2026
STATISTICS II
MADE EASY
KOEN HANEGREEFS
VUB
, Statistics II for Beginners
How to read this book
Welcome. This document is meant for someone who has never studied statistics before — or who studied it
once, forgot most of it, and is starting over with a friendly guide. The goal is to explain every important concept
of Statistics II in plain English, building intuition first and only then introducing formulas.
How each chapter is built
Every chapter follows the same rhythm:
• First, the big idea — what problem are we trying to solve, and why does it matter?
• Second, the intuition — a story or analogy that gives you a feel for what is going on.
• Third, the machinery — the formulas, conditions, and step-by-step procedures.
• Fourth, a worked example — a real-ish business problem solved from start to finish.
• Fifth, common traps — the mistakes that cost students points on the exam.
You can read it in order or jump around. If you do read it in order, the chapters build on each other: every later
chapter quietly assumes you have absorbed the ideas in the earlier ones.
A note on formulas
Statistics has a lot of formulas, and they can look intimidating. Try this trick: do not memorise the
formula. Memorise what each piece of the formula is doing. Almost every formula in this course has
the same shape:
Statistic ± Critical value × Standard error
Once you see that pattern, you have already understood half the course.
Page 1
, Statistics II for Beginners
Chapter 1 — The Big Picture
1.1 What statistics is actually for
Imagine you run a coffee shop and you want to know what fraction of your customers come back a second time.
You could ask every single customer — but that is impossible. You only ever see the customers who walk in
today, this week, this month. The real answer lives in a giant invisible group called the population (every
customer you have or could have), and what you actually see is a tiny slice of it called the sample.
Statistics is the art of making sensible statements about the invisible whole based on the small visible part. That
is it. The whole course is built around that one move: from sample to population. Every test, every confidence
interval, every formula is a different way of doing that move safely.
1.2 Population vs sample — the most important distinction
Let us be precise about these two words, because almost every formula in the course refers to them.
Term What it means
Population The entire group you would like to know about. Every customer,
every voter, every screw that the factory will ever produce.
Sample The subset you actually collect data from. A few hundred
customers, a thousand voters, a box of fifty screws.
Parameter A number that describes the population. Usually unknown. We use
Greek letters for it: μ (mu) for the mean, σ (sigma) for the standard
deviation, p for a proportion.
Statistic A number we compute from the sample. We use Latin letters for it:
x̄ (x-bar) for the sample mean, s for the sample standard deviation,
p-hat for the sample proportion.
Pay attention to the mental flip: parameters are facts about the world that we cannot see directly; statistics are
numbers we can compute from the data we collected. Statistics are our best guesses about parameters.
Tiny vocabulary helper
In Statistics II, every time you see a Greek letter, think “this is the truth we are trying to estimate”.
Every time you see a Latin letter or a hat (the little ^ symbol), think “this is what we computed from
the sample”.
1.3 Two kinds of variables
Before we can analyse data, we have to know what kind of data we have. Statistics splits variables into two big
families:
Type Examples
Categorical (qualitative) Eye colour, brand chosen, yes/no answers, country. We summarise
these with counts and proportions.
Quantitative (numerical) Height in cm, monthly income, time in seconds. We summarise
these with means and standard deviations.
Page 2
, Statistics II for Beginners
This split matters because it tells us which tools we are allowed to use. Proportions live in chapters about counts
and categories; means live in chapters about numbers. When we get to Chapter 8 (chi-square), we will be
working with categorical variables. When we get to regression in Chapter 10, we will be working with
quantitative variables. Mixing them up is a classic exam mistake.
1.4 The Normal distribution — the bell curve you keep hearing about
If you only remember one shape from Statistics I, make it the Normal distribution. It is the symmetric bell-
shaped curve that shows up everywhere in statistics. Why? Because of a beautiful piece of mathematics called
the Central Limit Theorem, which we will meet properly in Chapter 2. For now, just hold the picture in your
head: a hump in the middle, tails fading out on both sides, with most values close to the average and very few
far away.
Two numbers fully describe a Normal distribution:
• The mean μ — where the centre of the hump sits.
• The standard deviation σ — how wide the hump is. Small σ means tall and narrow; large σ means short
and spread out.
