Identifying Sources of Error - ACT Science

1. What is observer bias in data collection?

   Measurements influenced by the researcher's expectations. Researcher's expectations influence data interpretation.

2. What is a transcription error in recording results?

   Copying or entering data incorrectly from source to table. Writing down wrong numbers from original data.

3. What is the definition of precision in measurement?

   Closeness of repeated measurements to each other. Repeatability, not necessarily correctness.

4. Identify the error source if a table lists $0.52$ but the lab notebook shows $0.25$ for the same trial.

   Transcription/data entry error. Numbers don't match between original and final records.

5. What is a common source of random error in timing a reaction with a handheld stopwatch?

   Human reaction time variability. Delay between seeing event and pressing button.

6. Find the error in the slope calculation: $m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{2 - 2}{3 - 1} = 1$.

   Correct: $m = 0$. Equal $y$-values give zero rise: $\frac{0}{2} = 0$.

7. Which error type results from inconsistent data entry?

   Human error. Inconsistent recording introduces variability into datasets.

8. What is the definition of a random error in an experiment?

   Unpredictable variation that causes scatter around a true value. Reduces precision but doesn't bias the average.

9. Identify the error in the inequality: $-3 < -2$ implies $3 > 2$.

   No error; implication is correct. Multiplying by $-1$ reverses inequality direction correctly.

10. What type of error is a miscalculation due to human oversight?

    Human error. These mistakes result from carelessness or inadequate attention.

11. Identify the error source when a thermometer consistently reads $2\,^{\circ}\text{C}$ too high for all samples.

    Calibration error (systematic bias). Consistent offset indicates uncalibrated instrument.

12. Find the error in the multiplication: $6 \times 7 = 49$.

    Correct: $6 \times 7 = 42$. Basic multiplication: $6 \times 7 = 42$, not 49.

13. Identify the error source if lengths are recorded only to the nearest $1\,\text{cm}$ using a coarse ruler.

    Instrument resolution limitation (rounding uncertainty). Coarse scale limits measurement precision.

14. Identify the mistake in the derivative: $\frac{d}{dx}(x^3) = 2x^2$.

    Correct: $\frac{d}{dx}(x^3) = 3x^2$. Power rule: bring down exponent and subtract 1 from power.

15. What error occurs when measurements scatter around a true value?

    Random error. Multiple measurements fluctuate randomly around the true value.

16. Identify the likely source of error if light intensity changes while testing plant growth vs. fertilizer type.

    Confounding variable (light intensity not controlled). Two variables change together, obscuring causation.

17. Identify the error in the unit conversion: $1000 \text{ g} = 0.1 \text{ kg}$.

    Correct: $1000 \text{ g} = 1 \text{ kg}$. The decimal point was misplaced; 1000 g equals 1 kg, not 0.1 kg.

18. Which error type occurs if a scale is not zeroed before measuring?

    Systematic error. Unzeroed scales add a constant offset to all measurements.

19. What type of error is caused by random fluctuations in measurements?

    Random error. These errors vary unpredictably between measurements.

20. Identify the error in the equation: $3x + 2 = 11$ solved as $x = \frac{11}{3} - 2$.

    Correct: $x = \frac{11 - 2}{3}$. The subtraction order was incorrect; subtract 2 first, then divide by 3.

21. What is an outlier in a data set?

    A value far from the overall pattern of the other data. Unusually different from the general trend.

22. Find the error in the percentage calculation: \frac{50}{200} \times 100 = 50\text{%}.

    Correct: 25\text{%}. $\frac{50}{200} = 0.25 = 25\%$, not 50%.

23. Identify the error: Using $1.0 \times 10^3$ instead of $1.0 \times 10^6$.

    Error: Incorrect order of magnitude. Wrong power of 10 changes value by factor of 1000.

24. Identify the error in the formula: Volume of sphere $= \frac{4}{3}\text{π}r^3$

    No error; formula is correct. The sphere volume formula is mathematically correct.

25. Which type of error is indicated when measurements are widely scattered around the accepted value?

    Random error (low precision, potentially accurate on average). Scatter without consistent bias indicates random variation.

26. Identify the error in the reaction: $C_6H_{12}O_6 + 6O_2 \rightarrow 6CO_2 + 6H_2O$.

    No error; equation is balanced. Atom counts are balanced: 6C, 12H, 18O on each side.

27. What error occurs when ignoring range in data comparison?

    Error: Misleading conclusions. Range shows data spread; ignoring it hides variability.

28. Identify the error: Confusing precision with accuracy.

    Error: Precision is not accuracy. Precision is consistency; accuracy is correctness.

29. Find and correct the error in the formula: Area of circle $= 2\text{π}r^2$.

    Correct: Area $= \text{π}r^2$. Circle area uses $\pi r^2$, not $2\pi r^2$ which is circumference times radius.

30. Find the error in the function evaluation: $f(x) = x^2 - 3x$, $f(2) = 4 - 3 = 1$.

    Correct: $f(2) = 4 - 6 = -2$. Missed multiplying by $x$ coefficient: $f(2) = 4 - 3(2) = -2$.

