorize what type of data is being used and how it is ordered, and only then choose the suitable statistical technique. The next session will bring concrete examples of each data type and how to handle them.

1.3 Measures of Central Tendency

1.3.1 Mean, Median, and Mode

With any specific raw data, your initial analysis asks two questions: how does the central behavior of the data look, and what is the spreadness of the data. Mean, median, and mode answer the first question — they all convey the central behavior, but different situations demand different measures.

The mean is the average — it takes into consideration all the data points spread across the data set and takes the average. The median is the middle value when the data is ordered. The mode is the number that occurs the most — the value that keeps recurring.

The median needs one extra detail that the session left implicit and the textbook spells out: with an odd number of observations the median is the middle value itself; with an even number there is no single middle value, so the convention is to average the two middle values. For 445 students, the 223rd ordered value is the median.

1.3.2 Why the Mean Is the Most Reliable Measure

The session asked which of the three measures you would rank first, and why the mean is the best among them. The answer: the mean uses every single value in the data set, so every individual contributes to the result.

The mode, in contrast, can lie about the center:

So among all the measures of central tendency, the mean gives you the best picture of the data — which is exactly why the theory of estimation builds its inference on means.

1.3.3 From Means to Estimation Theory

Because the mean is the most reliable summary of the center, the standard inferential tools are built around it. This is a direct bridge to later parts of the course: in the theory of estimation you come across the Z test and the T test, and the inferences on means come into the picture. The Z test and T test both take a sample mean $\bar{x}$, compare it with a claimed population value, and ask whether the gap is too large to be chance — so the reliability of the mean is what makes those tests trustworthy.

That deeper message points straight at the next section: a single number that only says "where is the center" still cannot tell you whether the data hugs that center or scatters far from it.

1.4 Variance and Dispersion

1.4.1 Why the Mean Is Not Enough

The motivating question of this section: why is variance required when the mean is enough? The mean tells you where the center is; the second question of any initial analysis is the spreadness — the variability, or dispersion, of the data. Range and variance are the measures that talk about this variability. A student supplied the intuition:

That is the vocabulary of this section: spreadness, dispersion, and variability all describe how far the data points sit from the center. Variance is the single number that captures it. In manufacturing, the point is easy to feel: a chair that measures 40 cm on average is fine, but not if half the chairs come out at 30 cm and half at 50 cm — the average hides the scatter, and the scatter is what decides whether the product fits its purpose.

1.4.2 Worked Example — Old Drug vs New Drug

Here is the full setup that shows why variance is essential. A drug is used to maintain a steady heart rate in patients who have suffered a mild heart attack. Let $X$ denote the number of heartbeats per minute recorded per patient when the old drug is administered. On a similar note, suppose you wish to compare a new drug with the old one; let $Y$ denote the number of heartbeats per minute when the new drug is administered. So $X$ and $Y$ measure the same thing — heartbeats per minute — under two different drugs.

Both drugs were summarized by a density table: for each possible heartbeat rate, the probability that a patient lands on that rate. The average heartbeats per minute is then computed with the expected-value formula. The verbal description given was: "the expected value of X is summation X into f(x)."

The old-drug density table shown in the session starts with the rates 40 and 60 at probabilities 0.01 and 0.04; the middle rows of the table were shown on screen. The completed table below keeps every spoken number and reproduces every stated result:

The class was then asked to compute the average for the new drug in a similar fashion. The spoken working was "40 into 0.40, plus 16 into 0.05, and this is 100 into 0.04 — sorry, 0.40" — the "16 into 0.05" is a slip for 60 into 0.05, and the last term was corrected from 0.04 to 0.40:

Now the dilemma:

1.4.3 The Tolerance Argument and the Verdict

Physicians can tolerate, for a healthy individual, a heart rate within plus or minus two of the average heartbeats per minute — so around 70, the acceptable range is 68 to 72. Now inspect the two density tables against that band. With the old drug, 90% of the people fall within the average heartbeats per minute range (the probabilities at 69, 70, and 71 add to $0.10+0.70+0.10=0.90$), and only 2% of the people react extremely — the average of 70 coming down to 40 or shooting up to 100. With the new drug, the picture flips: about 10% of the people sit within the acceptable range, while 80% of the people react extremely, landing at the extreme rates.

How can you capture this information with a single number? Variance.

The old drug's heart rates cluster tightly around 70; the new drug's rates scatter out to the extremes. The lesser the variance, the better the drug — it is more efficient in the sense that it is not widely spread, and it does not give you extreme reactions. With the numerical values of the two variances, you recommend the old drug: not by simply writing it down, but justified with numbers. This is also the pattern you will meet whenever software output is on the exam table: the program gives you the averages and the measures of spread, and your job is the inference — which number is smaller, what that means, and which choice it supports.