The famous 68–95–99.7 rule says that, for a Normal distribution:
• About 68% of the values are within one standard deviation of the mean.
• About 95% are within two standard deviations.
• About 99.7% are within three standard deviations.
That rule is the secret behind almost every confidence interval you will see in this course.
1.5 Standard scores (z-scores) — measuring how unusual something is
A z-score answers the question: “How far from the average is this value, in units of standard deviation?”
z = (value − mean) / standard deviation
If your exam grade is 80 and the class average is 70 with a standard deviation of 5, your z-score is (80 − 70) / 5
= 2. That means you are two standard deviations above the average — pretty rare in a Normal world (the 95%
rule tells us that only the top 2.5% of students lie above z = 2).
Why bother with z-scores? Because they let us compare apples to oranges. A z of 2 in any context — exam
grades, heights, profits — always means the same thing: this observation is unusually large.
1.6 Sampling variability — the reason this course exists
Here is the single fact that motivates the entire rest of the book: if you take a new sample, you get a new
statistic. Pick 100 random voters today and 55 say they will vote for Party A. Pick another random 100 tomorrow
and only 51 say so. Neither answer is wrong — they are just two snapshots of the same population, each slightly
off because of who happened to land in your sample.
This wobble between samples is called sampling variability, and it is what makes inferential statistics hard. We
do not just want to know what our sample says — we want to know how much our sample might be lying
because of pure luck. The next chapter is about exactly this: when we look at all the possible samples we could
have drawn, what does the picture look like?
Page 3
, Statistics II for Beginners
Example — A small intuition pump
Suppose a bag has 10,000 marbles, half red and half blue (so the true proportion of red is 0.5). You
reach in blindfolded and grab 10 marbles. You might get 5 red, but you might also get 6 red, 4 red, 7
red — it would not be surprising. You would, however, be very surprised to grab 10 reds, because
that is wildly far from 50%.
Now grab 1,000 marbles instead of 10. You will get close to 500 red, every single time. Big samples
wobble less than small samples. That is the most important practical lesson in the whole course.
Page 4
, Statistics II for Beginners
Chapter 2 — Sampling Distributions
2.1 The thought experiment that changes everything
So far, you have one sample. You computed one proportion or one mean. But here is a thought experiment that
statisticians live by: imagine you could repeat your study a thousand times. Each time, you draw a fresh random
sample of the same size from the same population, and you write down the statistic you computed. What does
that long list of numbers look like?
That list — the collection of all the statistics you would get from all the possible samples — is called the sampling
distribution. It is the distribution of a sample statistic across many imaginary samples. You will never actually
compute it in practice, but knowing what it would look like is what lets us reason about how trustworthy any
single sample is.
2.2 The sampling distribution of a proportion
Let us make this concrete. Suppose 40% of all adults in a city own a pet (the true proportion p = 0.4). We pick a
random sample of n = 200 adults and find that 84 of them own a pet, so our sample proportion is p̂ =
= 0.42. Slightly different from the true 0.40.
Now imagine doing this experiment again and again — a thousand surveys of 200 adults each. We would get a
thousand values of p̂ : 0.39, 0.41, 0.38, 0.43, and so on. If we plot a histogram of those thousand values, three
things will be true:
• The histogram will be bell-shaped — it will look Normal.
• It will be centred on the truth, p = 0.40.
• It will have a specific, predictable spread.
That third bullet is gold. The spread is given by a famous formula:
Standard deviation of the sampling distribution of p̂
SD(p̂ ) = √( p × (1 − p) / n )
Read out loud: “square root of p times 1-minus-p, divided by n”. Bigger sample size n means smaller
spread, just as our marble example promised.
Why this matters
Because the sampling distribution of p̂ is approximately Normal with a known centre (p) and a known standard
deviation, we can do something almost magical: we can attach a probability to how far our single observed p̂
might be from the true p. That is the whole point of confidence intervals (Chapter 3) and hypothesis tests
(Chapter 5).
2.3 Conditions — when can we believe the Normal shape?
The bell-curve shape is not automatic. It depends on two assumptions about how we collected the data and
how big our sample is. The textbook lists four conditions, and we always check them in the same order.
