Quick wins for a faster PC:
Repair Windows errors before they cause bigger problemsFix Now →Scan for outdated or missing drivers - takes under a minuteDriver Scan →An error bar is a graphical interval around a measured or fitted value, but the label error bar does not identify one universal statistic. The bars might show standard deviation, standard error, a standard uncertainty, a confidence interval, an expanded uncertainty, or an instrument limit. Read the caption, legend, methods and units before deciding what the interval says. Then compare measurements with an uncertainty calculation that matches the experiment; visual overlap alone is not a significance test.
What an error bar actually represents
Suppose a graph plots a central value y with a vertical bar extending upward and downward. The point is the reported estimate; the bar is an interval associated with that estimate. Its meaning depends entirely on how the experimenter constructed it. A bar can describe the spread of individual observations, the uncertainty of a mean, a stated confidence level, a calibration limit, or a coverage interval for the measurand.
As an Amazon Associate I earn from qualifying purchases.
The National Institute of Standards and Technology (NIST) defines measurement uncertainty as a parameter associated with a measurement result that characterizes the dispersion of values reasonably attributable to the measurand. That parameter could be a standard deviation, a multiple of one, or the half-width of an interval with a stated coverage probability. Consequently, two figures that look identical can communicate different evidence if one uses standard deviations and the other uses 95% confidence intervals.
Start by identifying four things:
- Center: Is the point a single observation, an arithmetic mean, a weighted mean, a fitted parameter, or a model prediction?
- Quantity: What physical measurand is being reported, and in what units?
- Interval definition: What calculation produced the upper and lower ends?
- Coverage or confidence: If the interval is probabilistic, what level or multiplier was used?
If the caption does not define the bars, the graph is incomplete evidence. You can describe the visible interval, but you cannot responsibly call it a standard deviation, a 1-sigma uncertainty, or a 95% interval without that information.
#1 Best Overall
A reliable way to read any graph
- Read the axes and units. Confirm whether the plotted quantity is a raw measurement, a transformed quantity, a rate, a logarithm, or a fitted parameter. A symmetric bar in log space is not necessarily symmetric after converting back to the original units.
- Identify the central estimate. Note whether each point is one reading, a mean of repeated readings, a median, or a fit. A mean and a single reading need different uncertainty interpretations.
- Find the definition in the caption or methods. Look for phrases such as standard deviation, standard error, standard uncertainty, 95% confidence interval, expanded uncertainty, tolerance, accuracy, or confidence band.
- Check symmetry. Equal distances above and below the point often indicate a linearized or symmetric uncertainty, but unequal distances can be physically meaningful, especially for ratios, concentrations near zero, bounded quantities, and nonlinear transformations.
- Check sample size and degrees of freedom. A standard error based on five observations conveys less information than one based on 500. Confidence intervals for small samples may use a finite-sample distribution rather than a normal multiplier.
- Look for dependence. Repeated measurements may share a calibration, reference clock, detector gain, environmental drift, or other systematic contribution. Shared effects change how differences should be calculated.
- Compare using an uncertainty model. Compute the difference, propagate the relevant uncertainties and covariances, and state the convention used. Treat visual overlap only as a quick screening clue.
