A Confidence Interval Is Not a Margin of Error — How to Read the Two Numbers in the Parentheses

Retained placenta is a condition where the placenta remains inside the uterus after giving birth, and in such cases, the placenta is removed manually. In a review that gathered clinical trials on which anesthesia is best for this procedure, only one trial met the conditions, involving just 30 women. The proportion of cases where the operator judged pain control to be good was 1.50 times higher for one method than the other. Following that is a set of parentheses, and the two numbers inside the parentheses are 0.71 and 3.16. The proportion of women themselves who were satisfied with pain control leaned the other way at 0.82 times, and the parentheses were 0.49 to 1.37. Both sets of parentheses span across both the "increased" side and the "decreased" side, so the review notes that both types of satisfaction are uncertain as to which side it is.

The review authors concluded that a single small trial is insufficient evidence to evaluate the effectiveness and safety of these anesthesia methods, and rated the quality of the evidence as "very low." The reasons for downgrading the grade are two. One is limitations in design, and the other is imprecision. Imprecision means there is only a single small trial, there were few or no events, and the parentheses are wide. The final conclusion is that a well-designed multicenter randomized trial is needed (PMID 32529658).

The basics of reading such a set of parentheses were covered in the primer of the series, and are briefly revisited in the very next section. This article goes one step further from there with four questions. They are: how wide parentheses must be to be a problem, whether the bars drawn on a figure are parentheses, what can be said when the parentheses of two groups overlap, and which end of the parentheses is looked at in a trial trying to show that "it is not much worse." Finally, it details a dispute over whether even experts misread parentheses.

⚠️ The names of diseases and treatments appearing below are all examples to show how to read numbers. They are not judgments on whether a treatment is good or bad, and each number has meaning only within the design and subjects of that study.

Basics Revisited — What the Primer Established

A confidence interval is a range of values that do not contradict this data, written right next to the single value produced by a study. The parentheses in the retained placenta result we saw earlier are precisely the confidence interval. The position where the parentheses are placed tells us in which direction and by how much the effect is, and the width of the parentheses tells us how certainly we know that. The narrower the width, the more precise it is said to be, and the width is usually determined by the number of people gathered.

If we could count the entire target the study wants to know, the number that would come out is called the true value. The confidence level is the 95% attached in front of the parentheses, meaning that if the study is repeated endlessly in the same way, 95% of the parentheses made that way will contain the true value. However, this 95% does not tell us whether the single set of parentheses before our eyes has contained the true value.

"No difference" is 0 when the two values are subtracted and compared, and 1 when divided and compared. Whether the parentheses contain this "no difference" is usually the same story as whether the p-value is greater than 0.05. The p-value is the probability that a result as wide as, or wider than, the current one would occur assuming there is no difference at all. And the parentheses record only the wobble that comes from who happened to be sampled by chance. The parentheses do not tell us whether the study itself was biased in one direction. Up to this point is the basics established by "How Much Does a Number Counted Once Wobble? — The Two Numbers in the Parentheses, the Confidence Interval".

How Wide Must the Parentheses Be to Be a Problem? — The Threshold Where Judgments Flip

The 95% confidence interval for the retained placenta trial, 0.71 to 3.16, straddles 1, which represents "no difference." One end sits where clinician satisfaction drops by nearly 30%, while the other end sits where it increases more than threefold. This wide pair of parentheses was one of the reasons the review considered this evidence imprecise. GRADE answers how wide the parentheses must be to be judged "imprecise" using a tool called the decision threshold.

GRADE is an international system for grading the certainty of evidence. The decision threshold is the point where how we judge a result flips. An example is the spot where we cross from "no effect or trivial" to "slightly beneficial."