Page 5
, Statistics II for Beginners
Condition What to check Why we need it
Randomization Was the sample chosen Without randomness, the
randomly (or were people sample can be systematically
randomly assigned to groups)? biased and no formula will save
you.
10% condition Is n no more than 10% of the Sampling without replacement
population size? makes observations slightly
dependent. As long as the
sample is a small slice of the
population, the dependence is
negligible.
Independence Are the observations Each observation needs to give
independent of each other (no us its own piece of information.
clustering, no contagious
behaviour)?
Success/Failure Are both n × p ≥ 10 and n × (1 − The Normal shape only kicks in
p) ≥ 10? when the sample is large
enough relative to how rare the
event is.
How to remember the success/failure condition
Think of it as “you need at least ten successes and at least ten failures expected in the sample”. If
success is rare (small p), you need a big n to see enough of them. If success is common (large p), you
need a big n to see enough failures.
2.4 The sampling distribution of a mean
Everything we just said about proportions has a twin sister for means. Suppose the average height of Belgian
adults is μ = 174 cm with standard deviation σ = 8 cm. You sample n = 50 adults and compute the sample mean
x̄. If you could repeat this many times, the distribution of x̄ values would, again, be a bell curve.
Sampling distribution of the sample mean
Centre = μ (the population mean)
Spread = σ / √n (the standard error of the mean)
Notice the same idea: a larger n makes the spread smaller, because the square root of a bigger
number is a bigger number, and we divide by it.
Why we divide by the square root of n
This is not a typo. The standard deviation of one observation is σ. But we are looking at an average of n
observations. Averaging cancels out random ups and downs, and the maths works out so that the standard
deviation of the average is σ divided by the square root of n — not by n itself. Doubling your sample size only
shrinks the wobble by a factor of √2 ≈ 1.41, not by 2. Quadrupling cuts it in half.
Page 6
, Statistics II for Beginners
2.5 The Central Limit Theorem — the most beautiful result in statistics
Here is the punchline: the sampling distribution of the mean is approximately Normal even if the original data
are not Normal, as long as the sample size is big enough. This is called the Central Limit Theorem (CLT).
Why is this incredible? It means we can use Normal-based methods (confidence intervals, t-tests, z-tests) even
when the underlying population looks weird — skewed, lumpy, or bimodal — as long as we have collected
enough data. The “enough” depends on how skewed the population is, but a sample size around n ≥ 30 is usually
more than enough.
The intuition behind the CLT
Adding (or averaging) lots of random ups and downs tends to smooth them out. The peaks of one
observation get cancelled by the valleys of another, and what is left is a beautifully symmetric bell
shape. The CLT is the formal way of saying: averages behave Normally.
2.6 The standard error vs the standard deviation
This is a vocabulary trap that catches many students. Both words sound similar, but they mean different things.
Term Definition
Standard deviation How spread out individual observations are. A property of the data.
Standard error (SE) How spread out the sample statistic is across all possible samples. A
property of the statistic, not of the data.
In practice, the standard error is what you put inside confidence intervals. For a proportion, the standard error
of p̂ is √( p̂ × (1 − p̂ ) / n ). For a mean, it is s / √n, where s is the sample standard deviation. The standard error
tells us how trustworthy our statistic is.
2.7 Worked example — does the bell curve actually appear?
Example — Sampling distribution from a coin flip
Suppose I flip a fair coin 10 times and count the number of heads. The true proportion of heads is p
= 0.5. The standard deviation of p̂ is √(0.5 × 0.) = √0.025 ≈ 0.158.
If I check the success/failure condition: n × p = 10 × 0.5 = 5, which is less than 10. So the Normal
approximation is not great at n = 10. The 68–95–99.7 rule will overestimate the chance of unusual
results.
Now let me increase n to 100. Standard deviation becomes √(0.5 × 0.) = 0.05. Success/failure
check: n × p = 50. The Normal model now works beautifully. We expect about 95% of samples to land
in 0.5 ± 2 × 0.05 = [0.40, 0.60]. Anything outside that would be unusual.
That is the everyday logic of Chapter 3: turn the predictable shape of the sampling distribution into
an interval estimate.
Page 7
, Statistics II for Beginners
2.8 Common pitfalls
• Confusing the population standard deviation σ with the standard error σ / √n. The first describes
individual heights, scores, etc.; the second describes the wobble of the average across samples.