The main kinds of bars
| Bar definition | What it describes | Typical calculation | How to interpret it | Important limitation |
|---|---|---|---|---|
| Standard deviation (SD) | Observed spread of individual readings around their mean | s = √[Σ(xi − x̄)²/(n − 1)] for a sample | How variable the observations were in that data set | It is not automatically the uncertainty of the mean or a confidence interval. |
| Standard error (SE) | Sampling uncertainty of an estimated mean, under the stated model | SE = s/√n for independent observations | How precisely the sample mean estimates a population mean, given the assumptions | It shrinks with sample size and can hide substantial individual variability. |
| Standard uncertainty, u(y) | An estimated standard deviation for a measurement result | Obtained from relevant uncertainty components | Under approximately normal conditions and reliable u(y), ±u covers approximately 68% of the relevant distribution | The approximately 68% statement is conditional, not a universal property of every bar. |
| Confidence interval | An interval produced by a specified statistical procedure for an unknown parameter | Often estimate ± a multiplier based on a standard error and a chosen confidence level | The procedure has a stated long-run coverage under its assumptions | Its meaning is tied to the procedure and model; it is not simply a probability statement about one fixed parameter after the data are observed. |
| Expanded uncertainty | An interval intended to encompass a large fraction of attributable values | U = kcuc, where kc is a coverage factor | Reports a wider interval than the combined standard uncertainty | The coverage factor and intended coverage must be stated; do not relabel U as one standard deviation. |
| Instrument or specification interval | A manufacturer, calibration, resolution, or operating limit | Defined by the instrument specification or calibration procedure | What the device or procedure claims about performance or allowable range | It may not describe the random spread of the particular observations. |
Standard deviation: variability in the readings
For repeated observations x1, …, xn, the sample standard deviation measures how far readings typically lie from their sample mean. If a detector produces values with SD = 0.8 units, a plot with ±0.8 bars is showing data scatter when that is what the caption says. It is not saying that the mean is uncertain by 0.8 units. Increasing the number of repeats may leave the SD almost unchanged while making the mean much more precisely known.
Standard error: uncertainty in a mean
For independent, identically distributed observations, the standard error of the mean is s/√n. If 25 readings have SD = 2.0 units, SE = 0.4 units. A graph using ±0.4 bars is displaying precision of the estimated mean, not the 2.0-unit spread a new individual reading might show. The square-root relationship also means that quadrupling the number of independent observations halves the SE; collecting correlated or drifting observations does not provide the same gain.
Standard uncertainty and the conditional 68% interpretation
NIST states that the standard uncertainty u(y) of a measurement result y is the estimated standard deviation of y. If the relevant distribution is approximately normal and the estimate is reliable, y ± u(y) encompasses approximately 68% of the relevant distribution. That is a conditional coverage statement. A skewed distribution, a small or poorly characterized data set, truncation, a nonlinear model, or an unreliable uncertainty estimate can make the 68% approximation inappropriate.
Recommended Free Tools
Rank #2
Confidence intervals and confidence bands
A confidence interval is generated by a specified inferential procedure. The procedure might use a normal approximation, a Student distribution, a likelihood profile, resampling, or another model. A confidence band around a fitted curve can refer to uncertainty in the mean response, while a prediction band may include the variability of a future observation; the caption must distinguish them. Never infer the level from the visual size of a bar.
Expanded uncertainty
Laboratories often report a combined standard uncertainty uc and multiply it by a coverage factor to obtain an expanded uncertainty U. NIST describes expanded uncertainty as an interval expected to encompass a large fraction of attributable values. The multiplier and the intended coverage belong in the report. A bar of ±0.20 may be an expanded uncertainty with a coverage factor near two, not a one-standard-deviation bar of 0.20.
How uncertainty components are built
An experiment can have several contributions: repeatability, calibration, resolution, reference standards, environmental corrections, drift, and model parameters. NIST separates Type A evaluation, based on statistical analysis of repeated observations, from Type B evaluation, based on other information such as calibration data, prior measurements, specifications, or expert assessment. Type B does not mean less scientific; it means the component was evaluated by a route other than the statistical analysis of the current repeats.
Rank #3
- Introduction to Uncertainty Measurement for Students and Professionals Incorporating the GUM Guide to the Expression of Uncertainty in Measurement
List each component in the units of the measurand or convert it through the measurement model. For independent components, variances are combined, so a simplified combined standard uncertainty is:
uc = √(u1² + u2² + ···).