When GRADE judges precision, it looks at how many decision thresholds are spanned by the plausible effect sizes pointed to by the confidence interval. That count is used to decide how large to view the uncertainty and by how many steps to lower the certainty rating. This explanation comes from a commentary inspired by a recent concept from the GRADE working group. The commentary notes that such thresholds are likely applied unconsciously in settings like clinical practice guideline panels and vary from panel member to panel member. It also points out that a very small effect can be statistically significant yet unimportant to patients or the public. The commentary argues that there are multiple benefits to using thresholds, and making them explicit makes judgments more transparent and consistent. The commentary concludes that decision thresholds play multiple roles in the interpretation of results, the assessment of certainty, and the planning and design of studies (PMID 41314355).

In GRADE at least, when weighing whether the parentheses are problematically wide, what gets used is how many judgment-flipping thresholds fit inside them.

Are the Bars in a Figure Parentheses?

In research papers, it is common to plot the averages of two groups as bars or dots, with thin lines drawn vertically above and below them. These lines are called error bars. Error bars of the same appearance actually plot one of three different things.

The first is the standard deviation. The standard deviation is a value showing how much individual values are scattered from the average. When you measure the heights of 100 people, there are tall people and short people, and it measures that degree of scatter. Even if you gather more people, the degree to which people are inherently scattered remains the same, so the standard deviation does not shrink.

The second is the standard error. The standard error is a value showing how much the average value itself wobbles when the same survey is repeated with a new sample. Because it measures the degree to which a single average wobbles, it gets smaller the more people you gather.

The third is the confidence interval. A confidence interval bar is the width drawn by stretching the standard error to fit a preset confidence level, namely that percentage attached in front of the parentheses. A commonly used 95% confidence interval stretches out from the average on both sides by about twice the standard error. If the sample is small, it stretches out more than this, and with around ten people, it becomes noticeably wide.

Depending on which of the three you use to draw the same data, the length of the line varies greatly. Using an example created for this article, let us say that the height average of 100 people is 170 cm and the standard deviation is 10 cm. The standard deviation bar spans 160 to 180 cm. The standard error is 1 cm, obtained by dividing the standard deviation of 10 by 10, which is the square root of 100, and the standard error bar is 169 to 171 cm. The 95% confidence interval is roughly 168 to 172 cm. Ultimately, the standard deviation bar is five times longer than the confidence interval and ten times longer than the standard error bar.

A primer paper that spells out confidence intervals without formulas also explains confidence intervals by separating them from standard deviations, standard errors, and confidence levels. That paper concludes by stating that when drawing a conclusion, confidence intervals tell you much more than a single p-value (PMID 25742216).

There are also established tips for reading figures with your eyes. A 2005 paper laid out seven rules for reading error bar figures, starting from the diagnosis that psychologists rarely use confidence intervals and may not understand them well. Among the rules, three are general principles: finding the bar that directly corresponds to the effect of interest, examining the experimental design, and interpreting the interval. It also provides guidelines on what can be inferred when the confidence intervals attached to the averages of two independent groups overlap. The authors conclude that wider adoption of estimation by intervals could greatly improve how research findings are communicated (PMID 15740449).

Translating the first two principles—finding the bar corresponding to the effect and examining the design—into plain terms for this article goes like this. In a figure comparing two groups, what we are curious about is the difference between the two averages. However, the bars in figures are usually attached to the average of each individual group, so there is no separate bar corresponding to the difference. Moreover, if the design measured the same people twice, once before treatment and once after, the bars drawn for each group do not show how much each individual person changed.

Before reading the bars, all you need to do is look in the figure legend first to find out whether those bars are standard deviations, standard errors, or confidence intervals.

When Two Parentheses Overlap, Is There No Difference?

Let us start with a gonorrhea treatment trial. It is a phase 3 trial, which is a large-scale test right before approval, comparing a new oral drug to a standard treatment using a combination of an injection and an oral drug. The proportion of patients in whom the bacteria disappeared was 90.9% out of 506 people in the new drug group (95% confidence interval 88.1–93.3). In the standard treatment group, it was 96.2% out of 238 people (92.9–98.3). When the two parentheses are placed side by side, they overlap from 92.9 to 93.3. This article compares the two reported parentheses. Seeing them overlap like this, it is easy to read it as 「there is no difference between the two treatments」.