• Forgetting to check the conditions before using the Normal model. If randomization or the
success/failure rule fails, your interval and your p-value will be untrustworthy.
• Believing the sampling distribution is the same as the population. It is not. The population can be
skewed; the sampling distribution of the mean is approximately Normal.
Page 8
, Statistics II for Beginners
Chapter 3 — Confidence Intervals for a Proportion
3.1 From a point estimate to an honest range
Suppose a market researcher samples 400 people and finds that 240 of them have heard of a new product. The
sample proportion is p̂ = = 0.60. That number is called a point estimate — it is our single best guess
at the true proportion in the population. But it is just a guess, and we know from Chapter 2 that another sample
would have given a slightly different value.
A confidence interval (CI) replaces that single guess with an honest range. Instead of saying “we estimate that
60% of the market knows about the product”, we say something more useful: “we are 95% confident that the
true percentage is somewhere between 55.2% and 64.8%”. The range bakes in the wobble we know exists.
3.2 The shape of every confidence interval in this course
Almost every CI you will build in this course has the same skeleton. Get used to this shape and the formulas
later will feel like variations on a theme.
The universal confidence interval template
Estimate ± (critical value) × (standard error)
The estimate is what you computed from your sample (a proportion, a mean, a difference). The
critical value comes from a Normal or t-distribution table and depends on how confident you want to
be. The standard error is how much the estimate would wobble from sample to sample.
3.3 The one-proportion z-interval
Filling in the template for a single proportion gives us:
Confidence interval for a single proportion
p̂ ± z* × √( p̂ × (1 − p̂ ) / n )
p̂ is the sample proportion. n is the sample size. z* is the critical value from the standard Normal
distribution (we will pick it from a small table below).
Picking the right z*
The critical value z* depends only on the confidence level you choose. Most courses (and this one) use the
following standard values:
Confidence level z* Interpretation
90% 1.645 If we did this many times, 90% of
the intervals we build would
contain the true proportion.
95% 1.96 The classic default. 95% of
intervals would catch the truth.
Page 9
STATISTICS II
MADE EASY
KOEN HANEGREEFS
VUB
, Statistics II for Beginners
How to read this book
Welcome. This document is meant for someone who has never studied statistics before — or who studied it
once, forgot most of it, and is starting over with a friendly guide. The goal is to explain every important concept
of Statistics II in plain English, building intuition first and only then introducing formulas.
How each chapter is built
Every chapter follows the same rhythm:
• First, the big idea — what problem are we trying to solve, and why does it matter?
• Second, the intuition — a story or analogy that gives you a feel for what is going on.
• Third, the machinery — the formulas, conditions, and step-by-step procedures.
• Fourth, a worked example — a real-ish business problem solved from start to finish.
• Fifth, common traps — the mistakes that cost students points on the exam.
You can read it in order or jump around. If you do read it in order, the chapters build on each other: every later
chapter quietly assumes you have absorbed the ideas in the earlier ones.
A note on formulas
Statistics has a lot of formulas, and they can look intimidating. Try this trick: do not memorise the
formula. Memorise what each piece of the formula is doing. Almost every formula in this course has
the same shape:
Statistic ± Critical value × Standard error
Once you see that pattern, you have already understood half the course.
Page 1
, Statistics II for Beginners
Chapter 1 — The Big Picture
1.1 What statistics is actually for
Imagine you run a coffee shop and you want to know what fraction of your customers come back a second time.
You could ask every single customer — but that is impossible. You only ever see the customers who walk in
today, this week, this month. The real answer lives in a giant invisible group called the population (every
customer you have or could have), and what you actually see is a tiny slice of it called the sample.
Statistics is the art of making sensible statements about the invisible whole based on the small visible part. That
is it. The whole course is built around that one move: from sample to population. Every test, every confidence
interval, every formula is a different way of doing that move safely.
1.2 Population vs sample — the most important distinction
Let us be precise about these two words, because almost every formula in the course refers to them.
Term What it means
Population The entire group you would like to know about. Every customer,
every voter, every screw that the factory will ever produce.
Sample The subset you actually collect data from. A few hundred
customers, a thousand voters, a box of fifty screws.