Two closing remarks from the session. First, in any distribution — including the normal distributions that follow — these two parameters, mean and variance, are the ones you talk about; higher-order parameters exist but are out of scope for this course. Second, the intuition to remember: a low variance means the data hugs the average, a high variance means it scatters — and when two summaries have identical averages, the spread is the deciding factor.

1.4.4 Student Questions and Answers

The exchange repeats the core point: you do not compare the 90% and 80% figures directly — you feed the same density tables into the variance formula, get one number for each drug, and let the comparison of those two numbers decide. The 90% and 80% figures were already part of the reasoning that built the tables; the variance condenses all of that into the two numbers 26.2 and 730.06, and the smaller one wins.

1.5 The Normal Distribution Preview

1.5.1 Gauss and the Gaussian Name

The session paused before inferential statistics to revisit the normal distribution, calling it a very essential part of understanding estimations, hypothesis testing, and p-values. It is one of the most fascinating and widely used distributions among the discrete and continuous distributions. The distribution is associated with Carl Friedrich Gauss (1777–1855), the German mathematician who worked with it extensively in astronomy and geodesy — fitting curves to noisy measurements — which is why the name stuck. Real-world: in many industries, people even call the normal distribution the Gaussian distribution or a Gaussian. In engineering, finance, and machine learning, you will hear "assume a Gaussian" far more often than "assume a normal distribution" — the two names mean the same bell-shaped curve.

The session shared Gauss's own framing with a joke: God loves average looking, that is why he makes so many of them. Is it true? Yes. The two extremes are always rare while middlings are always common — that is the entire idea.

1.5.2 The Bell Curve: Extremes Are Rare, Middles Are Common

The normal distribution is a bell-shaped curve. The intuition was built with height: if X is height, the tallest person — maybe 16 feet — is very rare; one or two will be there. Two feet or three feet is also rare; very few will be there. So the majority of people lie in between a certain possible range.

To see the curve: draw the horizontal axis (the x-axis) for the value of the variable — height in feet — and the vertical axis (the y-axis) for how many people have that height. The curve rises smoothly from near zero at the far left, climbs to its highest point at the middle, and falls back to near zero on the far right. The peak sits at the average: most people cluster at a middling height. The two tails are the landmarks: the far left and far right are nearly flat because a 16-foot person and a 2-foot person are both nearly impossible. The one-sentence takeaway: the curve is tallest where the average is, and it dies out toward both extremes — the middlings are common, the extremes are rare.

The same joke applies to grading. The instructor loves average looking too — that is why many B, C, D grades are given while A and E are rare.

1.5.3 Large Samples and the Law of Large Numbers

The connection to sample size: this course's registered students number 445, while in the previous semester another course had a strength of just 26. Generally speaking, large samples will always follow a normal distribution; small samples may or may not follow a normal distribution. That is what the law of large numbers is all about.

The law of large numbers says, in plain words: as the number of observations grows, the behavior of the group settles down — averages stabilize near their true value, and the spread of the observations takes on a stable, bell-shaped pattern. With only 26 students, one or two unusual students can skew the whole picture; with 445, no single student can dominate, so the collection behaves like the smooth bell. With 445 students, once the evaluations are complete, we will be in a position to decide whether the class behaves like a normal distribution or a completely skewed distribution — but the number suggests it should tend toward the normal.

1.5.4 Why the Normal Distribution Matters in This Course

The normal distribution is the foundation for the inferential part of the course: estimations, hypothesis testing, and p-values all rest on it. A practical note for the entire course: all software output gives you the result directly — there is no explicit way of computing p-values by hand, because Python, SPSS, SAS, or R will give you a direct p-value. Your role is to interpret that output. Still, the course will show how the p-value is computed, because that understanding is also required.

Study advice: start reading chapters 6, 7, 8, and 9 — the material on the normal distribution, estimations, hypothesis testing, and p-values — before the coming sessions; these will be covered briefly, no need to mug it up. The next session takes 30 to 40 minutes to walk through the normal distribution and then introduces the actual course content.

1.6 Course Tools, Textbooks, and Evaluation

1.6.1 Software Tools: R, Python, SPSS

The course shows analyses in R — the sessions show how R runs and what the R output looks like. Students comfortable with Python may explore the same material in Python; the same results will come, so there is no issue. SPSS and SAS were also named as the industry-standard tools that produce direct p-values. The assignment component has no tool restriction: you can use R, Python, SPSS, or your own way — there is no compulsion.

The second exchange carries a rule that runs through the whole course: the software is a means, not the subject. The exam asks what the output means, not how the software computed it.

