Comparing one group's mean to a single known number — without knowing σ.
Last class, the z-test let us ask "is this group different from a benchmark?" — but it had a catch: it needed the population standard deviation (σ), a number we almost never actually have. The t-test fixes that. It uses the standard deviation we can compute, the one from our own sample. That one small swap is what makes this the test you'll reach for again and again — in your senior thesis, in published papers, on the exam. Master the one-sample t and the independent-samples and paired t-tests coming up are just variations on the same theme.
A one-sample t-test compares one group's mean to a single known or hypothesized number — we call that number μ₀ ("mu-naught," the value the null hypothesis claims the population mean equals). Same question as the z-test — is our group different from the standard? — but now we don't need to know the population spread. We use the sample standard deviation (s) instead, which is the one number our data actually hands us.
When should this be your test? Three things have to line up: you have one sample of quantitative scores; there's a fixed comparison value (μ₀) sitting out there — a spec, a published norm, a recommended amount; and you do not know σ, the population standard deviation. If a friend describes a study where one group's average is checked against a single agreed-upon number and nobody handed them the population SD, you're hearing a one-sample t.
| Symbol | Plain English | Where it comes from |
|---|---|---|
| x̄ (x-bar) | Sample mean | We compute it from our data |
| μ₀ (mu-naught) | The benchmark / hypothesized population mean | A known number we test against |
| σ (sigma) | Population standard deviation | We almost never know this |
| s | Sample standard deviation | We compute it from our data |
| df | Degrees of freedom | df = n − 1 |
The z-test formula needed σ, the spread of the whole population. But think about it — if we already knew that much about the population, why would we be testing a sample at all? In real research, σ is a mystery. So the z-test, as nice as it is, mostly can't be used. The fix is simple: swap in s, the standard deviation of our sample — the exact number Sheets hands us from our data. Now everything is computable.
But that swap isn't free. The sample SD (s) is itself only an estimate of σ, and a small sample can estimate it badly. Using an estimate where we used to have a known value adds a little wobble to the whole calculation. We can't just pretend that extra uncertainty isn't there — so statistics builds it in, with a slightly different reference curve. That curve is the t-distribution.
There's one price for estimating σ with s: a little extra uncertainty. The t-distribution builds that caution right in. It looks like the normal bell — same center at zero, same general hump — but with slightly fatter tails and a marginally shorter peak. Fatter tails just mean "extreme values are a bit more likely than the normal curve would say, so be a touch more cautious before calling something rare."
Because the tails are fatter, the cutoff for "rare enough to be surprising" sits a little farther from zero than it does for z. With z you needed to clear about ±1.96 for a two-tailed test at the .05 level; with t you have to clear a slightly larger number — and exactly how much larger depends on your sample size, through the degrees of freedom.
How fat the tails are depends on the degrees of freedom (df = n − 1). Small sample → fatter tails → more caution. As your sample grows, the tails thin out and t looks just like z. With our 200 students, t and z are nearly identical — but t is always the honest choice when you used s.
Why n − 1 and not n? Computing s uses the sample mean (x̄) you already calculated from the same data. Once the mean is fixed, the deviations from it are not all free to vary: if you know the mean and you know all but one of the scores, that last score is determined — it has to be whatever value makes the mean come out right. So only n − 1 of the deviations are genuinely "free." That lost piece of freedom is exactly what df = n − 1 is counting, and it's why s divides by n − 1 back in Unit 4.
Think of df as a dial for caution. Tiny df (say 4) → noticeably fatter tails → the critical t is large. Big df (say 199) → the tails have thinned almost to the normal curve → the critical t is barely above 1.96. More data, less penalty. The t-distribution isn't one curve — it's a family, one curve per df.
t = (x̄ − μ₀) / (s / √n)In words: take how far your sample mean (x̄) sits from the benchmark (μ₀), then divide by the estimated standard error, s / √n. It's the z-formula with s in place of σ — that's the whole change. Let's name the two parts, because every t-test you'll ever run has this same shape:
x̄ − μ₀ is the raw gap between your sample's mean and the benchmark, in the original units (hours, grams, points). Bigger gap → bigger t.s / √n is the standard error of the mean — how much sample means typically bounce around just by chance. Notice the √n: more data shrinks the standard error, so the same gap produces a larger t with a bigger sample.So t is a signal-to-noise ratio: how big is the difference, measured against how much wobble we'd expect by luck alone? A t near 0 means the mean is right about where μ₀ predicts; a t far from 0 (in either direction) means the gap is large relative to the noise.
s describes how spread out the individual scores are. s/√n (the standard error) describes how spread out the sample means would be if you repeated the study many times. The standard error is always the smaller of the two, and it shrinks as n grows — that's the engine behind every significance test.