Parameter A number that describes the population. Usually unknown. We use
Greek letters for it: μ (mu) for the mean, σ (sigma) for the standard
deviation, p for a proportion.
Statistic A number we compute from the sample. We use Latin letters for it:
x̄ (x-bar) for the sample mean, s for the sample standard deviation,
p-hat for the sample proportion.
Pay attention to the mental flip: parameters are facts about the world that we cannot see directly; statistics are
numbers we can compute from the data we collected. Statistics are our best guesses about parameters.
Tiny vocabulary helper
In Statistics II, every time you see a Greek letter, think “this is the truth we are trying to estimate”.
Every time you see a Latin letter or a hat (the little ^ symbol), think “this is what we computed from
the sample”.
1.3 Two kinds of variables
Before we can analyse data, we have to know what kind of data we have. Statistics splits variables into two big
families:
Type Examples
Categorical (qualitative) Eye colour, brand chosen, yes/no answers, country. We summarise
these with counts and proportions.
Quantitative (numerical) Height in cm, monthly income, time in seconds. We summarise
these with means and standard deviations.
Page 2
, Statistics II for Beginners
This split matters because it tells us which tools we are allowed to use. Proportions live in chapters about counts
and categories; means live in chapters about numbers. When we get to Chapter 8 (chi-square), we will be
working with categorical variables. When we get to regression in Chapter 10, we will be working with
quantitative variables. Mixing them up is a classic exam mistake.
1.4 The Normal distribution — the bell curve you keep hearing about
If you only remember one shape from Statistics I, make it the Normal distribution. It is the symmetric bell-
shaped curve that shows up everywhere in statistics. Why? Because of a beautiful piece of mathematics called
the Central Limit Theorem, which we will meet properly in Chapter 2. For now, just hold the picture in your
head: a hump in the middle, tails fading out on both sides, with most values close to the average and very few
far away.
Two numbers fully describe a Normal distribution:
• The mean μ — where the centre of the hump sits.
• The standard deviation σ — how wide the hump is. Small σ means tall and narrow; large σ means short
and spread out.
The famous 68–95–99.7 rule says that, for a Normal distribution:
• About 68% of the values are within one standard deviation of the mean.
• About 95% are within two standard deviations.
• About 99.7% are within three standard deviations.
That rule is the secret behind almost every confidence interval you will see in this course.
1.5 Standard scores (z-scores) — measuring how unusual something is
A z-score answers the question: “How far from the average is this value, in units of standard deviation?”
z = (value − mean) / standard deviation
If your exam grade is 80 and the class average is 70 with a standard deviation of 5, your z-score is (80 − 70) / 5
= 2. That means you are two standard deviations above the average — pretty rare in a Normal world (the 95%
rule tells us that only the top 2.5% of students lie above z = 2).
Why bother with z-scores? Because they let us compare apples to oranges. A z of 2 in any context — exam
grades, heights, profits — always means the same thing: this observation is unusually large.
1.6 Sampling variability — the reason this course exists
Here is the single fact that motivates the entire rest of the book: if you take a new sample, you get a new
statistic. Pick 100 random voters today and 55 say they will vote for Party A. Pick another random 100 tomorrow
and only 51 say so. Neither answer is wrong — they are just two snapshots of the same population, each slightly
off because of who happened to land in your sample.
This wobble between samples is called sampling variability, and it is what makes inferential statistics hard. We
do not just want to know what our sample says — we want to know how much our sample might be lying
because of pure luck. The next chapter is about exactly this: when we look at all the possible samples we could
have drawn, what does the picture look like?
Page 3
, Statistics II for Beginners
Example — A small intuition pump
Suppose a bag has 10,000 marbles, half red and half blue (so the true proportion of red is 0.5). You
reach in blindfolded and grab 10 marbles. You might get 5 red, but you might also get 6 red, 4 red, 7
red — it would not be surprising. You would, however, be very surprised to grab 10 reds, because
that is wildly far from 50%.
Now grab 1,000 marbles instead of 10. You will get close to 500 red, every single time. Big samples
wobble less than small samples. That is the most important practical lesson in the whole course.
