Off by 0.08: is that a lot?
A lab group measures the density of an aluminum block and gets 2.62 g/cm³. The accepted value for aluminum is 2.70 g/cm³. Being off by 0.08 g/cm³: is that good or bad? If the sample were gold, with a density of 19.3 g/cm³, the same 0.08 would hardly matter. To judge a result, you need to know not just how far off it is, but how large that difference is compared with the value itself.
Error and percent error
The trusted value of a quantity is the accepted value真实值（公认值）; the value you measure is the experimental value测量值.
The error has units and a sign: positive means too high, negative means too low. Percent error百分误差 uses the absolute value, so it’s always positive, and you state separately whether the result was high or low. For the aluminum block:
The result is 3.0% too low. Percent error lets you compare different measurements: missing aluminum by 0.08 is 3.0%, but missing gold by 0.08 is only about 0.4%.
Two conventions. Chinese textbooks, and some American ones, use a relative error相对误差 that keeps the sign: (2.62 − 2.70) ÷ 2.70 × 100% = −3.0%, where the minus sign itself says “too low.” Watch for which convention a problem uses.
Repeated trials: average and range
Any single measurement can be off by chance, so experiments are usually repeated and the results averaged. The closer the trials are to one another, the more precise the measurement. The range极差 (highest − lowest) is a quick measure of precision: the smaller the range, the more precise.
Keep lesson 1.1.5’s distinction in mind: percent error compares the average with the accepted value, so it measures accuracy; the range compares the trials with each other, so it measures precision. Averaging more trials reduces random error but does nothing to systematic error. If a balance isn’t zeroed and every reading is 1.00 g too high, every trial is too high by the same amount, no matter how many you do.
Error analysis: too high or too low?
Lab questions often ask whether a mistake makes a result too high or too low. Follow the formula:
- Write the formula, such as .
- Decide which measurement the mistake affects, and whether it makes that measurement too high or too low.
- Look at where that quantity sits in the formula: in the numerator, the result moves the same way; in the denominator, it moves the opposite way.
When you read a graduated cylinder, looking down from above the water line gives a reading that’s too high, and looking up from below gives one that’s too low.
Choose an error source, predict whether it makes the measured density too high or too low, then run a few trials to check:
Find the mass m of an aluminum block on a balance. Read the water level V₁ in a graduated cylinder, add the block, and read V₂. Density ρ = m / (V₂ − V₁). The accepted value for aluminum is 2.70 g/cm³.
1. Choose the conditions:
2. Predict the measured density:
| Trial | m (g) | V₁ (mL) | V₂ (mL) | V (cm³) | ρ (g/cm³) |
|---|
Example 1: Calculating percent error
Three trials give the density of an aluminum block as 2.62, 2.65, and 2.59 g/cm³. The accepted value is 2.70 g/cm³. Find the average, the range, and the percent error.
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Average. Add the three results and divide by 3:
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Range (a measure of precision). Largest minus smallest: 2.65 − 2.59 = 0.06 g/cm³. The trials are close together, so the measurement is fairly precise.
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Percent error (a measure of accuracy). Percent error = |measured − accepted| ÷ accepted × 100%, using the average as the measured value:
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Direction. The result is 3.0% too low; written as a signed relative error, that’s −3.0%.
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Conclusion. All three trials are low and close together: precise but not accurate, which points to a systematic error (an unzeroed balance, say). More trials can’t remove a systematic error.
Example 2: Which direction?
A metal block’s volume is measured by water displacement, and the student reads V₂, after adding the block, by looking up at the water line from below. Is the calculated density too high or too low?
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Write the formula and find the measured quantities.
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Which measurement is off, and which way? Looking up from below makes the reading lower than the real water level, so V₂ is too low.
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Carry it to the volume. V₁ is unaffected, so V = V₂ − V₁ is too low as well.
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Carry it to the result. V is in the denominator; a denominator that’s too low makes ρ too high.
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The method. Error analysis has three steps: which measurement is off → where it sits in the formula (numerator or denominator, added or subtracted) → whether the result comes out high or low.
- The balance isn't zeroed and reads 0.50 g when empty
The mass m is too high; m is in the numerator, so ρ is too high.
- Air bubbles cling to the metal block
The bubbles displace water too, so the volume V is too high; V is in the denominator, so ρ is too low.
- Some water splashes out when the block is dropped in
V₂ is too low, so V is too low and ρ is too high.
- V₂ is read from above the water line
Reading from above gives a high reading, so V₂ and V are too high and ρ is too low.
- The volume is measured first, and the wet block is then weighed
The water clinging to the block is weighed too, so m is too high and ρ is too high.
- V₁ is read from above the water line, before the block goes in
V₁ is too high, so V = V₂ − V₁ is too low and ρ is too high.
- Both V₁ and V₂ are read from above, by the same amount
Both readings are high by the same amount, so the difference V is barely affected.
- A larger block of the same metal is used
Density is a property of the substance, not of the block's size; a bigger block actually makes the percent error smaller.
- The cylinder starts with more water (but not enough to overflow)
V = V₂ − V₁, so the starting amount cancels out.
Common mistakes
Mistake 1: “Error means the experiment was done wrong.” Every measurement has some error. Blunders, like misreading a scale or a calculation slip, should be avoided; error can be reduced but never eliminated.
Mistake 2: “If my trials agree closely, my percent error must be small.” Close agreement shows precision, not accuracy. With a systematic error, every trial can miss the accepted value by the same amount.
Mistake 3: “Averaging more trials removes any error.” Averaging reduces only random error. Systematic errors, such as an unzeroed balance or always reading from above, have to be found and fixed.
Quick check
A metal's density is measured as 8.75 g/cm³; the accepted value is 8.96 g/cm³. What is the percent error?
|8.75 − 8.96| ÷ 8.96 × 100% = 2.34% ≈ 2.3%, and the result is too low. A signed relative error would be written −2.3%; 2.4% comes from dividing by the experimental value instead of the accepted one.
Which of these errors can be reduced by averaging more trials?
Variation in the estimated digit goes up and down at random, so averaging reduces it. The other three always push the result the same way: they're systematic errors.
Summary
- Error = experimental − accepted. Percent error = |error| ÷ accepted × 100%, always positive, with the direction stated separately; a signed relative error (used in Chinese textbooks) keeps the sign.
- How far the average is from the accepted value reflects accuracy; the range reflects precision. Averaging reduces random error but not systematic error.
- To find an error’s direction, follow the formula: a quantity in the numerator moves the result the same way; one in the denominator moves it the opposite way.
On to AP (for later)
Almost every year, AP lab questions ask whether a particular mistake makes a calculated result too high, too low, or unchanged, and why. Answer the way Example 2 does: name the measurement that’s affected and its direction, then carry it through the formula to the result. AP usually uses percent error as an absolute value and asks for the direction separately.