Understanding Measurement & Precision - ACT Science

Based on Figure 1, what volume of solution was delivered from the burette?
Answer: 14.2 mL The bottom of the meniscus reads 12.4 mL before the tap is opened and 26.6 mL after it is closed. The volume delivered is the difference, 26.6 − 12.4 = 14.2 mL. Choosing 26.6 mL takes the final reading as the answer, 39.0 mL adds the two readings instead of subtracting them, and 14.8 mL comes from reading the second meniscus at the 27.2 mL mark.
Why do the classroom thermometer's readings fail to show that all three cups are at different temperatures?
Answer: Its smallest division is larger than the differences between the cups. The cups span 21.4 °C to 22.0 °C, a range of 0.6 °C, and the classroom thermometer is marked only in whole degrees. Rounding to the nearest mark sends 21.4 to 21 and sends both 21.7 and 22.0 to 22, so two cups that really do differ come out identical. Nothing is wrong with the thermometer's accuracy — every reading is within half a degree of the laboratory thermometer — but an instrument cannot show a difference smaller than the divisions on its scale.
Which measurement shows the greatest precision in the data table?
Answer: 12.345 g Precision in measurements is indicated by the number of decimal places or significant figures recorded. Among the given measurements, 12.345 g shows the greatest precision with three decimal places, indicating the measuring instrument can detect differences as small as 0.001 g. The measurement 12.345 g demonstrates the highest level of precision compared to measurements with fewer decimal places. A measurement like 12 g shows the least precision with no decimal places indicated.
A student times a reaction three times using two different stopwatches. Stopwatch X shows time to the nearest 0.1 s; Stopwatch Y shows time to the nearest 0.01 s. The data table lists the recorded times. Which measurement set shows the greatest precision?
Answer: Stopwatch Y: 12.34 s, 12.36 s, 12.35 s Stopwatch Y shows the greatest precision because it records time to the nearest 0.01 s (hundredths), as evidenced by measurements like 12.34 s, 12.36 s, and 12.35 s. Precision refers to the smallest unit an instrument can measure, not the consistency of repeated measurements. Stopwatch X only measures to 0.1 s precision, making it less precise regardless of how consistent the readings are.
A student measures the time for a cart to travel a fixed distance using two timing methods. Method 1 uses a handheld stopwatch readable to 0.01 s; Method 2 uses a phone video analyzed frame-by-frame at 30 frames/s (time step $\approx 0.033$ s). The student records one trial from each method. Which recorded time is most consistent with appropriate precision for the method used?
Answer: Stopwatch: 2.34 s The handheld stopwatch is readable to 0.01 s, allowing time measurements to be recorded with two decimal places. Precision is determined by the instrument's smallest division; for the stopwatch, 0.01 s enables 2.34 s, while the video's 0.033 s frame step limits precision to about 0.03 s, making finer readings inappropriate. The recorded time 2.34 s for the stopwatch reflects appropriate precision because it aligns with the device's resolution for a single trial. Option C assigns 2.34 s to the video, which overstates precision given the coarser time step.
Four instruments are used to measure the same time interval. Which measurement shows the greatest precision (smallest increment implied by the recorded value)?
Answer: 12.000 s The measurement 12.000 s shows the greatest precision because it implies the instrument can measure to the nearest 0.001 s (millisecond). The number of decimal places in a measurement indicates the precision of the instrument used - more decimal places mean finer precision. While 12 s suggests measurement to the nearest second, 12.000 s indicates measurement to the nearest millisecond, a thousand times more precise. Each additional zero after the decimal point represents a tenfold increase in precision.
A student measures the time for a pendulum to complete one period using a phone app. The app reports 1.467 s. The instrument shown can measure to a precision of:
Answer: 0.001 s The phone timer displays 1.467 s with three decimal places, indicating it can measure to the nearest 0.001 s (millisecond precision). Digital instruments show their resolution through the smallest displayed place value. Choice C correctly identifies this precision level. The consistent display of three decimal places demonstrates the timer's capability to resolve time differences of one millisecond.
The digital balance reads 7.890 g. The precision of the balance is:
Answer: 0.001 g The precision of a digital balance is indicated by the number of decimal places displayed in its readout. A balance reading 7.890 g shows measurements to three decimal places, indicating precision to 0.001 g (1 milligram). This precision level is typical for analytical balances used in laboratory settings. The last digit displayed represents the smallest mass difference the balance can detect and measure reliably.
A sample's mass is measured on a digital balance that displays 12.340 g. Based on this display, the measurement should be recorded to the nearest:
Answer: 0.001 g The digital balance displays 12.340 g, showing three decimal places, so measurements should be recorded to the nearest 0.001 g (thousandth of a gram). Digital instruments display all digits they can reliably measure, and the last digit shown represents the instrument's precision limit. Since the display shows milligrams (thousandths), the balance can distinguish between 12.340 g and 12.341 g. Recording to only 0.01 g or 0.1 g would discard precision the instrument provides.
A student records repeated measurements of the same object's length using the same instrument. The instrument used most likely has a smallest marked division of:
Answer: 0.1 cm All recorded measurements show two decimal places (5.62, 5.63, etc.), indicating the instrument supports readings to the nearest 0.01 cm. For instruments with 0.01 cm precision, the smallest marked division is typically 0.1 cm (1 mm), allowing estimation to the hundredths place. Choice B correctly identifies 0.1 cm as the likely smallest marked division. The consistent decimal places in all trials confirm this precision level.