However, the same paper also attached parentheses to the difference between the two groups itself. The difference is 5.3 percentage points, and that parenthesis is 1.4–8.6 (PMID 41391465). This parenthesis does not contain 0, which means 「no difference」. The parenthesis for each group overlaps, but the parenthesis for the difference avoids 0. Read through the parenthesis attached to the difference, it means that the difference in cure rates between the two treatments is hard to dismiss as a coincidence.

This discrepancy happens for the following reason. When two groups consist of different people and wobble independently, they are called two independent groups. In this case, the range over which the difference wobbles is narrower than simply adding together the lengths that the two parentheses extend from the middle to the end. This is because the possibility of both groups simultaneously wobbling to opposite extremes is much smaller than the possibility of one group wobbling to one extreme. Therefore, looking only at the overlap makes it easy to miss a real difference.

Just because the confidence intervals attached to the averages of two independent groups overlap, a truly significant difference is not necessarily ruled out. A short report addressing the confusion caused by overlapping parentheses demonstrates this point using a recent randomized trial as an example. This report has two recommendations. For researchers, it recommends directly calculating the confidence interval for the difference between the two group averages to reduce missing real differences. For clinicians, it recommends interpreting overlapping parentheses with caution and not assuming that an overlap always means no difference in treatment effect when deciding whether to use a treatment (PMID 30119088).

How much is missed by using the overlap judgment method was measured by a toxicology simulation study. In toxicology, when comparing the median lethal concentrations of two groups—the concentration at which half die—some consider there to be no difference if their respective 95% confidence intervals overlap. This overlap test operated near 0.005, which is far below the threshold set at 0.05. This means that the rate of falsely judging that there is a difference when there is none was much lower than the set threshold, and correspondingly, its power to catch a real difference was weaker. The method of putting parentheses on the ratio of the two concentrations and seeing whether it includes 1 operated in accordance with the set 0.05 threshold, and its power to catch a real difference was also greater. The authors recommend this ratio method (PMID 16704080).

There are also matters where looking at the overlap is in its proper place. That is when judging how much the results of multiple studies disagree with one another. The Core GRADE guidelines instruct to consider together how different the values of each study are and how much their confidence intervals overlap, and also to look at where each study's value lies relative to the threshold of judgment (PMID 40328467). When gauging the degree to which multiple studies disagree, the overlap is useful. Where the overlap causes trouble is when judging whether there is a difference between two groups.

When comparing two groups, the parenthesis to read is the parenthesis attached to the difference between the two groups, rather than the parenthesis for each individual group.

A Trial Showing "Not Much Worse" — Which End of the Parentheses to Look At

The gonorrhea trial earlier concluded that the new drug was not inferior to standard treatment. This is true even though the difference's parentheses of 1.4 to 8.6 do not span zero and lie entirely on the side where standard treatment is better. That is because this trial asked a different question from the start. A non-inferiority trial is a trial aimed at showing that a new treatment is not worse than a standard treatment by more than a pre-specified amount.

Non-inferiority trials are often used when evaluating a new treatment that has other advantages besides efficacy. The non-inferiority margin is the line drawn before the trial stating, "We accept this much of being worse."

Such trials are frequently poorly reported and misunderstood. There are two core parts to the judgment: pre-specifying a clinically acceptable margin, and ensuring that the lower end of the confidence interval attached to the difference between the new and standard treatments lies above that margin. This judgment differs from the conventional analysis that sorts out whether a new treatment is better or worse than a standard treatment. Therefore, a treatment that meets the non-inferiority requirement may be significantly better, show no difference, or be significantly worse than the standard treatment in a conventional analysis. Here, the difference is written as "new treatment minus standard treatment," so the margin lies below zero. A review summarizing these points concludes that while non-inferiority comparisons are an important addition to clinical trial reporting, several factors not addressed by conventional superiority analyses must be considered in advance (PMID 36104512).