1.6.2 Textbooks and Study Resources

Three textbooks are used and will be referred to as T1, T2, and T3. T1 covers hypothesis testing — the fundamental material required before doing any analysis, which the course will revisit. About 60 to 70% of the course content comes from T1. T2 is the multivariate analysis textbook, aligned with the course notes. T3 has an SPSS focus. The edition matters: the reference is the 30th edition. If soft copies are available — on the e-library portal or through students who have access — they will be shared with everyone.

1.6.3 Evaluation Structure and Doubt Support

The evaluation plan for this course: two quizzes and one assignment, with the portions for quiz one, quiz two, and the assignment informed well in advance, and dates posted on the portal in a timely manner. With about 445 students, individual assignments are not feasible for checking, so the assignment will be done in groups: groups can be formed by the students or decided by the instructor, the group size will be mentioned, and each group performs a statistical analysis and submits a report. The point of the group format is to appreciate teamwork.

There are regular and makeup examination slots — two different slots. Makeups are given only for genuine cases, and for on-campus students the concern is scrutinized before approval. The personal advice was direct: do not go for the makeup exams, there is no point, by that time something else will be piled up; there is no such difference between the regular and makeup tests as such. The sessions number 16 contact sessions of two hours each.

The difficulty of the mathematics was addressed head-on: calculus is not severe — you have done algebra, and only a bit of linear algebra is required somewhere; no full calculus solving is expected.

Exam Guidance Summary

Everything examinable in this first session is collected here. The standing rule from the opening promise: what is taught is what will be examined, nothing more.

• Coverage policy (most important): the exams focus only on what is taught in the sessions. "Whichever is not taught, I will not ask." Do not search elsewhere — the delivered material is the boundary of the exam.
• Textbook requirement: the textbook (T1, T2, T3) is required because exam questions will be quoted as chapter, page number, and problem number rather than typed out in full. Sessions and note materials are enough for the taught content; the textbook adds practice problems.
• Assessment structure: two quizzes and one assignment. Portions for quiz one, quiz two, and the assignment are announced in advance; dates are posted on the portal in a timely manner.
• Assignment format: group work — groups formed by students or by the instructor, size to be announced; each group does a statistical analysis and submits a report. No tool restriction (R, Python, SPSS, or your own way); teamwork is the point.
• Exam answer style: expect to give your inference on software output. R, Python, SPSS, and SAS give direct p-values — your role is to interpret them; the course also shows how the p-value is computed, so that understanding matters too.
• Regular vs makeup: two different slots; makeup only for genuine cases. The advice given: do not plan for the makeup — there is no difference between the regular and makeup tests, and makeup time will pile up work.
• Prerequisites: no programming prerequisite for the course; calculus complexity is not severe — a bit of linear algebra is needed somewhere, no full calculus solving.
• Study advice: start reading chapters 6, 7, 8, and 9 (normal distribution, estimations, hypothesis testing, p-values) before the coming sessions; they will be covered briefly — no need to mug up. The normal distribution is the fundamental background for everything inferential in the course.
• Conceptual groundwork: the opening case studies (misleading bar charts, truncated axes, pie charts that do not total 100%) carry the course's recurring theme — accurate, honest representation of data — and the mean-versus-mode reasoning explains why inference is built on means (Z test, T test).

Key Industry Applications

The case studies and concepts of this session map directly onto real work in health communication, media, manufacturing, and pharmaceuticals:

• Health statistics and public communication: the Georgia Department of Public Health's COVID-19 bar chart (May 2020) shows how a government body can mislead with an unlabeled, non-chronological chart — and how the corrected chart restored honest communication.
• News media scrutiny: the truncated-axis chart about Americans identifying as Christians (77% in 2009 to 65% in 2019, plotted from 58 to 78) and the 193% pie chart of the 2020 presidential run show that even famous news networks publish misleading visualizations, deliberately or unknowingly.
• Crime statistics: the Federal Bureau of Investigation's (FBI) gun-death statistics were used to build per-person rates — a cautionary example of arithmetic without context leading to an absurd conclusion.
• Manufacturing and quality: variance matters when manufacturing units — estimating the size of chairs and tables — because you need to know how much measurements scatter around the target.
• Pharmaceutical trials: the old-drug vs new-drug comparison (heartbeats per minute after a mild heart attack) is a miniature clinical-trial decision: two drugs with identical averages are distinguished by variance, and the lower-variance drug is the efficient, recommended one.
• Software tools in industry: R, Python, SPSS, and SAS are the tools that compute p-values directly; the industry expectation is that analysts interpret the output, not compute it by hand.
• The Gaussian in industry: the normal distribution is widely called the Gaussian distribution in industry, and large-sample data (like a 445-student class) is expected to follow it via the law of large numbers.