Science Practice: Measurement and Uncertainty

Any instrument with marked lines, a buret, a ruler, a graduated cylinder, lets you read certain digits directly and then forces you to estimate one more. If a buret has lines every 0.1 mL, you read 23.4 confidently and then guess the next digit, say 23.47 mL, because your eye can judge roughly where between the lines the meniscus sits. That estimated digit isn't a mistake. It's the most honest digit in the number, because it admits exactly where your certainty runs out.

Calculations have to protect that honesty. When you add or subtract, line up decimal places and let the measurement with the fewest decimal places limit the answer, because you can't gain precision past the least precise decimal place. When you multiply or divide, count significant figures instead, and the input with the fewest significant figures limits the result, because that value carries the most relative uncertainty. Mixing up these two rules is the single biggest way results get reported wrong.

This connects directly to error analysis, the next practice. Significant figures communicate precision, the spread you'd see on repeated trials, while accuracy is closeness to a true value, and the two can disagree. A thermometer stuck two degrees high gives a systematic error: every reading shifts the same direction, consistent and traceable to the instrument. Random error comes from small unpredictable fluctuations, like estimating a meniscus slightly differently each time, and averages out over many trials. Recognizing which kind of error you're looking at tells you whether better technique or a better instrument is the fix.