When the output depends on several input quantities, sensitivity coefficients convert input uncertainties into output units. Covariances must be added when components are correlated. A component common to every point can dominate the uncertainty of each absolute value but cancel substantially in a difference between points. Conversely, treating correlated components as independent can overstate or understate the uncertainty.
Separate random scatter from systematic contributions
Random scatter changes from run to run and is often estimated from repeats. Systematic contributions shift results in a related direction, for example when the same calibration offset affects every measurement. Repeating the measurement reduces random uncertainty but does not automatically remove a common offset. A complete error bar can include both, provided the caption or methods explain the combination.
Rank #4
- An Introduction to Error Analysis
Do overlapping error bars mean two results are the same?
No. Overlap is neither a universal agreement test nor a universal difference test. Two bars may represent different quantities, such as SD in one series and SE in another, or different coverage levels. Even when both bars use the same convention, visual overlap does not account for sample size, the distribution of the estimates, or correlations between them.
Non-overlapping bars do not, by themselves, prove a statistically significant difference either. A formal conclusion requires a defined null hypothesis or measurement-comparison criterion, the relevant degrees of freedom or sampling model, and a stated confidence or coverage convention. In a measurement context, the practical question may be whether the difference is compatible with the combined uncertainty, not whether a p-value crosses an arbitrary threshold.
A defensible two-measurement calculation
- Write the estimates with their definitions: for example, x1 ± u1 and x2 ± u2, both as standard uncertainties or both under another explicitly stated convention.
- Compute the signed difference, d = x1 − x2.
- Propagate uncertainty with covariance: u(d) = √[u1² + u2² − 2 cov(x1, x2)].
- If the estimates are independent, the covariance term is zero and the expression reduces to √(u1² + u2²).
- Report the difference and its uncertainty, then state the comparison rule or coverage level used. Do not silently turn an expanded uncertainty into a standard uncertainty.
Example: independent measurements
Suppose one method gives 10.2 ± 0.3 units and another gives 9.7 ± 0.2 units, with both ± values identified as standard uncertainties and no correlation. The difference is 0.5 units. Its combined standard uncertainty is √(0.3² + 0.2²) = √0.13 ≈ 0.36 units. The difference is therefore about 1.4 combined standard uncertainties. That calculation shows the scale of the discrepancy; deciding whether it is statistically or scientifically significant still requires the stated distribution, degrees of freedom, and decision threshold.
Example: a shared calibration effect
Imagine two readings each contain a common calibration component of 0.08 units and an independent repeatability component of 0.06 units. Each absolute result has standard uncertainty √(0.08² + 0.06²) = 0.10 units. For their difference, the shared calibration offset cancels if it affects both readings in the same way. The difference uncertainty is then √(0.06² + 0.06²) ≈ 0.085 units, not √(0.10² + 0.10²) ≈ 0.141 units. The cancellation is valid only when the shared component and its correlation are justified by the measurement setup.
Independent reader supportYour contribution helps us test, update, and keep practical guides available for everyone.Asymmetric intervals and unusual plots
Asymmetric bars
Write asymmetric results as y+a−b, not as a single averaged error. Asymmetry can arise from a nonlinear conversion, a bounded parameter, a ratio, a logarithmic scale, or a confidence construction that is not symmetric around the estimate. Combining the upper and lower halves into one number can misrepresent the physically allowed range.
Horizontal and two-dimensional error bars
Horizontal bars may represent uncertainty in the independent variable, while vertical bars represent uncertainty in the dependent variable. If both are present, ordinary vertical-only comparisons are incomplete. Ellipses or covariance matrices show that the two coordinates are correlated; comparing each axis independently can miss the orientation of the joint uncertainty region.
What’s actually slowing this PC down?
Pick the symptom - the matching free tool is one click away.
Fitted curves
A shaded region around a fitted curve may be a confidence band for the estimated mean response, a prediction band for future observations, or a model envelope. Check whether parameter covariance, measurement noise, and calibration uncertainty were included. Do not assume that the band has the same interpretation as point-wise bars elsewhere in the figure.
