# Error

This page will support you in satisfying Writing Learning Outcome:

ANALYSIS - Analyze lab data by quantifying error.

## Learning Objectives

You should be able to

Define systematic and random error

Calculate the systematic error (aka bias) in a sample and explain its source

Calculate the random error (aka uncertainty) in a sample and recommend ways to reduce it

Differentiate systematic and random error

Present error in both absolute (as a quantity with units) and relative (as a percentage) terms

## What is Error?

Error is a difference between an expected value and a measured value and is categorized as either systematic (meaning it is repeatable) or random (not following a pattern). From the perspective of the statistics we have discussed previously (see Figure 1), a difference between a mean and expected value is a systematic error. The standard deviation is a measure of the random error.

Systematic errors can be attributed to problems in calibration or test configuration that can often be addressed or explained. They can also be the result of a bias introduced in a published value to ensure safety, as when the design strength of a material is specified at the low end of the distribution of tested strengths.

Random errors are often associated with the precision of instruments or operators (humans!) used in measurement and can be addressed only by improving the precision of equipment and standardization of procedures.

Figure 1. Depictions of accuracy and precision and relationships to statistical measures of mean and standard deviation.

## Why Does the Technical Audience Value Error Analysis?

Quantifying the error in reported values provides an indication of the precision of the result. This is conveyed commonly by correctly reporting significant figures. For example, a value of 5.3 mm indicates a precision of ± 0.1 mm (± 2%). However, a rigorous error analysis might show that the uncertainty is actually ± 0.7 mm (± 13%), which significantly impacts the confidence the audience might have in the result. If an error analysis is not provided, the audience will likely take the results at face value, or worse, question the work for lack of rigor.

## How is an Error Analysis Performed?

An error analysis can be conducted on either univariate or bivariate data. For univariate data, the analysis is simple:

Calculate the differences of a series of measured values from a single expected value.

Calculate the average of these differences. This is the systematic error, or bias, and it is either greater than or less than the expected value. Its sign is important.

Calculate the standard deviation of these differences. This quantifies the random error, or uncertainty, and it occurs on either side of the average measured value. Two standard deviations capture 95% of the likely error. Three standard deviations capture 99.7% of the likely error.

For bivariate data, the analysis is similar, but rather than comparing to a single expected value, you are comparing data to expected values estimated by a trendline.

This is all better explained with an example:

## What Expectations Does the Technical Audience Have for an Error Analysis?

Ensure accuracy of your procedure and results.

Report both the bias (or systematic error) and the uncertainty (or random error) in both absolute (with units) and relative (as a percentage) terms.

Describe a likely source or sources of the bias.

Describe the reasons for the random error.

Describe improvements to a testing procedure to reduce error.

## Common Mistakes

Error is not addressed at all. If this is the case, differences between values can be interpreted as significant, when in fact they are just within the bounds of random error.

Values without error bounds are presented with excessive and unrealistic levels of precision (e.g. 5.2343 mm). This gives a reader the impression that you have a capacity for precise measurement that, in fact, you do not.

Error analysis is not conducted; results stand alone without any discussion of bias or uncertainty.