31. Identify the error source if a graph's $y$-axis is labeled $\text{mL}$ but the data were measured in $\text{L}$.

    Unit mismatch (systematic scaling error). Scale doesn't match actual measurement units.

32. What error arises from extrapolating data beyond the range?

    Error: Unreliable predictions. Patterns may not continue outside observed range.

33. Identify the error: Misreading $2 \times 10^{-3}$ as $2 \times 10^{3}$.

    Error: Incorrect scientific notation. Sign error changes value by factor of million.

34. What is the definition of a systematic error in an experiment?

    A consistent bias that shifts results in one direction. Creates reproducible deviation from true values.

35. What error arises from using a small sample size?

    Increased variability. Small samples have less reliable estimates.

36. Identify the error source if a researcher rates outcomes more favorably for the treatment group they support.

    Observer bias (lack of blinding). Unconscious bias toward preferred hypothesis.

37. What is a placebo effect as a source of experimental error?

    A response caused by expectation rather than the treatment. Mind creates improvement without actual treatment effect.

38. What is the error source if temperature is recorded in $^{\circ}\text{C}$ but treated as K in calculations?

    Incorrect unit/scale conversion (systematic error). Wrong temperature scale affects all calculations.

39. Identify the error in the formula: Volume of sphere = $\frac{4}{3}\text{π}r^3$.

    No error; formula is correct. The sphere volume formula is mathematically correct.

40. Which type of error is most reduced by taking many trials and averaging the results?

    Random error. Multiple trials cancel out random fluctuations.

41. What does it suggest if a best-fit line systematically misses points in a curved pattern?

    Model mismatch (nonlinear relationship fitted as linear). Wrong mathematical model for the actual relationship.

42. Identify the error in the quadratic formula: $x = \frac{-b \text{±} \text{√}(b^2 - 4ac)}{2b}$.

    Correct: $x = \frac{-b \text{±} \text{√}(b^2 - 4ac)}{2a}$. Denominator should be $2a$, not $2b$ in the quadratic formula.

43. Identify the error in the molar mass calculation: $H_2O = 20 \text{ g/mol}$.

    Correct: $H_2O = 18 \text{ g/mol}$. $H_2O$: $(2 \times 1) + 16 = 18$ g/mol, not 20.

44. What is parallax error in reading a scale?

    Apparent shift from viewing the scale at an angle. Eye position affects perceived reading on graduated scales.

45. Identify the error: Mixing up dependent and independent variables.

    Error: Incorrect variable identification. Wrong variable assignment affects analysis.

46. Identify the design flaw if a procedure does not specify how "growth" is measured.

    Missing operational definition (measurement ambiguity). Undefined measurement creates unreproducible results.

47. Identify the error in the sequence sum: $1 + 2 + 3 + 4 = 9$.

    Correct: $1 + 2 + 3 + 4 = 10$. Simple arithmetic: $1 + 2 + 3 + 4 = 10$.

48. Identify the likely error source if multiple students start/stop a stopwatch and obtain scattered times.

    Random error from human reaction time differences. Each person has different response delays.

49. What is the independent variable in an experiment?

    The factor deliberately changed by the experimenter. The cause being tested in the experiment.

50. What is a common source of systematic error when a balance reads $+0.20\,\text{g}$ with nothing on it?

    Zero offset (instrument not tared/calibrated). Instrument reads mass when empty, creating bias.

51. Identify the error in the sequence: $1, 4, 9, 16, 24$.

    Correct: $1, 4, 9, 16, 25$. Perfect squares sequence: $5^2 = 25$, not 24.

52. Which type of error is indicated when all measurements are tightly clustered but far from the accepted value?

    Systematic error (high precision, low accuracy). Consistent offset from true value indicates bias.

53. Find and correct the error: $2^3 + 2^2 = 16$.

    Correct: $2^3 + 2^2 = 12$. $2^3 = 8$ and $2^2 = 4$, so $8 + 4 = 12$.

54. Identify the likely error if a meniscus is read from above the eye level each time.

    Parallax error causing a systematic volume bias. Wrong eye level creates consistent reading bias.

55. Which type of error is most reduced by calibrating instruments against a standard?

    Systematic error. Calibration corrects consistent instrument bias.

56. What is the purpose of a control group in an experiment?

    To provide a baseline for comparison without the treatment. Shows what happens without experimental treatment.

57. Find the error in the calculation: $9 - (3 \times 2) = 9 - 3 \times 2 = 3$.

    Correct: $9 - (3 \times 2) = 3$. Parentheses change order: $9 - 6 = 3$, not $9 - 3 \times 2$.

58. Which error type results from improperly calibrated instruments?

    Systematic error. Miscalibrated instruments produce consistent measurement bias.

59. Identify the source of error if a study compares two treatments but includes no untreated baseline condition.

    No control group (baseline missing). Cannot determine if treatments have any effect.

60. Identify the likely error source if participants know they received the new drug and report improved symptoms.

    Placebo/expectation effect due to lack of blinding. Knowledge of treatment affects reported outcomes.

61. Identify the mistake in the pH calculation: $\text{pH} = -\text{log}(1 \times 10^{-7}) = 8$.

    Correct: $\text{pH} = 7$. $-\log(10^{-7}) = -(-7) = 7$, not 8.

