General Chemistry I (CHEM-UA), Lab 2 | Source: Data Analysis Lab Handout (Geggier, NYU)
Tags: mean, average, range, standard deviation, normal distribution, percent error, uncertainty, data quality, summary statistics
Difficulty: Introductory | Prerequisites: Accuracy, Precision, and Experimental Errors; Significant Figures and Rounding Rules study notes.
Raw data from an experiment is just a list of numbers. Summary statistics turn that list into something you can reason about: a single best estimate (the mean), a measure of how spread out the data is (standard deviation), and a measure of how far the result is from the accepted value (percent error). These tools appear in every lab report from here on. If you skipped the notes on accuracy and precision, go back to those first, because the concepts of systematic and random error are what these statistics quantify.
The mean is your best single estimate of the true value. The standard deviation tells you how spread out your measurements are (precision). Percent error tells you how far your mean is from the accepted value (accuracy). Results are reported as mean ± uncertainty, where the uncertainty is the standard deviation rounded to 1 significant figure.
Mean (x̄)
The average of a series of measurements. Equal to the sum of all values divided by the number of measurements, N. The mean cannot be more precise than the original measurements (same number of SFs).
In simple terms, it is your best single guess at the true value.
Range
The difference between the largest and smallest values in a data set. A quick-and-dirty measure of spread, but it tells you nothing about how the data is distributed between those extremes.
Standard deviation (s)
A measure of how far the individual measurements are from the mean. A large standard deviation means the data is widely scattered (poor precision). A small one means the data clusters tightly (good precision). Uses N − 1 in the denominator for small data sets.
Think of it as: the typical distance a single measurement sits from the average.
Normal distribution
A bell-shaped curve that describes how measurements distribute around the mean when only random error is present. About 68% of all measurements fall within one standard deviation of the mean.
Percent error (% error)
A measure of accuracy. Compares the experimental result to the true (accepted) value. Can only be calculated when the true value is known.
In simple terms, it is the percentage by which you missed the bullseye.
Uncertainty
The standard deviation used to express the range within which the true value likely falls. Always rounded to 1 significant figure. Reported as mean ± uncertainty.
x̄ = (1/N) × Σxᵢ
Worked example (Table 1, density of 70% ethanol): (0.8614 + 0.8596 + 0.8679 + 0.8638) / 4 = 0.863175 → rounded to 0.8632 g/mL (4 SFs, matching the original measurements).
Range = largest value − smallest value.
For Table 1: 0.8679 − 0.8596 = 0.0083 g/mL. Simple to calculate but limited: it ignores how the intermediate values are distributed.
The sample standard deviation formula uses N − 1 in the denominator (not N) because one degree of freedom is consumed by calculating the mean.
Step-by-step procedure:
Compute the mean, x̄.
For each measurement, calculate the deviation: x̄ − xᵢ.
Square each deviation.
Sum all the squared deviations.
Divide by N − 1.
Take the square root.
Worked example (Table 1):
Trial | Density (g/mL) | x̄ − xᵢ | (x̄ − xᵢ)² |
|---|---|---|---|
1 | 0.8614 | −0.0018 | 3.24 x 10⁻⁶ |
2 | 0.8596 | −0.0036 | 1.296 x 10⁻⁵ |
3 | 0.8679 | +0.0047 | 2.209 x 10⁻⁵ |
4 | 0.8638 | +0.0006 | 3.6 x 10⁻⁷ |
Sum | 3.865 x 10⁻⁵ |
s = √(3.865 x 10⁻⁵ / 3) = √(0.00001288) = 0.003589 → rounded to 0.0036 g/mL.
If s rounds to zero, report it with 1 SF instead.
The standard deviation serves as the uncertainty. Round the uncertainty to 1 SF first, then round the mean to match.
For Table 1: 0.8632 ± 0.004 g/mL. But 0.004 means the value lies between roughly 0.859 and 0.867, so the trailing "2" in the mean is meaningless. The correct report is:
ρ(70% ethanol) = 0.863 ± 0.004 g/mL
The uncertainty of 0.004 tells the reader that about 68% of measurements would fall within ±0.004 of the mean.
% error = |true − experimental| / true × 100%
Worked example: CRC Handbook density of 70% ethanol = 0.8676 g/mL. Experimental mean = 0.863 g/mL.
% error = |0.8676 − 0.863| / 0.8676 × 100% = 0.5%
Percent error quantifies accuracy. It can only be calculated when a true (accepted) value is known.