The gonorrhea trial has the subtraction order reversed. Since the difference was defined as "standard treatment minus new drug," the worse the new drug is, the larger the difference, and the end to watch is the upper end. The trial team pre-determined that if the upper end of these parentheses fell below a 12 percentage point margin, it would be judged non-inferior, and the upper end of 8.6 was inside that. The authors concluded that the new drug was non-inferior, had a similar safety profile, and showed the potential to become an effective oral treatment option for uncomplicated urogenital gonorrhea (PMID 41391465). Although the authors did not write it this way, because the parentheses of the difference skirted zero, in the conventional reading we saw in the primer, it is the side where standard treatment is statistically significantly better. The "may be significantly worse" mentioned by the review is precisely this kind of case.

There are also examples in the opposite direction. This is a non-inferiority trial dividing children with severely displaced wrist bone fractures into a group receiving only a cast and a group receiving surgical bone realignment. 750 children aged 4 to 10 years participated across 49 hospitals in the UK. Because this trial defined the difference as "cast minus surgery," a more negative difference means that arm function in the cast group was worse than in the surgery group. The arm function score difference at 3 months was an adjusted mean of −1.64 points, and the 95% confidence interval was −2.84 to −0.44. The margin was −2.5 points, and the trial team noted that they set this value conservatively. The value of −1.64 produced by the study is inside the margin, but the lower end of the parentheses, −2.84, falls outside the margin. Therefore, non-inferiority was not proven, and the entire parentheses also lay favorably toward surgery. However, in children with completely displaced bones (329 out of 750), results consistent with non-inferiority were found for a wider margin pre-specified for this subgroup.

The trial team's conclusion was that they failed to show non-inferiority at 3 months for the conservative margin, and they did not stop there. The observed difference was small, below the threshold families considered meaningful, and did not persist after initial recovery. The surgery group cost more (the cast group cost an average of £1,665 less per patient), had complications early in the procedure, and the degree of improvement in appearance was not large. Citing this, the trial team concluded that a strategy of using a cast first is worth considering for most children. Meanwhile, children who re-fractured within 12 months were 9 in the cast group and 4 in the surgery group (PMID 41965242).

In the gonorrhea trial, we looked at the upper end of the parentheses, and in the fracture trial, we looked at the lower end. The logic is the same, only the direction is reversed. Depending on which way you subtract the difference, the worse end can be either the upper end or the lower end. Therefore, the first thing to check when reading a non-inferiority trial is which direction is the worse direction, rather than the number of the margin.

The next thing to look at is how the margin was determined. A commentary using inflammatory bowel disease trials as examples points out that the choice of margin at the design stage can be biased toward being set widely to favor non-inferiority claims. This is because the wider the margin, the easier it is to receive a "not worse" judgment. Therefore, the commentary recommends that readers judge based on a margin reflecting patients' values and preferences rather than the margin set by researchers (PMID 38159079).

While pre-specifying a margin is common, disclosing why that value was chosen is done less than half the time. Reviewing 94 non-inferiority trials with a placebo (fake drug) group, 96% of the trials pre-specified a margin. 41% of the trials justified that margin. Trials stating the reason for choosing a non-inferiority design were 22%, and trials displaying the margin and confidence interval graphically were 23%. The authors concluded that methodological transparency is missing in many trials, and recommended that future trials clearly state the reasons for choosing their design and margin (PMID 41940588).

In a non-inferiority trial, you simply need to see whether the end in the worse direction is inside the margin, and together look at why that margin is set to that value.

Do Even Experts Read the Parentheses Wrong?

There are six interpretive sentences written about a single confidence interval. A 2014 survey gave these sentences to 120 psychology researchers and 442 students, and asked them to mark whether each was true or false. By the researchers' classification, all six sentences were false. Yet both groups marked more than three on average as true. The researchers barely outperformed students who had never received any training in statistical inference.

In the same survey, the statistical experience reported by the researchers themselves was unrelated to their scores. The authors conclude that many researchers do not know the correct interpretation of a confidence interval, and write that this misunderstanding is particularly unfortunate because p-values and confidence intervals are the main tools psychologists use to draw conclusions from data (PMID 24420726).

