Confidence Intervals for a Population Mean (Small Sample)

You’ve just mastered INF-2, where you learned to build confidence intervals using the z-distribution — as long as n ≥ 30 or σ was known. But what happens when you’re a medical researcher running a pilot study with only 10 patients? Or a food scientist measuring a new ingredient in just 8 batches? The sample is too small for the CLT guarantee, and σ is unknown. Can you still build a reliable confidence interval?

In 1908, a statistician named William Sealy Gosset — writing under the pseudonym “Student” to protect his employer’s trade secrets — solved exactly this problem. He derived a new distribution that accounts for the extra uncertainty introduced by estimating σ from a small sample. We call it the t-distribution, and it is the tool that makes small-sample inference possible.

This lesson builds directly on INF-2. The CI formula keeps the same shape; we replace z* with t* and σ with s. The key new skill is knowing when to use t instead of z — and understanding why.

After this lesson, you will be able to:

Explain why the z-distribution is inadequate when σ is unknown and n is small
Construct a confidence interval for μ using the t-distribution:
Determine the correct degrees of freedom and look up t* from a t-table
Apply the t vs. z decision rule: which distribution to use and when
Interpret a t-interval correctly using the same frequentist logic from INF-2

Small-sample inference is a refinement of the large-sample methods you mastered in INF-2.

From INF-2: Confidence Interval Structure. Point estimate () ± Margin of Error ().
From INF-2: Standard Error (SE). when is unknown.
From INF-1: Normality Condition. For small samples (), the population must be approximately normal for our probability models to work.
Z-Values vs. T-Values: Both measure distance from the mean in standard errors, but t-values are larger to account for the extra uncertainty of small samples.

Success Factor:

The most common error in this lesson is using instead of when looking up values in the t-table. If your sample size is 15, your degrees of freedom are 14. Always subtract one.

Retrieval Warm-up — from earlier lessons

A random sample of apples from an orchard has a mean weight of g. The population standard deviation is known: g. What is the standard error of the sample mean?

From INF-2, a 95% confidence interval for a population mean uses the formula . A 99% CI from the same sample would be:

C1 — Why the -Distribution Falls Short

In INF-2, when was unknown and , we substituted for in the standard error formula. This worked because for large samples, reliably approximates — the approximation error is small. But for small samples (), can vary substantially from . Using in a -formula pretends we know exactly, which we don’t — and the -distribution has no way to account for that extra source of uncertainty.

The result: -intervals computed with small samples are too narrow. They appear more precise than they really are. The actual coverage rate — the fraction of such intervals that truly capture — falls below the stated confidence level. A “95% CI” might only capture less than 95% of the time — the actual shortfall depends on and can be substantial for very small samples.

This is not a minor correction. For , the difference between and the correct (, 95%) is large enough to make the -interval meaningfully misleading. The -distribution was invented precisely to fix this.

Figure 2: Each bar is a confidence interval from a different random sample of size n drawn from a normal population with μ = 0. Green bars capture μ; red dashed bars miss it. Switch the method from t to z with a small n to see the coverage rate fall below the nominal level — the defining flaw that the t-distribution corrects.

Try this: set n = 6, confidence 95%, method z. Watch the coverage rate — it will settle noticeably below 95%, because z-intervals are too narrow for small samples with unknown σ. Now switch to t: the rate climbs back to ~95%. Click New Simulation several times to see the sampling variability in the coverage rate itself.

C2 — The -Distribution: Shape and Properties

The -distribution (also called Student’s ) is a family of bell-shaped, symmetric distributions — one for each value of its parameter, the degrees of freedom (). Like the standard normal, it is centered at 0. Unlike the standard normal, it has heavier tails: more probability mass in the extremes.

The t-Distribution

The t-distribution with degrees of freedom is a symmetric, bell-shaped distribution centered at 0. Compared to the standard normal :

It has heavier tails — more probability in the extremes
Its peak is slightly lower than the normal
As , the t-distribution converges to

In practice: use the t-distribution whenever is unknown and the population is approximately normal.

The heavier tails reflect our extra uncertainty. When is small and we’re estimating with , extreme outcomes are more likely than the normal distribution would predict. The -distribution builds that in automatically.

Figure 1: t-distribution vs. standard normal. Adjust the df slider to watch the t-distribution converge toward the normal. Toggle critical values to see how t* is always larger than z* for the same confidence level — and by how much.