Common mistakes and how to correct them
| Mistake | Why it fails | Correction |
|---|---|---|
| Calling every bar an error or standard deviation | The graphic label does not define the statistic. | Use the caption’s exact term and report the units and convention. |
| Using SD when the question is precision of a mean | SD measures individual scatter, not uncertainty of the average. | Use an appropriate SE or full uncertainty budget for the mean. |
| Using SE to describe the range of future readings | SE can be much smaller than individual variability. | Show SD, a prediction interval, or another interval suited to that question. |
| Assuming ±1 bar always means 68% | The 68% result requires an approximately normal distribution and reliable standard uncertainty. | State the conditions and use the specified coverage instead. |
| Comparing bars by eye | Overlap ignores definitions, sample sizes and covariance. | Calculate the difference and its combined uncertainty. |
| Adding uncertainty widths directly | Independent random components combine through variances, not simple addition. | Use a valid propagation formula and include covariance. |
| Ignoring systematic or shared effects | Repeats do not remove a common calibration or drift contribution. | Include Type B components and correlations in the budget. |
| Mixing coverage conventions | An expanded uncertainty is wider than a standard uncertainty by a coverage factor. | Convert to a common convention only when the factor and assumptions are known. |
| Reporting excessive digits | More decimal places than the uncertainty supports imply false precision. | Round the uncertainty sensibly and match the value’s decimal place. |
From raw measurements to a transparent plot
- Define the measurand. State exactly what quantity, operating condition, range and units the experiment targets.
- Preserve the raw observations. Record repeats, exclusions, environmental conditions, instrument settings and calibration identifiers.
- Estimate repeatability. Calculate the mean and an appropriate measure of spread, while checking for drift, outliers and dependence rather than deleting inconvenient points automatically.
- List Type A and Type B components. Include calibration, resolution, reference values, corrections, model inputs and any known environmental effects.
- Convert components to output units. Apply sensitivity coefficients or the measurement model before combining them.
- Record correlations. Mark components shared between runs, and identify paired measurements or common references.
- Choose the interval for the reader’s question. Use SD for observed spread, an uncertainty of a mean for estimated precision, or an expanded interval when a stated coverage is required.
- Plot and label completely. Put the definition, sample size, coverage or multiplier, and any exclusions in the caption or legend.
- Report the comparison separately. When two points are compared, provide their difference and propagated uncertainty rather than asking readers to infer a result from bar overlap.
A compact reporting checklist
- Central estimate and physical units are identified.
- Each error bar has a named definition.
- Sample size and, where relevant, degrees of freedom are given.
- Confidence level or coverage factor is stated.
- Asymmetric intervals are shown with separate upper and lower values.
- Type A and Type B components are distinguished.
- Systematic contributions and shared correlations are addressed.
- The comparison uses a difference and combined uncertainty, not visual overlap alone.
- Rounding does not imply more precision than the uncertainty supports.
What to do when the caption is silent
Ask the author or consult the methods before assigning a statistical meaning. You may safely say that the plotted points have intervals of a stated numerical width, but you should not translate those intervals into percentages or significance claims. If the underlying data are available, request the raw observations, the uncertainty budget, the coverage convention and the covariance information. Without them, a visual comparison can be descriptive but not a defensible measurement statement.
The practical decision rule
Read the definition before reading the size. Treat ±1 standard uncertainty as approximately 68% coverage only when the normality and reliability conditions are stated or justified. Distinguish spread from precision, include both Type A and Type B information, and compare two results through their difference with the correct covariance. Those steps turn an attractive graph into evidence that can be interpreted and reproduced.
Quick Recap
Product prices and availability are accurate as of the date/time indicated and are subject to change. Any price and availability information displayed on Amazon at the time of purchase will apply.
Free tools Windows power users keep installed
One-click scans. No signup required.
