Measurement and Uncertainty

Understanding the Measuring Process¶

Measurement lies at the heart of our scientific understanding. Though perhaps overstated, this sentiment captures an essential truth - proper measurement forms the foundation of meaningful experimentation.

Let’s begin with a simple example to understand the fundamental nature of measurement. Imagine measuring the height of a coffee mug with a ruler marked in millimeters. You might report “87 mm,” but does this mean the mug is exactly 87.00000... mm tall? Of course not. What you’re really doing is determining that the height falls within some interval - perhaps between 86.5 mm and 87.5 mm.

Through this dual process of approaching from above and below, we identify an interval - the smallest range within which we’re confident the true value lies. This reveals measurement’s essential nature: we don’t determine exact values but rather intervals of possibility.

We must assess each situation individually rather than following oversimplified rules (like assuming uncertainty equals half the smallest scale division). A well-defined object under perfect conditions might allow precision well beyond the smallest marked division, while a poorly defined object might create uncertainty spanning several divisions.

Understanding Digital Readouts and Rounding¶

Digital instruments present their own interpretive challenges. When a digital multimeter displays “3.82 V,” what exactly does this mean? The answer depends on the instrument’s design.

However, some digital timers might operate differently, showing “10:15” for any time between exactly 10:15:00 and 10:15:59. Each instrument type requires understanding its specific operation.

This highlights a broader concept: rounding introduces its own form of uncertainty. When we write π = 3.14, we understand this isn’t exactly true. Rather, we mean the value lies between 3.135 and 3.145.

With modern calculators, it’s wise to maintain extra digits throughout calculations, rounding appropriately only at the final step. Similarly, statements like “measured to the nearest millimeter” inadequately convey measurement uncertainty, as they establish only minimum bounds for the measurement interval.

Absolute and Relative Uncertainty¶

Measurements should represent the range within which we believe the true value lies. For instance, we might determine a tabletop’s length lies between 152.7 cm and 153.1 cm. While this interval representation is entirely valid, we often restate it as 152.9 ± 0.2 cm.

This uncertainty value (±0.2 cm) represents the absolute uncertainty of our measurement. However, the significance of any uncertainty depends on the measurement’s magnitude. An uncertainty of ±0.2 cm would be:

Identifying Systematic Errors¶

The uncertainties discussed so far arise from natural limitations in measurement processes. However, another category - systematic errors - affects all measurements in a consistent way.

These systematic errors, particularly calibration errors, require vigilance. Always check instrument zeros before measurement and verify calibration when possible.

Calculating Uncertainty in Derived Quantities¶

Rarely does a single measurement complete our work. Usually, we need to calculate some quantity based on multiple measurements or apply mathematical operations to our measured values.

When calculating uncertainties in derived quantities, we will focus on finding the maximum possible uncertainty by considering the absolute values of all component uncertainties. This approach ensures we account for the worst-case scenario where all uncertainties combine to produce the largest possible error in our final result.

Uncertainty in Single-Variable Functions¶

Consider a measured quantity $x$ with uncertainty $\pm\delta x$, and a calculated result $z = f(x)$. The maximum possible uncertainty in $z$ is determined by considering how much $z$ could change when $x$ varies by $\pm\delta x$.

For example, if $z = \frac{x}{x^2+4}$:

Let’s examine several common function types:

Powers and Roots¶

This reveals an important principle: the relative uncertainty in the result equals the relative uncertainty in the measurement multiplied by the power. This applies to both positive powers (multiplication) and negative powers (division/roots).

Exponential Functions¶

Logarithmic Functions¶

Trigonometric Functions¶

Uncertainty in Multi-Variable Functions¶

When dealing with functions of multiple variables, we calculate the maximum possible uncertainty by taking the sum of the absolute values of all contributing uncertainties. This approach ensures we account for the worst possible case where all uncertainties combine to maximize the final uncertainty.

Sum and Difference of Variables¶

Products and Quotients¶

General Approach for Multi-Variable Functions¶

Complex Functions¶

For more complex functions, break them down into simpler components and apply the chain rule, always using absolute values to ensure maximum uncertainty:

Understanding Significant Figures: Purpose Over Rules¶

When working with measurements, significant figures serve a critical purpose that goes beyond mere rule-following. They communicate the quality and reliability of your measurements to others. While textbooks often present lengthy lists of rules about significant figures, it’s more valuable to understand their fundamental purpose.

At their core, significant figures represent the digits that are known with certainty, plus one additional digit that represents your best estimate. This approach emerges naturally from the measurement process itself.

Consider how you might record a measurement from a graduated cylinder. When the liquid level falls between markings, you don’t simply write down the nearest mark. Instead, you estimate to one digit beyond what the scale directly shows. That estimated digit—the last significant figure—carries valuable information about your measurement.

Rather than memorizing a complex set of rules about zeroes and calculations, focus first on the fundamental principle: significant figures reflect the precision of measurement. When you understand this purpose, many of the rules become intuitive rather than arbitrary.

# Let's write a simple function to estimate significant figures in a measurement
def count_sig_figs(measurement_str):
    """Estimate the number of significant figures in a measurement"""
    # Remove any units that might be present
    measurement_str = measurement_str.split()[0]

    # Handle scientific notation
    if 'e' in measurement_str.lower():
        base, exponent = measurement_str.lower().split('e')
        return count_sig_figs(base)

    # Count significant digits according to basic rules
    digits = ''.join(c for c in measurement_str if c.isdigit())
    if '.' in measurement_str:
        # With decimal point, trailing zeros are significant
        # Remove leading zeros
        digits = digits.lstrip('0')
        return len(digits)
    else:
        # Without decimal point, trailing zeros might not be significant
        # This is ambiguous without more context
        digits = digits.lstrip('0')
        # Remove trailing zeros as they're ambiguous
        digits = digits.rstrip('0')
        return len(digits)

# Test with some examples
examples = ["12.34", "0.0056", "1200", "1200.0", "0.1200"]
for example in examples:
    print(f"Measurement: {example}, Significant figures: {count_sig_figs(example)}")

When propagating significant figures through calculations, focus on mastering the multiplication rule first:

• The result of multiplication or division should have the same number of significant figures as the measurement with the fewest significant figures.

Other rules for addition, logarithms, and special functions become easier to learn once you’ve established this foundation.

A Practical Approach to Zeroes¶

Zeroes often cause the most confusion when determining significant figures. Instead of memorizing rules, consider where the number came from:

• A measurement of 4.70 mL from a graduated cylinder has three significant figures because you estimated the last digit (confirming zero tenths)
• A measurement of 4.7 mL has two significant figures, indicating you didn’t estimate beyond the tenths place
• A measurement of 470 mL is ambiguous without context—it could have two or three significant figures

When teaching or learning significant figures, focusing on their purpose—communicating measurement quality—provides a more meaningful framework than simply memorizing rules. This understanding helps you make appropriate judgments when recording and working with experimental data.

Calculations often produce more digits than are justified by our measurement precision. We must quote results sensibly.

Glossary¶

Problems¶