Every formula in MAT 300 in one place — what each letter means, when to use it, and a worked nursing example. Four of the boxes are live calculators.
Greek letters describe the whole population. Roman letters describe your sample. Nearly every mistake on an exam is using a sample formula where the question asked about a population, or the other way round. Check the letter first, every time.
| Symbol | Say it | Means |
|---|---|---|
μ | mew | Population mean — the average if you could measure everyone |
x̄ | x-bar | Sample mean — the average of the ones you measured |
σ | sigma | Population standard deviation |
s | ess | Sample standard deviation |
p | pee | Population proportion (a fraction of the group) |
p̂ | p-hat | Sample proportion |
Σ | sum | “Add all of these up” |
n | en | Sample size — how many you measured |
σx̄ | standard error | How much sample means bounce around |
Add every value, divide by how many there are.
Use it when the data is roughly symmetric with no wild outliers.
Example. Five patients wait 12, 15, 9, 22 and 17 minutes. Σx = 75, n = 5, so x̄ = 75 / 5 = 15 minutes.
One extreme value drags the mean with it. One 300-minute wait would push that mean to 72 minutes, which describes nobody.
Odd n → the single middle value, at position (n+1)/2.
Even n → the average of the two middle values.
Use it when the data is skewed or has outliers — length of stay, income, wait times. This is why hospitals report median length of stay.
Example. 9, 12, 15, 17, 22 → median is 15. Add a 300 and you get 9, 12, 15, 17, 22, 300 → median is (15+17)/2 = 16. The mean would have jumped to 62.5. The median barely moved.
Use it for categories, where a mean makes no sense. The most common blood type on your unit, the most frequent admitting diagnosis.
Data can have no mode, one mode, or two (bimodal). Bimodal usually means two different groups got mixed together — worth noticing.
Use it when some values count more than others — this is exactly how your course grade is calculated.
Example. Exams 60% at 88, homework 20% at 95, final 20% at 80.(0.6·88 + 0.2·95 + 0.2·80) / 1 = 52.8 + 19 + 16 = 87.8
Quick, but it uses only two numbers and both are the most extreme ones, so a single strange value ruins it.
Roughly: the average distance from the mean. Small s means everyone is bunched near the average; big s means they are spread out.
The n − 1 matters. Sample formulas divide by n − 1, population formulas divide by N. Dividing by the smaller number makes s slightly bigger, which corrects for a sample under-guessing the true spread.
How to do it by hand:
The standard deviation before you square-root it. Its units are squared, so it is hard to picture — it is mostly a stepping stone to s.
Use it to compare spread between things measured in different units. Is weight more variable than height? You cannot compare kilograms to centimetres directly, but you can compare their CVs.
Example. Systolic BP: x̄ = 120, s = 12 → CV = 10%. Potassium: x̄ = 4.0, s = 0.4 → CV = 10%. Equally variable, relative to their own size.
The IQR is the spread of the middle half of the data, so outliers cannot touch it. The 1.5·IQR rule is the standard test for whether a value is a genuine outlier.
Example. Q1 = 12, Q3 = 20 → IQR = 8. Fences: 12 − 12 = 0 and 20 + 12 = 32. A stay of 40 days is an outlier; a stay of 30 is not.
How many standard deviations a value sits from the mean. Positive is above, negative is below.
It turns any measurement into a common scale, so you can compare a blood pressure to a lab value to an exam score.
Example. Sodium 148 where μ = 140 and σ = 4.z = (148 − 140) / 4 = +2.0 — two standard deviations high, which is roughly the top 2.5% of people.
Rule of thumb: |z| under 2 is ordinary. |z| over 3 is genuinely unusual.
Type a value, the mean and the standard deviation.
The percent of the group that scored below you. The 90th percentile means 90% are below you — not that you got 90%.
Example. A child at the 25th percentile for weight weighs more than 25 of every 100 children the same age. That is normal, not a problem.
Q1 cuts off the bottom quarter, Q3 the top quarter. Together with the min and max they make the five-number summary that a box plot draws.
Always between 0 and 1. If you get a negative number or something above 1, you have made an arithmetic slip.
Use it whenever a question says “at least one”. Working out “at least one” directly means adding up many cases; the complement is one subtraction.
Example. P(at least one of 5 patients has the allergy) = 1 − P(none of them do).
You subtract the overlap so you do not count it twice. If A and B cannot both happen (mutually exclusive) the overlap is 0 and it is just P(A) + P(B).
Example. 40% have hypertension, 25% have diabetes, 10% have both.P(either) = 0.40 + 0.25 − 0.10 = 0.55
Independent means the first result does not change the second — two different patients. Dependent means it does — drawing two cards without putting the first back.
Read the bar as “given that”. P(A | B) is the probability of A once you already know B happened.
This is what sensitivity and specificity are: sensitivity = P(test positive | disease present).
