Chapter 3: The Margin of Error

Chapter 3: The Margin of Error#

Uncertainty, Sampling, and Confidence Intervals#

Dataset: Framingham Heart Study teaching subset — framingham_teaching.csv, n = 500 participants, baseline examination.

Learning Objectives

By the end of this chapter, you will be able to:

Distinguish between a population parameter and a sample statistic
Calculate the standard error of the mean
Construct and correctly interpret a 95% confidence interval
Distinguish random error from systematic bias

Before You Begin: From Description to Inference#

Chapter 2 described our 500 Framingham participants. The scientific goal is broader: to use this sample to draw conclusions about the wider population — all middle-aged Americans of that era, and by extension, populations at cardiovascular risk today.

This is the shift from descriptive to inferential statistics. It requires accepting one fact: every sample statistic carries uncertainty. This chapter gives you the tools to quantify that uncertainty: the standard error and the confidence interval.

Section 1: Populations and Samples#

1.1 Key Definitions#

Population parameter — the true value for the entire population (e.g., the true mean systolic BP of all middle-aged Americans in the 1950s). Denoted $\mu$ (mean) and $\sigma$ (SD).
Sample statistic — the value calculated from our specific 500 participants. Denoted $\bar{x}$ and $s$.
Sampling error — the inevitable, random difference between a sample statistic and the population parameter it estimates. Not a mistake — a mathematical certainty.

Our 500 participants are a sample. Their mean SYSBP ($\bar{x}$) estimates the population mean ($\mu$) — but $\bar{x} \neq \mu$ exactly. How far off might it be? The standard error answers this.

1.2 Accuracy vs Precision#

../_images/ch03_accuracy_precision.png — Fig. 5 **Figure 3.1** Accuracy vs precision illustrated with target diagrams. High precision with low accuracy (systematic bias) is more dangerous than low precision alone — it produces confident, reproducible, and consistently wrong results. In epidemiology, a miscalibrated blood pressure cuff produces this pattern.#

Framingham example: If the blood pressure cuffs used in the 1950s Framingham exams were systematically over-reading by 5 mmHg (a calibration error), every measurement would be precise but inaccurate. A larger sample would not fix this — it would just give a more precisely wrong answer.

Section 2: The Standard Error#

2.1 Standard Deviation vs Standard Error#

💡 Plain English first: If you drew 100 different samples of 500 Framingham participants, each would give a slightly different mean SYSBP. The SE measures how much those sample means typically vary — it is the standard deviation of the sampling distribution of the mean.

Statistic	What it measures	Formula
SD ($s$)	Spread of individual observations	$s = \sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}$
SE	Precision of the sample mean	$SE = \frac{s}{\sqrt{n}}$

Worked example — SYSBP:

$\bar{x}$ = 131.6 mmHg
$s$ = 21.9 mmHg
$n$ = 500

\[SE = \frac{21.9}{\sqrt{500}} = \frac{21.9}{22.36} \approx 0.98 \text{ mmHg}\]

Our sample mean of 131.6 mmHg estimates the population mean, and we expect that estimate to be off by about 1 mmHg on average.

../_images/ch03_sample_size_se.png — Fig. 6 **Figure 3.2** The relationship between sample size and standard error. The Framingham study’s large sample (here $n=500$) yields a very small SE — the mean blood pressure estimate is highly precise. Halving the sample to 250 would increase SE by a factor of $\sqrt{2} \approx 1.41$, not by double.#

Key relationship: SE = s/√n. To halve the SE, you must quadruple n. This has major implications for clinical trial design — precision is expensive.

Section 3: The 95% Confidence Interval#

3.1 Construction#

\[95\% \text{ CI} = \bar{x} \pm 1.96 \times SE\]

Worked example — mean SYSBP: $$95\% \text{ CI} = 131.6 \pm 1.96 \times 0.98 = 131.6 \pm 1.92$$= [129.7 \text{ mmHg}, 133.5 \text{ mmHg}]$$

3.2 Correct Interpretation#

We are 95% confident that the true mean systolic blood pressure in the population from which this sample was drawn is between 129.7 and 133.5 mmHg.

⚡ Common mistake — the two most frequent CI misinterpretations:

❌ “There is a 95% probability the true mean lies in this interval.” ✅ The 95% refers to the procedure — 95% of intervals constructed this way will contain the true mean. This specific interval either contains it or does not.

❌ “95% of individual participants have SYSBP between 129.7 and 133.5 mmHg.” ✅ The CI is about the mean, not about individuals. The range of individual SYBPs is much wider (86–228 mmHg in our dataset).

3.3 What Affects CI Width?#

Factor	Effect on CI	Clinical implication
Larger n	Narrower	The Framingham study’s long run produced very narrow CIs
Smaller SD	Narrower	More homogeneous population → more precise estimate
99% vs 95% confidence	Wider	More certainty requires a wider net

3.4 Bias: What a CI Cannot Fix#

A CI quantifies random error — unavoidable sampling variation. It cannot fix systematic bias — a consistent, directional error in data collection.