Before reading further, use the slider to explore — then check your observations against the questions below.

Set . Record the 95% critical value shown above the curves.
Increase to , then , then .
At which does the gap between and first drop below 0.10?

You will revisit this question in Challenge 2 — record your answer before continuing. As grows, the heavier tails of the -distribution lighten toward the normal, which is why the -approximation becomes acceptable for large samples.

Quick Check — Small-Sample Critical Values

For a 95% confidence interval with and unknown , how does the correct compare with ?

C3 — Degrees of Freedom: Why ?

The -distribution is indexed by degrees of freedom: . But why and not ?

Here’s the intuition. To compute (the sample standard deviation), we first compute the sample mean and then measure how far each observation deviates from it. But once we know and of the data values, the last value is completely determined — it’s not free to vary. So there are only truly independent pieces of information about the spread. We’ve “used up” one degree of freedom by estimating with .

Figure: Five data points with x̄ = 10 (already computed). Drag any blue point — the orange point moves automatically to keep the mean at 10. Only 4 of the 5 values are free; the last is determined. That is why df = n − 1 = 4.

Degrees of Freedom

For a one-sample t-interval, the degrees of freedom are:

where n is the sample size. This is the row you look up in the t-table.

The single most common arithmetic error in this lesson is using instead of . For : , not 12. Always subtract 1 before looking up . Getting this wrong gives you the wrong critical value and therefore the wrong CI.

Quick Check — Degrees of Freedom

A researcher uses observations to construct a t-interval. How many degrees of freedom does she have?

C4–C5 — Reading the -Table

The critical value depends on two things: the degrees of freedom () and the confidence level. The table below gives critical values for a wide range of and confidence levels. As increases, decreases toward the value — you can see the convergence in the bottom rows (the row matches the standard normal ).

Table de la loi t de Student

Valeurs critiques (t*) pour les degrés de liberté (ddl) et l'aire de queue donnés.

ddl	Confiance
	80%	90%	95%	98%	99%	99.9%
	0.10 (1) 0.20 (2)	0.05 (1) 0.10 (2)	0.025 (1) 0.05 (2)	0.01 (1) 0.02 (2)	0.005 (1) 0.01 (2)	0.0005 (1) 0.001 (2)
1	3.078	6.314	12.706	31.821	63.657	636.619
2	1.886	2.920	4.303	6.965	9.925	31.599
3	1.638	2.353	3.182	4.541	5.841	12.924
4	1.533	2.132	2.776	3.747	4.604	8.610
5	1.476	2.015	2.571	3.365	4.032	6.869
6	1.440	1.943	2.447	3.143	3.707	5.959
7	1.415	1.895	2.365	2.998	3.499	5.408
8	1.397	1.860	2.306	2.896	3.355	5.041
9	1.383	1.833	2.262	2.821	3.250	4.781
10	1.372	1.812	2.228	2.764	3.169	4.587
11	1.363	1.796	2.201	2.718	3.106	4.437
12	1.356	1.782	2.179	2.681	3.055	4.318
13	1.350	1.771	2.160	2.650	3.012	4.221
14	1.345	1.761	2.145	2.624	2.977	4.140
15	1.341	1.753	2.131	2.602	2.947	4.073
16	1.337	1.746	2.120	2.583	2.921	4.015
17	1.333	1.740	2.110	2.567	2.898	3.965
18	1.330	1.734	2.101	2.552	2.878	3.922
19	1.328	1.729	2.093	2.539	2.861	3.883
20	1.325	1.725	2.086	2.528	2.845	3.850
21	1.323	1.721	2.080	2.518	2.831	3.819
22	1.321	1.717	2.074	2.508	2.819	3.792
23	1.319	1.714	2.069	2.500	2.807	3.768
24	1.318	1.711	2.064	2.492	2.797	3.745
25	1.316	1.708	2.060	2.485	2.787	3.725
26	1.315	1.706	2.056	2.479	2.779	3.707
27	1.314	1.703	2.052	2.473	2.771	3.690
28	1.313	1.701	2.048	2.467	2.763	3.674
29	1.311	1.699	2.045	2.462	2.756	3.659
30	1.310	1.697	2.042	2.457	2.750	3.646
40	1.303	1.684	2.021	2.423	2.704	3.551
50	1.299	1.676	2.009	2.403	2.678	3.496
60	1.296	1.671	2.000	2.390	2.660	3.460
80	1.292	1.664	1.990	2.374	2.639	3.416
100	1.290	1.660	1.984	2.364	2.626	3.390
∞	1.282	1.646	1.962	2.330	2.581	3.300

The bottom row () shows the normal critical values — notice that approaches from above as increases. Hover over any cell to highlight its row and column, making it easy to read the correct value.