Once you have your t, there are two equivalent ways to decide whether it's surprising enough to reject the null hypothesis (H₀: the population mean equals μ₀). They always agree — pick whichever your software or table makes easy.
(a) Compare t to a critical value. Look up the critical t for your df and significance level (α, usually .05, two-tailed). That cutoff marks where the fat tails begin. If your calculated |t| is bigger than the critical value, your result lands out in the tail — too rare to chalk up to chance — so you reject H₀. If it's smaller, you fail to reject H₀.
(b) Read the p-value. The p-value is the probability of getting a t at least this far from zero if H₀ were true. Small p (conventionally < .05) = "this would be a weird fluke if nothing were going on" → reject H₀. Larger p → fail to reject. The two-tailed p counts both tails, since the mean could be off in either direction.
When p is not small enough, we "fail to reject" H₀ — we never "accept" it or "prove there's no difference." Absence of evidence isn't evidence of absence; maybe the effect is real but our sample was too small to catch it. This wording isn't fussiness — it's the honest description of what the test can and can't show.
Question: A wellness coach claims her clients average 7 hours of sleep. You sample n = 25 of them and find a mean of x̄ = 6.2 hours with s = 1.0. Is their true average different from 7? Test at α = .05, two-tailed. Predict each step before revealing it.
Notice the rhythm: standard error → gap → t → df → compare → decide. Every one-sample t-test runs through those same six beats. The negative sign on t just means the sample fell below μ₀; the size (4.0) is what made it significant.
Question: Do SJNY students sleep less than the recommended 8 hours? We'll test sleep_hours (column F) against the benchmark μ₀ = 8. The numbers from our 200 students:
Plug in:
t = (6.82 − 8) / (1.02 / √200) = −1.18 / 0.0721 = −16.48With df = 199, the critical t (two-tailed, α = .05) is only about 1.97, and |t| = 16.48 blows past it — that gives p < .001. The mean sits a huge distance below 8. Yes — SJNY students sleep significantly less than 8 hours.
Here's the live one-sample t-test. Paste your own numbers and a benchmark (μ), or hit Load class data to run the sleep-vs-8-hours example above and watch the t and p drop out.
Unit 10. Is your sample mean different from a known value (μ)? E.g., do students sleep less than 8 hours?
The one-sample t leans on a couple of conditions. They don't have to hold perfectly — the test is fairly forgiving — but knowing them tells you when to be careful.
Mild non-normality is usually fine, especially with larger n. Non-independence is not. If your observations are linked — repeated measures, matched pairs, clustered sampling — a one-sample t can be badly misleading, and you want a different design (often the paired t-test in Unit 12).
A tiny p-value tells you the difference is real, not how big it is. With 200 students, even a trivial gap turns up "significant" because the huge n shrinks the standard error to almost nothing. So we also report Cohen's d — the gap measured in standard deviations, which doesn't care how big your sample is:
d = (x̄ − μ₀) / s = (6.82 − 8) / 1.02 = −1.16SJNY students sleep about 1.16 standard deviations below the 8-hour mark. By the usual rule of thumb (0.2 small, 0.5 medium, 0.8 large), 1.16 is a large effect. The sleep gap isn't just real — it's big. Notice the difference from t: Cohen's d divides by s (raw spread), while t divides by s/√n (standard error). That's why a giant sample can make t huge while d stays modest.
A hypothesis test hands you a verdict: different from 8 hours, or not. A confidence interval (CI) answers the better question hiding underneath: so what IS the true mean, roughly? Instead of one number, you report a range that very likely contains the population mean. The recipe recycles pieces you already have:
95% CI = x̄ ± t_critical × SE
sleep: 6.82 ± 1.97 × 0.0721 → [6.68, 6.96]Read it aloud: "we're 95% confident the true average sleep of SJNY students lands between 6.68 and 6.96 hours." The multiplier is the same critical t from §6 (1.97 for df = 199), and the SE is the same noise term from the test — nothing new was computed, the ingredients were just reassembled from "verdict" into "estimate."
The Confidence Interval simulation draws sample after sample and stacks up the intervals — watch roughly 1 in 20 of the 95% CIs miss the true mean entirely. Two minutes in that sim makes the "95% of intervals capture it" idea stick better than any paragraph.
