Plots, error bars and uncertainty ranges

A plotted point is an estimate of a defined quantity

Before reading a trend, name the horizontal and vertical quantities, their units and the scale of each axis. A straight line on logarithmic axes can describe a different relation from a straight line on linear axes. Binning or normalization can also change what the vertical number means.

A point often marks an estimate rather than an exactly known value. A bar beside it may show a standard uncertainty, a sample spread, a confidence interval, a probability interval or a limit. These are different objects. The drawing’s appearance cannot supply the missing definition.

Measurement error is the realized difference between an estimate and the relevant true value. That difference is generally not known. Uncertainty describes the incomplete knowledge represented by an analysis. Calling an uncertainty 1 does not say that the actual error has been measured to be exactly +1 or −1.

Repeated detector readings may reveal random scatter, while calibration or modelling issues require other information. A small scatter alone does not establish that every source of uncertainty is small. A graph should state what was included and what the plotted range means.

One standard uncertainty need not mean 68 percent

Choose a hypothetical quantity q in arbitrary units, with estimate q₀ = 10. Assign a uniform probability model from 10 − √3 to 10 + √3, approximately 8.268 to 11.732. Uniform means equal probability per unit interval throughout this support and zero outside it.

For this particular uniform model, the standard deviation is the half-width divided by √3. Its half-width is √3, so its standard uncertainty is 1. The interval one standard uncertainty on either side of the estimate is therefore [9, 11].

The probability in an interval inside a uniform distribution is its width divided by the full support width. The interval [9, 11] has width 2, while the full support has width 2√3. Its coverage is 2/(2√3) = 1/√3, about 57.7 percent.

This counterexample shows why the familiar approximately 68 percent statement requires the appropriate normal-distribution conditions. A standard uncertainty is a measure of spread; its value alone does not determine a universal coverage percentage. No real cosmological error distribution has been asserted here.

A coverage interval needs its construction

For the chosen uniform model, a central 95 percent interval keeps 95 percent of the full width and removes equal tails. Its half-width is 0.95√3 ≈ 1.645, giving about [8.355, 11.645]. This is wider than [9, 11], as required to contain more of the assigned probability.

That is a probability statement within an explicitly assigned model. A frequentist confidence interval has a different construction and interpretation: a specified repeated-sampling procedure covers the fixed target value in a stated fraction of repetitions, under its assumptions. One realized interval should not be relabelled an unconditional probability for a fixed parameter without supplying a probability model for it.

A Bayesian credible interval also has its own conditional meaning, using a posterior distribution and stated modelling choices. The detailed comparison comes later in the learning plan. For now, preserve the label the analysis supplies instead of treating all intervals as interchangeable.

An expanded uncertainty can be reported as a stated multiple of a standard uncertainty. The corresponding coverage depends on the distribution and estimation conditions. A multiplier of two is not a universal proof of exactly 95 percent probability.

More points do not always supply more independent information

Nearby plotted estimates can share observations, calibration errors or processing steps. Their uncertainties may therefore be correlated: an error source that moves one estimate can also move another. Separate bars do not reveal those relationships unless the analysis reports them.

The uncertainty in a difference or a combined estimate depends on those correlations. Adding two quoted errors as if they were independent can be wrong even if each individual bar is accurately drawn. A later lesson in this chapter works through a concrete difference calculation.

Nor does bar overlap alone define a complete consistency test. One must know the target quantities, units, distributional assumptions and shared error structure. A point outside a one-standard-uncertainty bar is not automatically a discovery or an impossible measurement.

In cosmology, instrumental noise, foreground modelling and the finite sky can contribute different limits. Which term belongs in a particular error bar is an analysis question. The figure legend and method description carry information that a central value cannot replace.

Read a range as part of a claim

A useful comparison names the estimate, target quantity, unit, interval convention and assumptions. Ask which changes would alter the result: more independent measurements, better calibration, a different foreground model or a changed physical model. These changes address different sources of uncertainty.

For a first reading, keep the uniform example in mind. The exact same standard uncertainty of 1 can have a coverage different from the familiar normal case. When a plot omits its convention, the honest next step is to find that definition rather than infer it from the bar’s length.

Worked reasoning

Try the reasoning

1. Does a standard uncertainty of 1 establish that the realized measurement error is exactly 1?

   Show the answer and reasoning

   No.

   Uncertainty summarizes incomplete knowledge or spread under stated assumptions; the realized signed error is generally unknown.

2. May an unlabeled error bar be assumed to be a 68 percent interval?

   Show the answer and reasoning

   No. Its convention must be supplied.

   It might represent a standard deviation, another interval, a bound or a different quantity entirely.

3. Is the central 95 percent interval wider or narrower than [9, 11] in the chosen uniform example?

   Show the answer and reasoning

   Wider.

   [9, 11] contains only about 57.7 percent; the central 95 percent interval is approximately [8.355, 11.645].

What to carry forward

• An uncertainty bar needs a quantity, units and a stated convention.
• Standard deviation does not determine a universal coverage percentage without a distribution.
• Correlations and modelling assumptions matter when comparing or combining estimates.