Three groups measured the density of a metal (accepted value: 10.4 g/cm³):
Group A | Group B | Group C | |
|---|---|---|---|
Mean (g/cm³) | 11.06 | 10.3 | 10.33 |
s (g/cm³) | 0.03 | 0.2 | 0.06 |
% error | 6% | 1% | 0.7% |
Key takeaways:
Group A is the most precise (smallest s) but the least accurate (largest % error). This is the hallmark of systematic error: consistent readings that are consistently wrong.
Groups B and C have similar accuracy, but Group C is more precise.
Group B's larger standard deviation likely reflects lower-precision instrumentation (e.g. a 50 mL graduated cylinder rather than a 10 mL one).
In real experiments you usually do not know the true value, which makes systematic errors difficult to detect from the data alone.
Mean: x̄ = (1/N) × Σxᵢ
Standard deviation: s = √[ Σ(x̄ − xᵢ)² / (N − 1) ]
Percent error: % error = |true − experimental| / true × 100%
Reporting format: result = mean ± uncertainty (units), where uncertainty = s rounded to 1 SF, then the mean rounded to match.
"A small standard deviation means my data is accurate." Standard deviation measures precision (spread), not accuracy (closeness to the true value). Group A in Table 2 had the smallest s but the worst accuracy.
"I can use percent error to check my precision." Percent error measures accuracy, not precision. Standard deviation is the tool for precision.
"The mean should have more decimal places than my measurements because it is calculated." The mean cannot be more precise than the original measurements and should have the same number of SFs.
"The uncertainty should have lots of decimal places for accuracy." Always round uncertainty to exactly 1 significant figure. Then round the mean to the same decimal place.
⚠️ Worksheet II (Question 5) requires a full standard deviation calculation with 10 data points (caffeine in coffee). Practise the step-by-step procedure until it is automatic.
⚠️ Worksheet I Question 4 is a straightforward percent error problem. Do not forget SF rules when reporting the answer.
⚠️ You must report results in the format: mean ± uncertainty (units), with uncertainty rounded to 1 SF.
⚠️ Be able to interpret what standard deviation and percent error tell you about a data set (the Table 2 analysis pattern).
True or False: The standard deviation uses N − 1 in the denominator for small data sets. (True)
Fill in the blank: Uncertainties should always be rounded to __________ significant figure(s). (1)
True or False: Percent error can be calculated without knowing the accepted value. (False)
Fill in the blank: About __________% of measurements fall within one standard deviation of the mean in a normal distribution. (68)
True or False: A high percent error combined with a low standard deviation suggests systematic error. (True)
Q: A scientist measures the caffeine content of coffee in 10 trials (mg): 136, 145, 128, 147, 151, 138, 143, 129, 135, 141. Calculate the mean and report the result with uncertainty.
A: Mean = (136 + 145 + 128 + 147 + 151 + 138 + 143 + 129 + 135 + 141) / 10 = 1393 / 10 = 139.3 mg. For s: compute each (x̄ − xᵢ)², sum them, divide by 9, take the square root. The sum of squared deviations = 530.1. s = √(530.1/9) = √58.9 = 7.674... → rounded to 8 mg (1 SF). The mean is then rounded to match: 139 mg. Report as 139 ± 8 mg.
Q: Using the data from Table 2, explain why Group A's results are problematic despite having the smallest standard deviation.
A: Group A has the smallest s (0.03 g/cm³), meaning the measurements are very tightly clustered. But their mean (11.06 g/cm³) is far from the accepted value (10.4 g/cm³), giving a 6% error. High precision with low accuracy is the signature of a systematic error. Something was consistently wrong with their method.
Q: Calculate the standard deviation of density for an unknown liquid given masses of 4.89 g, 4.53 g, and 4.95 g, each with a volume of 5.00 mL.
A: Densities = 4.89/5.00 = 0.978, 4.53/5.00 = 0.906, 4.95/5.00 = 0.990 g/mL. Mean = 0.958 g/mL. Deviations: 0.020, −0.052, 0.032. Squared: 0.000400, 0.002704, 0.001024. Sum = 0.004128. s = √(0.004128/2) = √0.002064 = 0.04543 → 0.045 g/mL.
Q: Why does the standard deviation formula use N − 1 rather than N?
A: Because one degree of freedom is "used up" in calculating the mean from the same data set. Dividing by N − 1 (Bessel's correction) gives a better estimate of the true population standard deviation when working with small samples.
Mean and standard deviation will appear in every lab report this semester when you analyse experimental data. Percent error will be calculated whenever you compare your results to a literature value. The normal distribution concept reappears in more advanced statistics courses and is the theoretical foundation for confidence intervals and hypothesis testing. Understanding how to interpret s and % error together (as in the Table 2 analysis) is the core skill for the "Discussion" section of any lab report.
Related Terms / Search Tags
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