When this result goes into a paper or lab report, it gets one standardized APA (7th ed.) sentence: the test, the descriptives, the statistic with its df, the p, and the effect size. The template for a one-sample t:
t(df) = value, p = value, d = valueHouse rules: statistical letters (t, p, M, SD, d) are italicized; statistics get two decimals; p-values drop the leading zero (.032, not 0.032) and are never written "= .000" — below .001, write p < .001. Our sleep result, written out:
"SJNY students slept significantly less (M = 6.82 hours, SD = 1.02) than the recommended 8 hours, t(199) = −16.48, p < .001, d = −1.16, 95% CI [6.68, 6.96]."
The CI rides along in square brackets — APA style encourages it, because it upgrades the sentence from verdict to estimate: not just "less than 8," but "between 6.68 and 6.96."
Notice the anatomy: the direction in plain words ("slept significantly less"), the descriptives that anchor the claim (M, SD), then the full statistical sentence with the effect size riding along. Every inferential test from here to the end of the course reports in exactly this rhythm.
The one-sample t-test shows up anywhere there's a known benchmark to measure against: a factory checks whether its bolts hit the 5 cm spec; a school asks whether its students' average IQ differs from the national 100; a renter asks whether this city's rent differs from the national average; a clinic checks whether patients' average wait time beats the 15-minute promise.
The pattern: one group, one fixed number everyone agrees on, and the question "are we different from it?" Whenever you have a benchmark sitting there — a spec, a standard, a published average, a recommended amount — this is your test.
The one-sample t needs a benchmark — and some benchmarks carry weight beyond the arithmetic. Take animal agriculture. Producing a kilogram of beef emits roughly 60 kg of CO₂-equivalent, while a kilogram of peas emits about 1 kg (Poore & Nemecek, 2018). Imagine sampling the carbon footprint of n students' typical dinners and asking whether the average differs from a climate-budget benchmark — say, the per-meal footprint we'd need to hit a 1.5 °C target. That's a one-sample t: one sample of numbers, one fixed standard, σ unknown.
Our class dataset lets us ask quieter versions of the same question. We could test whether students' average animal_compassion score differs from the scale's neutral midpoint, or whether weekly meat servings differ from a dietary-guideline target. In every case the logic is identical to the sleep example — x̄ versus μ₀, divided by the standard error — but the benchmark turns a dry calculation into a question with stakes. A "single known number" can be a spec, a norm, or a line we've collectively decided matters.
You don't run a t-test by hand — Sheets does it. Open the SJNY Student Life dataset. The slickest way is the built-in function, which compares a range against a benchmark and hands back the p-value directly:
=T.TEST(F2:F201, ARRAYFORMULA(F2:F201*0 + 8), 2, 1) → two-tailed p for sleep vs 8The arguments are (your data, the benchmark repeated, tails, type): 2 = two-tailed, 1 = one-sample. (The little ARRAYFORMULA(…*0 + 8) trick just builds a same-length column of 8's to compare against — a one-sample t is mathematically a paired test against a constant.) If you'd rather see the machinery, build t from its three ingredients and then convert to p:
=(AVERAGE(F2:F201) - 8) / (STDEV.S(F2:F201) / SQRT(COUNT(F2:F201))) → t = −16.48
=T.DIST.2T(ABS(-16.48), 199) → p < .001Here =AVERAGE gives x̄, =STDEV.S gives the sample s (the ".S" version divides by n − 1, which is what you want), and =COUNT gives n. =T.DIST.2T turns your t and df into the two-tailed p-value — the chance of a t this far from zero in either direction. And to find the critical t — the cutoff you'd compare |t| against — use:
=T.INV.2T(0.05, 199) → critical t ≈ 1.97 (for df = 24 it returns ≈ 2.06)=T.INV.2T(α, df) is the inverse of T.DIST.2T: feed it your significance level and df, and it returns the boundary where the rejection region begins. If your |t| exceeds it, reject H₀.
z needs σ; t needs only s — don't know the population SD? Use t (you almost always will). The formula is t = (x̄ − μ₀) / (s/√n), and the denominator is the standard error, not the plain SD. Remember df = n − 1, not n: with n = 200, df = 199. When p isn't small enough you "fail to reject" H₀ — never "accept" it or claim you've proven no difference. A small p means "real," not "big," so always pair it with Cohen's d. The benchmark must be a single known number — a spec, a standard, a published value — not another group's mean (that's the independent-samples t in Unit 11). And the sign just shows direction: t = −16.48 means below μ₀; the size is what matters.
Eleven quick true/false items with instant explanations. (Different from your Canvas quiz — real practice.)