62. What is an operational definition in experimental design?

    A precise description of how a variable is measured or defined. Eliminates ambiguity in measurement procedures.

63. What error occurs when assuming linearity in non-linear data?

    Error: Model misfit. Linear models don't fit curved relationships well.

64. Which error type involves a consistent deviation from the true value?

    Systematic error. These errors consistently shift all measurements in one direction.

65. Find the error in the simplification: $x^3 \times x^2 = x^5$.

    No error; simplification is correct. Exponent addition rule: $x^3 \times x^2 = x^{3+2} = x^5$.

66. Find the error in the angle sum: $45^\text{o} + 45^\text{o} = 180^\text{o}$.

    Correct: $45^\text{o} + 45^\text{o} = 90^\text{o}$. Two $45°$ angles form a right angle, not straight angle.

67. Find and correct the error: $7 - (3 + 2) = 7 - 3 + 2$.

    Correct: $7 - (3 + 2) = 2$. Distribution of negative sign was incorrect; $-(3 + 2) = -5$.

68. Identify the error in the arithmetic: $8 \times 5 - 3 = 40 - 3 = 43$.

    Correct: $8 \times 5 - 3 = 37$. Order of operations: multiplication before subtraction gives $40 - 3 = 37$.

69. Identify the error in the chemical equation: $H_2 + O_2 \rightarrow H_2O_2$.

    Correct: $2H_2 + O_2 \rightarrow 2H_2O$. Water formation requires 2 hydrogen molecules and proper atom balance.

70. Identify the error in the expression: $\frac{1}{x} + \frac{1}{y} = \frac{1}{x + y}$.

    Correct: $\frac{1}{x} + \frac{1}{y} = \frac{x + y}{xy}$. Common denominator needed: $\frac{y + x}{xy}$, not $\frac{1}{x + y}$.

71. What error type is reduced by increasing sample size in experiments?

    Random error. Larger samples average out random fluctuations more effectively.

72. Find and correct the error: $\text{sin}^2(x) + \text{cos}^2(x) = 1.5$.

    Correct: $\text{sin}^2(x) + \text{cos}^2(x) = 1$. Pythagorean identity always equals 1 for any angle.

73. Identify the error in the averaging: $(5 + 10 + 15) / 3 = 10$.

    No error; calculation is correct. The arithmetic mean calculation is performed correctly.

74. Identify the calculation error if $3.0\times 10^2$ is recorded as $3.0\times 10^3$ in the results.

    Exponent/place-value error in scientific notation. Wrong power of ten changes value dramatically.

75. What is a confounding variable?

    An uncontrolled factor that varies with the independent variable. Creates false correlation between variables.

76. What is the definition of accuracy in measurement?

    Closeness of a measurement to the true or accepted value. Correctness, not necessarily repeatability.

77. What is a source of error when a sensor has low resolution (large smallest scale division)?

    Quantization/rounding uncertainty from limited resolution. Cannot read values between scale markings precisely.

78. Identify the most likely cause if one data point is extreme and contradicts all repeats taken the same way.

    Procedural mistake or transient disturbance (outlier). Single anomaly suggests measurement error occurred.

79. Which error type occurs if experimenter bias affects results?

    Systematic error. Experimenter expectations can unconsciously influence data collection.

80. Find and correct the error: $x^2 + 4x + 4 = 0$ solved as $x = -2$ or $x = 2$.

    Correct: $x = -2$ (repeated root). The perfect square trinomial has only one solution at $x = -2$.

81. Identify the sampling error if fish are counted only near the shoreline to estimate the whole lake population.

    Selection bias (nonrepresentative sampling region). Shoreline fish don't represent whole lake.

82. Identify the likely error if the dependent variable is measured with different instruments across groups.

    Inconsistent measurement method (systematic group bias). Different tools create systematic measurement differences.

83. What is the dependent variable in an experiment?

    The measured response that depends on the independent variable. The effect being measured and recorded.

84. Identify the error: Mislabeling axes on a graph.

    Error: Misinterpretation of data. Wrong labels make data uninterpretable.

85. Identify the error in the integration: $\frac{d}{dx}(e^x) = e^x + C$.

    Correct: $\frac{d}{dx}(e^x) = e^x$. Derivative doesn't include constant of integration; that's for integrals.

86. What error arises when ignoring systematic errors in measurement?

    Bias in results. Systematic errors create consistent bias in all measurements.

87. Identify the mistake in the chemical balance: $Fe + O_2 \rightarrow Fe_2O_3$.

    Correct: $4Fe + 3O_2 \rightarrow 2Fe_2O_3$. Iron oxidation requires balancing: 4 Fe atoms and 3 $O_2$ molecules.

88. What is selection bias in sampling?

    A nonrandom sampling method that misrepresents the population. Sample doesn't reflect the target population.

89. Find the error in exponentiation: $(2^3)^2 = 2^5$.

    Correct: $(2^3)^2 = 2^6$. Power rule: $(a^m)^n = a^{mn}$, so $3 \times 2 = 6$.

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