Two branches of criticism attached to this conclusion. The first branch is the criticism that respondents may have read the sentences in their own way yet correctly, so it does not serve as evidence of misunderstanding. The authors of the original survey replied with three points. One is that even if that alternative interpretation is right on its own, it is not an acceptable reading to give to the survey sentences. This is because of a well-known problem (the reference class problem) that it is unsettled what counts as a "repetition of the same sort." Another is that the data lack any evidence that the participants had such interpretations in mind. The last is that those interpretations are merely a trivial restatement of the confidence interval definition, and say nothing about where the true value lies (PMID 26620955). These three arguments are the replies of the original survey's authors, and the original texts of the critics are not included in the data verified by this article.

The second branch is a paper by a different team of researchers containing both criticism and replication. The first argument of this paper is the interpretation itself. Since there are two interpretations of a confidence interval, one cannot infer that someone misunderstood just because they agreed with a certain sentence. The second argument is the design. Since all six sentences were sentences containing misunderstandings and respondents had to answer every sentence, the result may have been produced by the design. Reanalyzing the original data also showed hard-to-explain differences between first-year students and master's degree students.

So this research team broadened the survey and investigated again. They added two sentences containing correct interpretations. They also had master's degree students mark sentences they would not have answered if they could have chosen not to. This is a device to separate agreement born of knowledge from agreement born of ignorance. First-year students, who had not yet learned, chose the correct sentences and the sentences the researchers classified as wrong in equal proportions. This means the two kinds of sentences looked equally plausible on the surface. Master's degree students chose the correct sentences markedly more often once responses born of ignorance were stripped away. However, instances of admitting uncertainty were greater than expected. The authors conclude by discussing implications for teaching methods (PMID 27458424).

Arguments went back and forth between 2014 and 2016 over "how many people are wrong" and "did the survey measure it properly," and in the data verified by this article, there is no trace of a settled conclusion. The first-year student results of the 2016 replication show that to someone who has not learned, correct and incorrect interpretations cannot be told apart on the surface. How well they can be told apart after learning is the point where the two sides dispute.

One question remains that cannot be answered even by reading the parentheses correctly. When someone choosing a treatment is curious, it is often about "the probability that this treatment will benefit." P-values and confidence intervals are established tools for indirectly evaluating treatment effects, but they do not directly estimate that probability. There is another statistical framework that combines prior evidence and observed data to directly yield that probability. A perspective paper introducing this framework to rehabilitation research concludes that this method complements traditional analysis. The same paper adds that its conclusion depends on assumptions about prior evidence, and cannot replace rigorous trial design, conventional-style inference, and repeated verification (PMID 42366482).

There are reports that interpreting confidence intervals correctly is not easy even for trained researchers, and there has been a dispute over the extent of it.

When You Meet Parentheses, Ask

The primer had you ask two questions when you meet parentheses: "How small could it get?" and "How large could it get?". This article adds four more questions on top of that.

1. See how many thresholds where a judgment changes the parentheses cross.
2. If it is a bar in a figure, find in the figure caption whether that bar is a standard deviation, a standard error, or a confidence interval.
3. If it is the result of comparing two groups, look for the parentheses attached to the difference rather than whether the parentheses of each group overlap.
4. If it is a non-inferiority trial, check which side the difference was subtracted from, hold the end in the bad direction against the margin, and see if the reason for setting that margin is written.

How to read the parentheses of multiple studies gathered in a single figure is covered in "A Meta-Analysis Does Not Take an Average — How the Diamond in a Forest Plot Is Built". What the p-value is the probability of is covered in "What Is p = 0.03 the Probability Of? — The One Question a p-Value Answers", and how many people significance depends on is covered in "Where Did 0.05 Come From? — Counting How Many Patients a Finding Rests On".

Reading parentheses goes from checking two numbers to asking whose parentheses they are and which end divides the judgment.

