From sum
to mean
One question, two parts, and one difference that trips up most of the class: when you add observations up, the spread grows; when you average them, the spread shrinks. This page builds that from scratch — until you can solve a question like this with your eyes closed.
The problem we're solving
A population is normally distributed with mean 80 and variance 16. A researcher draws 180000 samples of size 16 each. Given:
The four options on the exam were:
What every symbol means
Before any calculation — let's make sure every character in the question is clear. These are all the symbols you'll meet, and nothing more.
What a normal distribution actually is
Imagine measuring the height of every student in your year. Most values will land close to the average, and only a few will be extreme. Draw it — and you get a bell. That's the normal distribution.
Two numbers define it completely: where the bell is centered (that's μ) and how wide it is (that's σ). Play with the sliders and see exactly what each one does.
The 68−95−99.7 rule of thumb
In every normal distribution, whatever μ and σ are:
It's a great sanity check in the exam: if you end up with a probability of 0.4 for something three standard deviations from the mean — you went wrong somewhere.
Two rules for expectation and variance
The whole question rests on four lines. They're worth memorizing exactly as written:
Rule A — a sum of independent variables
Both the expectation and the variance simply add up. Each extra observation adds another μ to the expectation and another σ² to the variance.
Rule B — multiplying by a constant
This is the biggest trap in the question: the expectation is multiplied by c, but the variance is multiplied by c squared. Why? Because variance measures squared distances. Stretch every value by a factor of 3, and every distance from the mean grows 3 times — so every squared distance grows 9 times.
The safe way to remember it: the standard deviation behaves "normally" —
and you can always get the variance from it by squaring. If you get confused, work with the standard deviation and square at the end.
Sum vs. mean
The two parts of the problem look alike, but they pull in opposite directions. Here's the line that has to stick:
Sum — the spread grows
Every new observation adds noise of its own. The pile gets bigger and wilder.
Mean — the spread shrinks
The deviations cancel each other out. The bigger the sample, the steadier the mean.
Where it comes from — one line
The mean is just the sum divided by n, that is, the sum times the constant 1/n. Apply Rule B:
That's it. The whole difference between the two parts comes from that square on the n.
Watch it happen
Below we simulate real samples from the problem's population. Single observations and means of 16 observations are both centered around 80, but look at the width.
Standardizing: how everything becomes Z
There are infinitely many normal distributions — one for every pair of μ and σ. Nobody can print a table for each of them. The solution: translate every question into the language of one standard bell, centered at 0 with a standard deviation of 1.
The two steps this formula performs:
1. Subtract the expectation
Shifts the bell so its center sits on 0. Now the value says "how far above or below the mean I am".
2. Divide by the standard deviation
Squeezes or stretches the width to 1. Now the value says "how many standard deviations above the mean I am".
What a z-score means: the number 0.25 we'll get later simply says "this value is a quarter of a standard deviation above the mean". That's all Z is — a uniform ruler for distance.
From Z to a probability
The Z table always gives the area to the left, that is
Every other case follows from it. These three lines cover any question:
Why is the third line true? Because the standard bell is perfectly symmetric around 0, so the left tail to the left of (−z) has the same area as the right tail to the right of z. Most tables print only positive values, so you need this flip.
The exact table values we'll use:
Full solution, step by step
You now have all the tools. The order is fixed and doesn't change from question to question: identify the expression → find the expectation → find the variance → take the square root → standardize → table.
Part A — computing a
What's the expression? Call the sum S; then the expression inside the probability is 3S.
The distribution of S — by Rule A:
Multiply by 3 — by Rule B. The expectation times 3, the variance times 9:
Standardize and look it up in the table:
Part B — computing b
The distribution of the mean — this time the variance gets divided:
Standardize. This time z comes out negative, so we use symmetry:
ab ≈ 0.0268 — that is, option c.
Traps and the shortcut
The shortcut — for the confident
Notice that the sum and the mean are really the same variable in disguise. Since S = 16 · X̄, we have:
Same z, same answer, in a third of the work. It's also a great check: if the two methods disagree — one of them has a mistake.
The memory card
Where you stand
The four check questions are spread across the relevant slides. Here's where you are — click a row to jump to its question.
Three questions to ask yourself before the exam
If you can answer these without scrolling up — you're ready for this question:
And if something still doesn't sit right — mark it on the slide with "Mark what I didn't get", or ask for a different explanation. Don't leave a gap.