Ask: would swapping two people give a different answer? Yes → permutation (who is charge nurse vs who is float). No → combination (which 3 nurses are on the committee).
Example. Choosing 3 nurses from 8 for a committee: 8C3 = 8! / (3!·5!) = 56 ways.
Use it when all four are true: a fixed number of trials, each is pass/fail, the probability is the same each time, and the trials are independent.
Example. A drug works for 80% of patients. In 10 patients, the expected number helped is np = 10 × 0.8 = 8, with SD = √(10 × 0.8 × 0.2) = 1.26.
Worth memorising — it answers a lot of questions without any calculation.
Example. Adult heart rate μ = 72, σ = 8. About 95% of people fall between 72 − 16 = 56 and 72 + 16 = 88.
This does what a z-table does, without the table.
The SD describes how spread out the people are. The standard error describes how much the sample mean bounces around. Mixing these two up is the single most common error in this half of the course.
Because you divide by √n, bigger samples give a smaller SE — more data, steadier estimate. To halve the SE you need four times the sample.
Example. σ = 12, n = 36 → SE = 12 / 6 = 2.
Why the whole course works. Even if the thing you are measuring is skewed, the averages of big enough samples pile up into a normal curve. That is what lets you use z and t on almost anything.
n ≥ 30 is the usual cutoff. Below that you need the original data to be roughly normal.
Every interval in this course is that one sentence. Only the middle and right pieces change.
Use t whenever you calculated s from your own data, which is nearly always. Use z only if the question hands you the population σ.
Example. 25 patients, x̄ = 7.2 days, s = 2.5. For 95%, df = 24 so t* = 2.064.SE = 2.5/5 = 0.5, E = 2.064 × 0.5 = 1.03
Interval: 7.2 ± 1.03 = 6.17 to 8.23 days.
Uses t and picks the right df for you.
Proportions always use z, never t.
Example. 48 of 200 readmitted, so p̂ = 0.24. For 95%, z* = 1.96.SE = √(0.24 × 0.76 / 200) = 0.0302E = 1.96 × 0.0302 = 0.059 → 0.181 to 0.299, about 18% to 30%.
It is about the method, not this one interval. If you repeated the whole study 100 times, about 95 of the intervals you built would contain the true value.
It does not mean there is a 95% chance the true mean is in your interval. Your interval either contains it or it does not — you just cannot tell which.
Three things make an interval narrower: a bigger sample, less variable data, or accepting less confidence (90% instead of 99%).
Higher confidence always costs width. A 99% interval is wider than a 95% one, because being more certain means casting a bigger net.
Always round up, whatever the decimal. 24.1 people means you need 25.
If you have no idea what p̂ will be, use 0.5 — it gives the biggest answer, so you are safe.
For estimating a mean.
Every test below is that sentence. It asks: how many standard errors away from the claim did my sample land?
| Testing | Formula | Use when |
|---|---|---|
| A mean, σ known | z = (x̄ − μ₀) / (σ/√n) | Rare — only if handed σ |
| A mean, σ unknown | t = (x̄ − μ₀) / (s/√n) | The usual case, df = n − 1 |
| A proportion | z = (p̂ − p₀) / √[p₀(1−p₀)/n] | Counts and percentages |
Say it as “if the p is low, the null must go”.
You never “accept” H₀ and you never “prove” anything. Not finding evidence is not the same as showing there is no effect.
Type I: you rejected H₀ when it was actually true. You claimed an effect that is not there.
Type II: you failed to reject H₀ when it was actually false. There was a real effect and you missed it.
Power = 1 − β is the chance of catching a real effect. A bigger sample raises power.
Nursing way to remember: Type I is telling a healthy patient they are sick. Type II is sending a sick patient home.
Two-tailed when the question says different from, changed, not equal. Split α between both tails.
One-tailed when it says greater than, less than, improved, reduced. All of α goes in one tail.
Read the wording of Hₐ — that is what decides it.
Always between −1 and +1. The sign is the direction, the size is the strength.
Near 0 means no straight-line relationship. There could still be a curved one, which is why you always look at the scatterplot.
b is the slope: how much y changes for each 1-unit rise in x.
a is the intercept: the predicted y when x is 0 — often meaningless in real terms, but needed for the line.
ŷ (y-hat) is a prediction, not a real measurement.
Never predict outside the range of your data. A line fitted on BMIs of 19 to 35 says nothing about a BMI of 60.
Example. r = 0.8 → r² = 0.64, so 64% of the variation in y is explained by x and the other 36% is down to everything else.
Squaring kills the sign, so r² cannot tell you the direction — only r can.
A strong r means two things move together. It does not tell you which causes which, or whether a third thing causes both.
Only a randomised controlled trial licenses a causal claim. An observational study, however big, does not.
1. Population or sample? Greek letters or Roman letters.
2. Do I know σ? Almost never — so use t, not z, for means.
3. Is it a mean or a proportion? Means use s/√n. Proportions use √[p(1−p)/n] and always z.