A spring scale is used to measure force. The scale has major labels every $1,\mathrm{N}$ and minor tick marks every $0.2,\mathrm{N}$ . The pointer rests between $3.4,\mathrm{N}$ and $3.6,\mathrm{N}$ , approximately at $3.5,\mathrm{N}$ . Based on the measuring instrument shown, the measurement should be recorded to the nearest:
Answer: $0.1\,\mathrm{N}$ The spring scale has minor ticks every 0.2 N, so measurements should be recorded to the nearest 0.1 N by estimating between these divisions. To determine the appropriate precision, identify the smallest marked gradation (0.2 N) and include one estimated digit beyond it, allowing for half the interval or finer estimation. The correct answer, 0.1 N, reflects this precision since the pointer at the midpoint between 3.4 N and 3.6 N would be read as 3.5 N. Choice D is incorrect as it suggests 0.01 N, which exceeds the scale's resolution.
Based on Figure 1, which of the following is the most appropriate way to record this voltage?
Answer: 6.3 V The needle sits between the 6.0 V and 6.5 V marks, a little closer to 6.5 V than to 6.0 V, so the certain digits are 6 and the estimated digit is 3: 6.3 V. Recording 6.0 V discards the estimated digit and 6.5 V just reads the nearer mark. The needle is not at 6.35 V either; a scale divided every 0.5 V does not support reading to the hundredths place.
Which measurement represents an overestimation of precision?
Answer: 3.45678 m Overestimation of precision occurs when measurements are recorded with more decimal places than the instrument can reliably measure. The measurement 3.45678 m with five decimal places likely represents false precision unless using an extremely precise instrument. Most common measuring devices cannot justify this many significant figures. Appropriate precision should match the instrument's actual capability, typically fewer decimal places for standard measuring tools.
Based on Figure 1, what is the length of the block?
Answer: 5.6 cm The left end of the block sits at the 2.0 cm mark and the right end at the 7.6 cm mark, so the length is the difference, 7.6 − 2.0 = 5.6 cm. Reading the right-hand end alone gives 7.6 cm, which overstates the length by exactly the 2.0 cm offset at the left end; 2.0 cm is the left-hand reading on its own; and 5.0 cm comes from reading the right-hand end as 7.0 cm instead of 7.6 cm.
A student measures 10.0 mL of a solution using a 10 mL graduated cylinder with 0.2 mL markings. The meniscus is shown at 9.8 mL. Which value is recorded with appropriate precision?
Answer: 9.80 mL The graduated cylinder has 0.2 mL markings as its smallest division, so measurements should be estimated one digit beyond that (0.02 mL precision). With the meniscus at the 9.8 mL mark, the measurement should be recorded as 9.80 mL to show the appropriate precision level. Choice C correctly includes the estimated digit in the hundredths place. Choice D (9.8 mL) shows insufficient precision for this instrument's capabilities.
The digital thermometer reads 36.58 °C. The precision of the thermometer is:
Answer: 0.01 °C Digital thermometer precision is indicated by the number of decimal places in its display. A reading of 36.58°C shows measurement to two decimal places, indicating precision to 0.01°C. This precision level is typical for quality digital thermometers used in laboratory or medical applications. The displayed resolution reflects the instrument's ability to detect small temperature differences reliably.
Based on Figure 1, which rifle is precise but not accurate?
Answer: Target 2 The five shots on Target 2 land within a small area, so that rifle is precise, but the whole group sits well up and to the right of the center, so it is not accurate. Target 1 is both precise and accurate, Target 3 is scattered around the center, and Target 4 is both scattered and off-center.
A student measures the same mass five times on a digital balance. Which statement about measurement precision is supported by the data?
Answer: The balance resolution is 0.001 g because values report three decimals. Each measurement reports to the thousandths place (three decimal places), indicating the digital balance has 0.001 g resolution. The consistent decimal places across all trials demonstrate the instrument's precision capability, not measurement error. Choice B correctly identifies the balance resolution based on the displayed decimal places. The small variation between readings (2.499-2.502 g) represents normal measurement uncertainty within the instrument's precision.
A student records temperature using the glass thermometer shown. The smallest marked division is 1 $^{\circ}$ C, and the liquid column ends slightly above 22 $^{\circ}$ C, about 0.3 of the way to 23 $^{\circ}$ C. Based on the measuring instrument shown, the temperature should be recorded as:
Answer: 22.3 $^{\circ}$ C The glass thermometer has smallest marked divisions of 1°C, enabling temperature readings estimated to the nearest 0.1°C. To determine precision, note the smallest gradation and estimate one-tenth of the interval by observing the liquid column's position; here, 0.3 of the way from 22°C to 23°C yields 22.3°C. The correct answer, 22.3°C, reflects appropriate precision as it includes one estimated decimal place consistent with the instrument's scale. Option C, 22.30°C, overstates precision with an extra zero not justified by the 1°C divisions.
Which statement about the ruler's precision is correct?
Answer: It measures to the nearest 1 mm. The precision of a ruler depends on its smallest marked divisions. A standard ruler typically has millimeter markings, allowing measurements to be read directly to the nearest 1 mm. While some estimation between marks is possible, the fundamental precision is limited by the 1 mm gradations. Claiming precision to 0.1 mm or 0.5 mm would require finer gradations than typically found on standard rulers.

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