Page 4
, Statistics II for Beginners
Chapter 2 — Sampling Distributions
2.1 The thought experiment that changes everything
So far, you have one sample. You computed one proportion or one mean. But here is a thought experiment that
statisticians live by: imagine you could repeat your study a thousand times. Each time, you draw a fresh random
sample of the same size from the same population, and you write down the statistic you computed. What does
that long list of numbers look like?
That list — the collection of all the statistics you would get from all the possible samples — is called the sampling
distribution. It is the distribution of a sample statistic across many imaginary samples. You will never actually
compute it in practice, but knowing what it would look like is what lets us reason about how trustworthy any
single sample is.
2.2 The sampling distribution of a proportion
Let us make this concrete. Suppose 40% of all adults in a city own a pet (the true proportion p = 0.4). We pick a
random sample of n = 200 adults and find that 84 of them own a pet, so our sample proportion is p̂ =
= 0.42. Slightly different from the true 0.40.
Now imagine doing this experiment again and again — a thousand surveys of 200 adults each. We would get a
thousand values of p̂ : 0.39, 0.41, 0.38, 0.43, and so on. If we plot a histogram of those thousand values, three
things will be true:
• The histogram will be bell-shaped — it will look Normal.
• It will be centred on the truth, p = 0.40.
• It will have a specific, predictable spread.
That third bullet is gold. The spread is given by a famous formula:
Standard deviation of the sampling distribution of p̂
SD(p̂ ) = √( p × (1 − p) / n )
Read out loud: “square root of p times 1-minus-p, divided by n”. Bigger sample size n means smaller
spread, just as our marble example promised.
Why this matters
Because the sampling distribution of p̂ is approximately Normal with a known centre (p) and a known standard
deviation, we can do something almost magical: we can attach a probability to how far our single observed p̂
might be from the true p. That is the whole point of confidence intervals (Chapter 3) and hypothesis tests
(Chapter 5).
2.3 Conditions — when can we believe the Normal shape?
The bell-curve shape is not automatic. It depends on two assumptions about how we collected the data and
how big our sample is. The textbook lists four conditions, and we always check them in the same order.
Page 5
, Statistics II for Beginners
Condition What to check Why we need it
Randomization Was the sample chosen Without randomness, the
randomly (or were people sample can be systematically
randomly assigned to groups)? biased and no formula will save
you.
10% condition Is n no more than 10% of the Sampling without replacement
population size? makes observations slightly
dependent. As long as the
sample is a small slice of the
population, the dependence is
negligible.
Independence Are the observations Each observation needs to give
independent of each other (no us its own piece of information.
clustering, no contagious
behaviour)?
Success/Failure Are both n × p ≥ 10 and n × (1 − The Normal shape only kicks in
p) ≥ 10? when the sample is large
enough relative to how rare the
event is.
How to remember the success/failure condition
Think of it as “you need at least ten successes and at least ten failures expected in the sample”. If
success is rare (small p), you need a big n to see enough of them. If success is common (large p), you
need a big n to see enough failures.
2.4 The sampling distribution of a mean
Everything we just said about proportions has a twin sister for means. Suppose the average height of Belgian
adults is μ = 174 cm with standard deviation σ = 8 cm. You sample n = 50 adults and compute the sample mean
x̄. If you could repeat this many times, the distribution of x̄ values would, again, be a bell curve.
Sampling distribution of the sample mean
Centre = μ (the population mean)
Spread = σ / √n (the standard error of the mean)
Notice the same idea: a larger n makes the spread smaller, because the square root of a bigger
number is a bigger number, and we divide by it.
Why we divide by the square root of n
This is not a typo. The standard deviation of one observation is σ. But we are looking at an average of n
observations. Averaging cancels out random ups and downs, and the maths works out so that the standard
deviation of the average is σ divided by the square root of n — not by n itself. Doubling your sample size only
shrinks the wobble by a factor of √2 ≈ 1.41, not by 2. Quadrupling cuts it in half.
Page 6
, Statistics II for Beginners
2.5 The Central Limit Theorem — the most beautiful result in statistics
Here is the punchline: the sampling distribution of the mean is approximately Normal even if the original data
are not Normal, as long as the sample size is big enough. This is called the Central Limit Theorem (CLT).