Now practice the lookup yourself. Set a sample size and a confidence level — the tool derives for you and highlights the matching . Watch the and values side by side: that single subtraction is the step students most often skip.

Sample size

n = 6 df = n − 1 = 5

Confidence level

Figure 4: Pick a sample size and confidence level. The lookup always goes n → df = n − 1 → confidence column → t* — you read the table at row df, never row n. The highlighted cell and the readout above update together so the off-by-one step stays visible.

Quick Check — Reading the t-Table

A sample of observations has unknown. You are constructing a 95% CI. What is the correct critical value ?

C6 — Confidence Interval Formula

The -interval has the same structure as the -interval from INF-2. The only changes are replacing with and with :

Confidence Interval for μ (Small Sample, σ Unknown)

which gives the interval:

where is the critical value from the -table with and the desired confidence level.

Figure 3: Adjust n and s to watch every piece of the formula update live. Notice that df = n − 1 (not n) controls which row of the t-table you look up, and that t* is always larger than z* — making the t-interval always wider than the z-approximation for the same data.

C7 — When Can You Use the -Interval?

Three conditions must hold:

Random sample — observations must be independent (stated or assumed)
unknown — if were known, use the -interval regardless of
Population approximately normal — required because the CLT guarantee doesn’t apply for small . Check with a histogram or dot plot; for the assumption matters most; for mild departures are acceptable.

What does “approximately normal” look like in practice? When you inspect a histogram or dot plot, look for:

Single peak — the data should mound in one place, not two or three
Rough symmetry — neither tail should be dramatically longer than the other
No extreme outliers — one or two points far beyond the rest can distort the t-interval, especially when is very small

You do not need a perfect bell curve. For , mild skew is generally tolerable. For , even moderate skew is enough to call the normality assumption into question and should be reported explicitly.

When is the t-interval valid? Population shape by sample size.
Shape of population	n = 6	n = 12	n = 20
Approximately normal Symmetric, single peak	Valid	Valid	Valid
Moderate skew One tail longer, no outliers	Not valid	Borderline — report skew	Acceptable
Strong skew / outliers Heavy tail or extreme values	Not valid	Not valid	Borderline — report caveat

✓ Valid — t-interval is appropriate ⚠ Borderline — proceed with caution and report limitations ✗ Not valid — t-interval assumptions are not met

Figure 5: When can you trust the t-interval? Normality matters most at very small n. By n = 20, mild departures are generally tolerable; strong skew or outliers require explicit caveats or a different method regardless of n.

Quick Check — Checking the Conditions

A researcher has measurements of reaction time (ms). is unknown. A histogram of the sample looks roughly bell-shaped. Which statement best describes whether a t-interval is valid?

C8 — When to Use vs. : The Practical Decision Rule

C7 was a stop sign — three conditions that must hold before any inference is valid. C8 is the fork in the road — once you’ve confirmed those conditions, which distribution do you reach for?

Is known? → Use
unknown? → Use — always. The t-distribution is the correct procedure whenever must be estimated from the data.
unknown + ? → Use (the -approximation is also acceptable here, since for large , but is the principled choice)
unknown + + population clearly non-normal? → Neither method is valid without more information

The rule is simple: if is unknown, use . For large samples (), the -approximation is acceptable — and converge at high degrees of freedom, so the practical difference is small. But is the approximation; is the correct procedure. The distinction matters most when , where the gap between and is large enough to meaningfully affect the interval.

Figure 4: Work through six scenarios. At each step the flowchart lights up the path you have taken — reinforcing the habit of asking "Is σ known?" first, then "Is the population approximately normal?"

C9 — Interpreting the -Interval

The frequentist interpretation from INF-2 applies unchanged:

We are confident that the true population mean lies between and . This means: if we repeated this procedure many times, about of intervals constructed this way would contain the true .