Source Papers

• "Anaesthesia/analgesia for manual removal of retained placenta." (2020) — PMID 32529658 — Cochrane review. 1 trial, 30 participants, 1.50 (0.71–3.16). Evidence insufficient and of very low certainty (design limitations, imprecision); multicenter trials needed.
• "The many roles of decision thresholds…" (2026) — PMID 41314355 — Commentary. In GRADE imprecision rating, the number of decision thresholds crossed by a confidence interval is used to determine the level of downgrading. Thresholds play multiple roles in interpretation, certainty assessment, and study design.
• "A primer on confidence intervals in psychopharmacology." (2015) — PMID 25742216 — Primer. Distinguishes confidence intervals from standard deviations, standard errors, and confidence levels, concluding that confidence intervals provide much more information than p-values.
• "Inference by eye: confidence intervals and how to read pictures of data." (2005) — PMID 15740449 — Seven rules for reading error bar figures and guidelines for independent group overlap. Concludes that interval estimation can greatly improve research communication.
• "Zoliflodacin versus ceftriaxone plus azithromycin…" (2026) — PMID 41391465 — Phase 3 noninferiority trial. Cure rate 90.9% (88.1–93.3) vs 96.2% (92.9–98.3), difference 5.3% (1.4–8.6), upper limit within the 12% margin. Noninferior, similar safety, potential oral treatment option.
• "A Tale of Confusion From Overlapping Confidence Intervals." (2019) — PMID 30119088 — Short report. Even if intervals for two independent group means overlap, a true difference is not necessarily ruled out. Recommends calculating the interval for the difference itself and not assuming overlap means no difference.
• "Comparing median lethal concentration values using confidence interval overlap or ratio tests." (2006) — PMID 16704080 — Simulation study. Overlap tests break down near 0.005 and lose power, whereas ratio tests conform to nominal levels and have greater power.
• "Core GRADE 3: rating certainty of evidence-assessing inconsistency." (2025) — PMID 40328467 — Examines point estimate differences and confidence interval overlap together when rating inconsistency across studies.
• "Interpreting the results of noninferiority trials-a review." (2022) — PMID 36104512 — Review. The lower limit of the difference interval must lie above the margin, and treatments meeting this requirement may still be significantly inferior. An important addition, but multiple factors must be weighed in advance.
• "Non-surgical casting versus surgical reduction… (the CRAFFT Study)" (2026) — PMID 41965242 — 750 participants. Difference −1.64 (−2.84 to −0.44) exceeded the conservative margin of −2.5, failing to establish noninferiority. The difference was small and not sustained, while surgery carried higher costs and early complications, leading to the conclusion that a casting-first strategy is worth considering.
• "The Clinical Interpretation of Noninferiority Trials." (2024) — PMID 38159079 — Commentary. Because margin selection can lean toward broader limits, recommends judging based on margins that reflect patient values and preferences.
• "The evolving role of placebo in non-inferiority trials: A scoping review." (2026) — PMID 41940588 — 94 placebo-controlled non-inferiority trials. 96% pre-specified margins, 41% justified them. Lack of transparency; design and margin justification needed.
• "Robust misinterpretation of confidence intervals." (2014) — PMID 24420726 — 120 researchers and 442 students endorsed an average of more than 3 out of 6 false statements, with researchers barely outperforming students.
• "Continued misinterpretation of confidence intervals: response to Miller and Ulrich." (2016) — PMID 26620955 — Original authors' reply. Alternative interpretations are not acceptable readings of the items, lack evidence, and do not imply the location of the parameter.
• "The Interpretation of Scholars' Interpretations of Confidence Intervals…" (2016) — PMID 27458424 — Critique and replication. Questions two interpretations and the forced-choice design; in the replication, first-year students endorsed correct and incorrect items at equal rates, while master's students endorsed correct items more after removing "don't know" responses.
• "Bayesian Thinking in Rehabilitation Research." (2026) — PMID 42366482 — Perspective. p-values and confidence intervals do not directly estimate the probability that a treatment is beneficial. Methods yielding that probability are complementary but depend on prior assumptions and cannot replace design, conventional inference, and replication.