Why is this incredible? It means we can use Normal-based methods (confidence intervals, t-tests, z-tests) even
when the underlying population looks weird — skewed, lumpy, or bimodal — as long as we have collected
enough data. The “enough” depends on how skewed the population is, but a sample size around n ≥ 30 is usually
more than enough.
The intuition behind the CLT
Adding (or averaging) lots of random ups and downs tends to smooth them out. The peaks of one
observation get cancelled by the valleys of another, and what is left is a beautifully symmetric bell
shape. The CLT is the formal way of saying: averages behave Normally.
2.6 The standard error vs the standard deviation
This is a vocabulary trap that catches many students. Both words sound similar, but they mean different things.
Term Definition
Standard deviation How spread out individual observations are. A property of the data.
Standard error (SE) How spread out the sample statistic is across all possible samples. A
property of the statistic, not of the data.
In practice, the standard error is what you put inside confidence intervals. For a proportion, the standard error
of p̂ is √( p̂ × (1 − p̂ ) / n ). For a mean, it is s / √n, where s is the sample standard deviation. The standard error
tells us how trustworthy our statistic is.
2.7 Worked example — does the bell curve actually appear?
Example — Sampling distribution from a coin flip
Suppose I flip a fair coin 10 times and count the number of heads. The true proportion of heads is p
= 0.5. The standard deviation of p̂ is √(0.5 × 0.) = √0.025 ≈ 0.158.
If I check the success/failure condition: n × p = 10 × 0.5 = 5, which is less than 10. So the Normal
approximation is not great at n = 10. The 68–95–99.7 rule will overestimate the chance of unusual
results.
Now let me increase n to 100. Standard deviation becomes √(0.5 × 0.) = 0.05. Success/failure
check: n × p = 50. The Normal model now works beautifully. We expect about 95% of samples to land
in 0.5 ± 2 × 0.05 = [0.40, 0.60]. Anything outside that would be unusual.
That is the everyday logic of Chapter 3: turn the predictable shape of the sampling distribution into
an interval estimate.
Page 7
, Statistics II for Beginners
2.8 Common pitfalls
• Confusing the population standard deviation σ with the standard error σ / √n. The first describes
individual heights, scores, etc.; the second describes the wobble of the average across samples.
• Forgetting to check the conditions before using the Normal model. If randomization or the
success/failure rule fails, your interval and your p-value will be untrustworthy.
• Believing the sampling distribution is the same as the population. It is not. The population can be
skewed; the sampling distribution of the mean is approximately Normal.
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, Statistics II for Beginners
Chapter 3 — Confidence Intervals for a Proportion
3.1 From a point estimate to an honest range
Suppose a market researcher samples 400 people and finds that 240 of them have heard of a new product. The
sample proportion is p̂ = = 0.60. That number is called a point estimate — it is our single best guess
at the true proportion in the population. But it is just a guess, and we know from Chapter 2 that another sample
would have given a slightly different value.
A confidence interval (CI) replaces that single guess with an honest range. Instead of saying “we estimate that
60% of the market knows about the product”, we say something more useful: “we are 95% confident that the
true percentage is somewhere between 55.2% and 64.8%”. The range bakes in the wobble we know exists.
3.2 The shape of every confidence interval in this course
Almost every CI you will build in this course has the same skeleton. Get used to this shape and the formulas
later will feel like variations on a theme.
The universal confidence interval template
Estimate ± (critical value) × (standard error)
The estimate is what you computed from your sample (a proportion, a mean, a difference). The
critical value comes from a Normal or t-distribution table and depends on how confident you want to
be. The standard error is how much the estimate would wobble from sample to sample.
3.3 The one-proportion z-interval
Filling in the template for a single proportion gives us:
Confidence interval for a single proportion
p̂ ± z* × √( p̂ × (1 − p̂ ) / n )
p̂ is the sample proportion. n is the sample size. z* is the critical value from the standard Normal
distribution (we will pick it from a small table below).
Picking the right z*
The critical value z* depends only on the confidence level you choose. Most courses (and this one) use the
following standard values:
Confidence level z* Interpretation
90% 1.645 If we did this many times, 90% of
the intervals we build would
contain the true proportion.
95% 1.96 The classic default. 95% of
intervals would catch the truth.
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