Error type	Cause	Fixed by more data?	Example
Random error	Sampling variation	✓ Yes — larger n shrinks SE	Different nurses get slightly different BP readings
Selection bias	Unrepresentative sample	✗ No	Framingham enrolled only White participants
Information bias	Systematic mismeasurement	✗ No	Miscalibrated BP cuff over-reads by 5 mmHg consistently

Information bias is a specific form of systematic error where the measurement of exposure or outcome is systematically wrong. In the Framingham study, if participants under-reported cigarettes per day (social desirability bias), the true smoking–CHD association would be underestimated. No sample size correction can fix a recording error that happens for every participant.

Epidemiological example: The Framingham cohort was almost entirely White, drawn from one Massachusetts town. The mean SYSBP may be a precise estimate of this population whilst being a biased estimate of the broader American — let alone global — population. No amount of additional Framingham participants can fix selection bias.

🔬 Lab Manual — Chapter 3#

Objective#

Calculate the SE and 95% CI for mean SYSBP and mean TOTCHOL. Compare CI widths and interpret the results.

Option A — PSPP#

Open framingham_study.sav.
Analyze → Compare Means → One-Sample T Test.
Move SYSBP and TOTCHOL to the Test Variables box. Set Test Value = 0.
Click OK — the CI appears in the output table.

Option B — R / RStudio#

# -------------------------------------------------------
# Chapter 3 Lab: Standard Error and Confidence Intervals
# -------------------------------------------------------

fram_data <- read.csv("data/framingham_teaching.csv")

# ── Manual SE and 95% CI for SYSBP ───────────────────
n     <- length(na.omit(fram_data$SYSBP))
x_bar <- mean(fram_data$SYSBP, na.rm = TRUE)
s     <- sd(fram_data$SYSBP,   na.rm = TRUE)
SE    <- s / sqrt(n)

CI_lower <- x_bar - 1.96 * SE
CI_upper <- x_bar + 1.96 * SE

cat("SYSBP: mean =", round(x_bar,2), "| SD =", round(s,2),
    "| SE =", round(SE,3), "\n")
cat("95% CI: [", round(CI_lower,2), ",", round(CI_upper,2), "] mmHg\n")

# ── Using t.test() — more accurate for any sample size ──
t.test(fram_data$SYSBP)
t.test(fram_data$TOTCHOL)

# ── Compare CI widths ─────────────────────────────────
# Which variable has more uncertainty in its mean estimate?
ci_sysbp <- t.test(fram_data$SYSBP)$conf.int
ci_chol  <- t.test(fram_data$TOTCHOL)$conf.int

cat("SYSBP CI width:  ", round(diff(ci_sysbp), 3), "mmHg\n")
cat("TOTCHOL CI width:", round(diff(ci_chol),  3), "mg/dL\n")

# ── CI for SYSBP by smoking group ─────────────────────
by(fram_data$SYSBP,
   factor(fram_data$CURSMOKE, labels=c("Non-smoker","Smoker")),
   function(x) t.test(x)$conf.int)

🧪 Test Your Knowledge#

A clinician reads your report and says: “Your 95% CI for mean SYSBP is [129.7, 133.5] mmHg. That means 95% of your participants have blood pressure in that range.” (a) Identify the error. (b) Write the correct interpretation. (c) What is the actual range of individual SYSBP values?

Key Terms#

Term	Definition
Population parameter	True value for the entire population. Denoted μ, σ.
Sample statistic	Value calculated from a specific sample. Denoted x̄, s.
Sampling error	Inevitable random difference between a sample statistic and the population parameter.
Standard error (SE)	SE = s/√n. How precisely the sample mean estimates the population mean.
95% Confidence interval	x̄ ± 1.96 × SE. The plausible range for the true population mean.
Random error	Sampling variation — reduced by increasing n.
Systematic bias	Consistent directional error — not fixed by larger sample size.

Review Questions#

In the Framingham dataset (n=500), the SD of AGE is 8.7 years. Calculate the SE and construct the 95% CI for mean age. Interpret the result.
If the Framingham study had sampled only n=50 participants instead of 500, what would happen to the SE and CI width for mean SYSBP? Show the calculation.
Explain in plain English why a narrow CI does not guarantee the estimate is correct.
A published study reports a very narrow 95% CI for mean blood pressure measured in a study where all participants were recruited from a single hospital clinic. A critic says this precision is misleading. Explain why.
Run t.test(fram_data$TOTCHOL) in R. Identify the 95% CI and write a correct one-sentence interpretation.

Key Takeaways

SE = s/√n — measures precision of the sample mean estimate.
95% CI = x̄ ± 1.96 × SE — plausible range for the true population mean.
Wider CI = less precision. Narrower = more precision (achieved by larger n).
Bias is the silent threat — a narrow CI around the wrong value is worse than a wide CI around the right one.
In R: t.test(variable) returns the CI directly.

Next: Chapter 4 — The Laws of Chance introduces the probability distributions that underpin all hypothesis testing.

Part I — Describing the World in Numbers

Statistic	What it measures	Formula
SD (\(s\))	Spread of individual observations	\(s = \sqrt{\frac{\sum(x_i-\bar{x})^2}{n-1}}\)
SE	Precision of the sample mean	\(SE = \frac{s}{\sqrt{n}}\)