What it does NOT mean: “There is a probability that lies in this interval.” The mean is a fixed constant — it either is or isn’t in your interval.

C10 — Why the -Interval is Wider (and That’s Correct)

For the same data, confidence level, and sample size, the -interval is always wider than the -interval, because for any finite . For example, at 95% confidence with : vs. .

Students sometimes worry that the -interval “must be wrong” because it’s wider than they expected. This is backwards. The -interval is wider because it honestly reflects our uncertainty about . The -interval applied to small samples would be artificially narrow — falsely precise. Wider = more honest when is estimated, not worse.

Example 1 — Computing a t-Interval (Fully Worked)

A physician tests a new pain medication on randomly selected patients. The reduction in pain score (0–10 scale) after treatment yields points and points. The physician does not know the population SD . Construct a 95% confidence interval for the true mean reduction .

Step 1 — Check Conditions:

Random sample: ✓ (stated)
σ unknown: ✓ → use t-distribution
Population approximately normal: assumed ✓ (pain score reductions are typically symmetric)
→ CLT does not guarantee normality; the normality assumption of the population is required

Step 2 — Find df and t:*

From the t-table, for df = 11 and 95% confidence:

Step 3 — Compute SE and Margin of Error:

Step 4 — Build the Interval:

Interpretation: We are 95% confident that the true mean reduction in pain score is between 2.91 and 4.69 points. Notice that this interval does not include 0 — suggesting the medication has a real effect, though this is not a formal hypothesis test.

Note on z vs. t: If we had incorrectly used z* = 1.96, we would get E = 1.96 × 0.404 = 0.792, CI = (3.008, 4.592) — a narrower, but falsely precise interval.

Example 2 — Constructing a 90% t-Interval (Partially Scaffolded)

An environmental scientist collects 8 soil samples near an industrial site and measures lead concentration. The results yield x̄ = 22.5 ppm and s = 4.2 ppm. Construct a 90% CI for the true mean lead concentration. (σ unknown.)

Before seeing the solution: what is df here? Which row of the t-table will you look up? Take a moment to answer before continuing.

Step 1 — Conditions: n = 8 < 30, σ unknown → use t. Assume approximately normal population.

Step 2 — df and t:*

. From t-table: (90%, df = 7).

Step 3 — SE and Margin of Error:

Step 4 — Interval:

Interpretation: We are 90% confident the true mean lead concentration at this site is between 19.69 and 25.31 ppm.

Example 3 — 99% Confidence Interval (Minimally Scaffolded)

A nutritionist measures daily caloric intake for randomly selected college students. She finds calories and calories. is unknown. Construct a 99% CI.

Hint: df = 19. Look up the 99% critical value in the t-table.

Show Solution

. From t-table: (99%, df = 19).

Interpretation: We are 99% confident the true mean daily caloric intake is between 1952 and 2348 calories.

Example 4 — Choosing the Right Distribution (Application Twist)

A quality control engineer measures shaft diameters from n = 25 machined parts: x̄ = 50.3 mm and s = 0.8 mm. σ is unknown. A colleague suggests using z* = 1.96 because “n = 25 is close enough to 30.” Evaluate this claim and construct the correct 95% CI.

The colleague’s reasoning is incorrect. The rule for using z is that σ is known — not that n is “large enough.” Here σ is unknown, so we must use t.

. From t-table: (95%, df = 24).

Correct t-interval:

Incorrect z-interval (for comparison):

The difference is small here (n = 25, close to 30), but the principle is absolute: σ unknown → use t. The error grows as n decreases. For n = 6, df = 5 and t* = 2.571 — a 31% wider margin of error than z* = 1.96.

Work through each problem step by step. The dropdowns give immediate feedback — wrong answers explain what went wrong. The generator problems (GP4) create fresh problems each time you click.

Problem 1 — Degrees of Freedom and Critical Values (C1, C2)

Start with a fully guided procedure. Each numeric step is a cold response, so the feedback can identify the step that needs repair.

Problem 2 — Distribution Selection and CI Interpretation (C3, C4)

For each scenario, choose the defensible procedure and then the correct interpretation.

Scenario: A nutritionist records sodium content for n = 20 randomly selected prepared meals. She does not know σ. She finds mg and mg. Using the correct method, she computes a 95% CI of (1,742, 1,938) mg.