AP Chemistry Percent Error

Calculating percent error, telling accuracy from precision, and reasoning about how a lab mistake changes a result, in four worked examples.

Lab-based free-response questions often end with an error-analysis prompt: compare your value with the accepted one, say whether your replicates agree, or explain how a specific mistake would change the calculated result. The arithmetic is short, and the reasoning about direction is what earns the points. This page covers both. It sits next to the calculation skills in calorimetry and significant figures, which produce the numbers being compared.

Percent Error, Accuracy, and Precision Defined

Percent error = |experimental − accepted| ÷ accepted × 100.
Accuracy: how close a result is to the accepted value.
Precision: how close repeated measurements are to one another.
Systematic error shifts every result the same way and hurts accuracy; random error scatters results and hurts precision.

The chart shows four titration trials for three different sets of results, plotted against an accepted concentration of 0.1000 M. Tight clustering means good precision; sitting on the green line means good accuracy. The two properties are independent, which is the point most questions test.

Worked Example 1: Calculating Percent Error From an Experimental and an Accepted Value

A student measures the heat of neutralization as −52.1 kJ/mol; the accepted value is −55.8 kJ/mol. A second student finds the density of aluminum to be 2.64 g/cm³, against an accepted 2.70 g/cm³. Find each percent error.

Neutralization: |−52.1 − (−55.8)| = 3.7, and 3.7 ÷ 55.8 × 100 = 6.6%.
Aluminum: |2.64 − 2.70| = 0.06, and 0.06 ÷ 2.70 × 100 = 2.2%.

Divide by the accepted value, not the experimental one, and give the answer to the number of significant figures the data support. The first value is lower in magnitude, which is what heat escaping from the cup would cause; naming the direction, as in “6.6% low,” is usually worth saying. A percent error compares a measurement with a reference; it is a different calculation from the percent yield in limiting reactant and percent yield.

Worked Example 2: Judging Accuracy and Precision From Replicate Titrations

The accepted concentration of a standard NaOH solution is 0.1000 M. Three students each run four titrations (0.0998, 0.1001, 0.1000, 0.1002 M; 0.1011, 0.1009, 0.1012, 0.1010 M; 0.0955, 0.1048, 0.0972, 0.1025 M). Describe the accuracy and precision of each set.

Set 1: mean 0.1000 M, range 0.0004 M. The trials agree and the mean matches: accurate and precise.
Set 2: mean 0.1011 M, range 0.0003 M. The trials agree, but the mean is about 1% high: precise but not accurate, which points to a systematic error, such as a consistently over-read buret.
Set 3: mean 0.1000 M, but the trials range from 0.0955 to 0.1048 M, or about 9% of the value: the average is accurate, but the results are not precise, which points to random error.

The last case shows why accuracy of the mean can hide poor technique: any single trial could be off by 4 to 5%. When trials are the data for a concentration, the same replicates feed the pH and equilibrium calculations in titration curve pH calculations.

Worked Example 3: Predicting How a Lab Mistake Changes the Result

A student heats 2.500 g of CuSO₄·5H₂O (36.08% water by mass) to find its percent water, but stops heating too early and the solid loses only 0.812 g. Find the percent water found and the percent error, and say how the mistake affected the result.

Percent water found: 0.812 ÷ 2.500 × 100 = 32.5%.
Percent error: (36.08 − 32.48) ÷ 36.08 × 100 = 10.0% low.
Reason: some water was still in the solid, so the measured mass loss was too small and the calculated percent water too low.

The reasoning pattern is the same for any error: say which measured quantity is too high or too low, and carry that through the formula to the final answer. For example, if the sample were not fully dried before weighing, the extra mass of water would make a percent-water result too high. The 36.08% reference value comes from the formula mass in molar mass.

Worked Example 4: Classifying Errors as Systematic or Random

Classify each error as systematic or random, and say whether it mainly reduces accuracy or precision.

(a) A balance that reads 0.05 g too high on every mass: systematic; it reduces accuracy and not precision, since every reading shifts equally.
(b) Reading a buret meniscus slightly differently each time: random; it reduces precision.
(c) A thermometer that reads 1.0°C high at every temperature: systematic; it affects accuracy. A temperature change is unaffected if the offset is constant.
(d) Wind drafts that cause the mass on a sensitive balance to fluctuate: random.

Averaging more trials reduces random error because the highs and lows cancel, but it does nothing for systematic error, which is the same in every trial; that has to be found and corrected. This is why a result can be repeatable and still wrong. In part (c), the offset cancels in a difference, which links to the way a change in temperature is used in temperature conversions and calorimetry.

Five Percent Error and Accuracy Practice Questions With Answers

A student finds the boiling point of water to be 98.5°C; the accepted value is 100.0°C. What is the percent error?
Show answer
|98.5 − 100.0| ÷ 100.0 × 100 = 1.5%.
Three trials give 12.1 g, 12.2 g, and 12.1 g. The accepted mass is 15.0 g. Describe the result.
Show answer
Precise, since the trials agree, but not accurate, since they are about 19% low. This suggests a systematic error.
Would adding extra trials to the data in question 2 fix the problem? Why or why not?
Show answer
No. More trials reduce random scatter, but a systematic error is the same in every trial, so the average remains off.
A hydrate sample is weighed while still slightly damp. How does that affect the calculated percent water?
Show answer
It is too high, because the extra water in the starting mass is lost on heating and counted as water of hydration.
Can an average be accurate even when the individual measurements are imprecise?
Show answer
Yes. If the errors are random and spread evenly above and below the accepted value, the mean can land close to it, although each single trial is unreliable.

Common Percent Error and Error Analysis Mistakes

Dividing by the experimental value. Divide by the accepted value.
Forgetting the absolute value when the question wants a positive percent error, or forgetting to say high or low when direction is asked.
Mixing up accuracy and precision. Accuracy compares to the accepted value; precision compares trials to each other.
Blaming “human error.” A specific error with a direction, such as “some water remained, so the mass loss was too small,” is what earns credit.
Saying more trials fix everything. They reduce random error only.
Reporting too many digits. A percent error does not need more significant figures than the data support.

Error-analysis answers are short but need precise wording. To see how a stronger lab section would shift your overall estimate, the AP Chem Score Calculator turns section scores into a 1–5 range.

Frequently Asked Questions

What is the formula for percent error?

Percent error = |experimental value − accepted value| ÷ accepted value × 100. The absolute value makes it positive; if you keep the sign, the sign shows whether your result was too high or too low.

What is the difference between accuracy and precision?

Accuracy is how close a measurement or average is to the accepted value. Precision is how close repeated measurements are to each other. A result can be precise without being accurate if there is a consistent error.

What is the difference between a systematic and a random error?

A systematic error pushes every measurement the same way, such as a balance that always reads high, and it lowers accuracy. A random error scatters measurements unpredictably, such as inconsistent reading of a meniscus, and it lowers precision. Averaging more trials reduces random error but not systematic error.

Can percent error be negative?

The usual definition uses the absolute value, so it is never negative. Some courses keep the sign to show direction. Check what the question asks for and state whether your result was high or low.

Is percent error the same as percent yield?

No. Percent yield compares the actual amount of product with the theoretical amount, as in a limiting reactant problem. Percent error compares a measured value with an accepted value, and it measures how far off the measurement was.

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