PHY 113 · 114 · 121 · 123 · 181
Every measurement you make has an uncertainty. Understanding that uncertainty is what separates a result from a guess.
Each bar carries a ±3% margin of error
⚠ The difference (3%) equals the margin of error — this "surge" is statistically meaningless.
Why It Matters
A poll shows Harris at $49\% \pm 3\%$ and Trump at $46\% \pm 3\%$. The headline screams a "surge" — but the two numbers overlap completely within their uncertainties.
Without understanding error, you cannot tell signal from noise. This applies in politics, climate science, medicine — and every physics lab you will ever do.
Why It Matters
Global average temperature in 2024 was reported as $+1.60^\text{o}\text{C}$ above pre-industrial levels — about $0.12^\text{o}\text{C}$ warmer than the previous record set in 2023.
But the instrumental uncertainty in global temperature averages is roughly $\pm0.10^\text{o}\text{C}$. That $0.12^\text{o}\text{C}$ difference barely exceeds the measurement error.
The record is likely real — but quantifying uncertainty is precisely how science makes that claim responsibly.
Why It Matters
Student A measures \(g = 9.85 \pm 0.20 \ \text{m/s}^2\). Student B measures \(g = 9.75 \pm 0.15 \ \text{m/s}^2\).
The central values differ by 0.10 m/s² — but their uncertainty ranges overlap substantially. Both are consistent with the accepted value \(g = 9.81 \ \text{m/s}^2\).
Knowing the error is what lets you make that judgment.
Measurement Basics
When you measure the same pencil $10$ times with a ruler, you don't get $12.7$ cm every time. You get $12.5$, $12.8$, $12.6$, $12.9$, etc.
This scatter isn't "bad" — it's data. It tells you about the limits of your measurement tool and technique.
Random error: ruler markings are discrete, your eye position varies, you press differently each time.
Systematic error: a bent ruler always reads high; a cold ruler contracts; parallax bias from one direction.
Both shape your result. Understanding which is which helps you improve.
Accuracy vs. Precision
Accuracy tells you how close your measurements are to the true value. An accurate measurement is right on target.
Precision tells you how reproducible your measurements are. A precise measurement is tightly clustered.
You can be precise but inaccurate (a broken clock is precise but usually wrong). You can be accurate but imprecise (lucky guesses). The goal is both: tight clusters at the true value.
Distributions & Mean
Measure a pendulum's period $N=50$ times. Each measurement has random error. Plot all values as a histogram — they cluster around $2.018$ seconds in a bell-shaped (normal) distribution.
The population mean $μ$ is the true period (unknown). The sample mean $\bar{x}$ is what you calculate from your N measurements:
$\displaystyle \bar{x} = \frac{1}{N} \sum_{i=1}^{N} x_i$
The more measurements you take, the closer $\bar{x}$ gets to $\mu$. The bell curve shape is universal: most measurements cluster near the true value, with fewer far away. This fact is known as the central limit theorem
Standard Deviation
The population standard deviation, $\sigma$, is how spread out the population is. $\sigma^2$ is called the population variance. The spread in your data reflects the spread in the population. Sample variance is computed:
$\displaystyle \bar{s}^2 = \frac{1}{N-1} \sum_{i=1}^{N} (x_i - \bar{x})^2$
Data with a larger value of $\bar{s}$ is more spread out than data with a small value of $\bar{s}$. The more measurements you take, the closer $\bar{s}$ gets to $\sigma$.
Poisson Statistics
When you count discrete events (radioactive decays, photons, particles), the results follow a Poisson distribution with a unique property:
\(\sigma = \sqrt{\lambda} = \sqrt{N}\)
The standard deviation equals the square root of the count itself. If you observe 100 decays, σ ≈ 10. If you observe 10,000, σ ≈ 100.
This is not a coincidence — it emerges naturally from the random process of counting independent events. Poisson statistics apply whenever you're counting things with random, constant rates.
Error in the Mean
Each data sample will have its own mean, $\bar{x}$ and standard deviation, $\bar{s}_x$. When you combine multiple samples, you will get a distribution of sample means.
The standard error of the mean (SEM) is:
$\bar{s}_{\bar{x}} = \frac{\sigma}{\sqrt{N}}$
Each time you double N, the error shrinks by √2. Taking 10 measurements reduces error by √10 ≈ 3.16×. This is why experiments repeat measurements: averaging is powerful.
Error Propagation
Once you've measured something with uncertainty, what happens when you use that measurement in a formula?
For a small uncertainty δx in your measurement, the uncertainty in the result is:
\(\sigma_f \approx \left| \frac{df}{dx} \right| \sigma_x\)
The derivative amplifies or dampens the input error. If the derivative is small, errors shrink. If it's large, errors grow.
This is why careful measurement at the right scale matters: a small error in radius becomes a larger relative error in area, because area depends on r².
Summary
This is your toolkit for understanding and quantifying measurement uncertainty.
Always ask: How big is my error? Does my measurement answer the question? Do my results agree with others?
Error analysis turns raw data into reliable conclusions. When you report a result, report its uncertainty too. That's how science works.
Ready to measure?