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Measurements and uncertainties (GCSE)

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KS4 National Curriculum Statement(s) covered

  • Recognising the importance of scientific quantities and understanding how they are determined. (WS)
  • Using SI units and IUPAC chemical nomenclature unless inappropriate. (WS)
  • Using prefixes and powers of ten for orders of magnitude (e.g., tera, giga, mega, kilo, centi, milli, micro, and nano). (WS)
  • Interconverting units. (WS)
  • Using an appropriate number of significant figures in calculations. (WS)
  • Applying the cycle of collecting, presenting, and analysing data [...] (WS)

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Rounding: decimal places and significant figures
SI Units in chemistry
Measurements in chemistry
Uncertainties in measurements

Understanding how to round numbers and measure accurately is crucial in chemistry. Whether you're reporting data from an experiment or calculating values for a reaction, the precision of your numbers matters. This entry covers the basics of rounding to decimal places and significant figures, as well as the use of SI units and how to convert between them. Additionally, we'll discuss the importance of uncertainties in measurements and how to estimate and report them accurately.

Rounding: Decimal Places and Significant Figures

In chemistry, rounding values is essential for clarity and precision. Measurements in experiments are not perfectly precise due to limitations in instruments and methods. Quoting recurring values or fractions as answers can imply a level of precision that doesn't actually exist, leading to misleading results. 

By rounding, we ensure that our reported values accurately reflect the precision of our measurements. This helps in maintaining consistency and reliability in scientific communication, making it easier to compare and replicate experiments. Additionally, rounded values simplify calculations and interpretations, aiding in clearer and more effective data presentation.

Decimal places

Decimal places (d.p.) refer to the number of digits to the right of the decimal point. Rounding to a certain number of decimal places can make numbers easier to read and work with, especially when dealing with very precise measurements.

How to round to decimal places:

  1. Identify the digit at the required decimal place: For example, if rounding to 2 decimal places, look at the second digit to the right of the decimal point.
  2. Check the digit immediately to the right of this place: If this digit is 5 or greater, round the last retained digit up by one. If it is less than 5, leave the last retained digit as it is.
  3. Drop all digits to the right of the rounded place.

Worked Example - Rounding by decimal place

Round 3.146 to 2 decimal places.

  1. The digit at the second decimal place is 4
  2. The digit immediately to the right is 6 (which is greater than 5)
  3. Therefore, round up the 4 to 5
  4. The rounded number is 3.15

Significant figures

Significant figures (s.f.) include all the meaningful digits in a number, which contribute to its precision. This method is crucial in scientific measurements, where the precision of the data is important.

Rules for significant figures:

  • All non-zero digits are significant.
  • Any zeros between significant digits are also significant.
  • Leading zeros (zeros before the first non-zero digit) are not significant.
  • Trailing zeros (zeros at the end of a number) in a decimal number are significant.

How to round to significant figures:

  1. Count the number of significant figures needed: For example, to round to 3 significant figures.
  2. Identify the digit at the last required significant figure place.
  3. Check the digit immediately to the right of this place: If this digit is 5 or greater, round the last retained digit up by one. If it is less than 5, leave the last retained digit as it is.
  4. Replace all digits to the right of the rounded place with zeros if they are before the decimal point, or drop them if they are after the decimal point.

Worked Example - Rounding by significant figures 1

Round 0.0045678 to 3 significant figures.

  1. The first three significant figures are 4, 5, and 6
  2. The digit immediately to the right is 7 (which is greater than 5)
  3. Therefore, round up the 6 to 7
  4. The rounded number is 0.00457

Worked Example - Rounding by significant figures 2

Round 12345 to 2 significant figures.

  1. The first two significant figures are 1 and 2
  2. The digit immediately to the right is 3 (which is less than 5)
  3. Therefore, leave the 2 as it is
  4. The rounded number is 12000

Sometimes you will not be told how many significant figures to round to, so you will need to determine an appropriate number based on the context of the data. Here are some guidelines:

  1. Look at the precision of the instruments used to measure the data. For example, if a burette measures to the nearest 0.05 cm³, the data should reflect this precision.
  2. Ensure that all numbers in a calculation or report have a consistent number of significant figures. Do not mix numbers with different levels of precision.
  3. In many scientific contexts, it is common to use 3 significant figures for most measurements and calculations, unless more precision is necessary.
  4. In a calculation involving measurements, it is best practice to use the smallest number of significant figures from the data involved. For instance, if adding 12.34 cm (4 significant figures) and 0.05678 cm (5 significant figures), the result should be rounded to 4 significant figures.

Tips for rounding

  1. Always perform rounding as the last step in your calculations. This helps maintain precision during intermediate steps. 
  2. Pay attention to context: Rounding rules can vary slightly depending on the specific requirements of an experiment or calculation.
  3. Practice: The more you practice rounding, the more intuitive it will become.

Common mistakes to avoid:

  • Rounding too early: This can lead to a loss of precision in your final result, and make it incorrect.
  • Misidentifying significant figures: Remember the rules, especially for zeros in different positions.
  • Forgetting to drop digits after rounding: Ensure that digits to the right of your rounded place are correctly removed or replaced with zeros.

SI Units in Chemistry

SI units, or International System of Units, provide a standardised way of measuring and reporting scientific data. Chemists use the International System of Units (SI) to ensure measurements are standardised and accurate.

Here is a comprehensive list of SI units commonly used in chemistry:

quantity unit symbol notes
amount of substance mole mol 1 mol = 6.022 × 10²³ particles (Avogadro's number)
length metre m The basic unit of length in the SI system
mass kilogram kg The basic unit of mass; 1 kg = 1000 g
pressure pascal Pa 1 Pa = 1 N/m² (newton per square metre)
temperature kelvin K The basic unit of temperature; 0 K = -273°C
time second s The basic unit of time
volume cubic metre m³ 1 m³ = 1000 dm³ = 1,000,000 cm³

SI units come with prefixes that denote different orders of magnitude, making it easier to express very large or very small numbers.

Here is a table showing the common SI prefixes and their corresponding powers of ten. Note how each step (except for centi) involves multiplying or dividing by 1000:

prefix symbol power of ten example
tera T 10¹² 1 terametre (Tm) = 10¹² m
giga G 10⁹ 1 gigametre (Gm) = 10⁹ m
mega M 10⁶ 1 megametre (Mm) = 10⁶ m
kilo k 10³ 1 kilometre (km) = 10³ m
centi c 10⁻² 1 centimetre (cm) = 10⁻² m
milli m 10⁻³ 1 millimetre (mm) = 10⁻³ m
micro µ 10⁻⁶ 1 micrometre (µm) = 10⁻⁶ m
nano n 10⁻⁹ 1 nanometre (nm) = 10⁻⁹ m
pico p 10⁻¹² 1 picometre (pm) = 10⁻¹² m

In chemistry, we often use standard form (also known as scientific notation) to express very large or very small numbers conveniently. Standard form is written as  𝑎 × 10ⁿ , where 𝑎 is a number between 1 and 10, and 𝑛 is an integer. This makes it easier to handle the large and small quantities commonly encountered in scientific measurements.

For example:

  • The distance of 1 gigametre (Gm) can be written in standard form as 1 × 10⁹ metres (m).
  • The size of a water molecule is about 0.275 nanometres (nm), which can be written in standard form as 2.75 × 10⁻¹⁰ metres (m).

Measurements in Chemistry

Understanding the different types of measurements in chemistry is crucial for conducting experiments and analysing data accurately. Here is a table summarising the key types of measurements, their definitions, and common units used in chemistry:

measurement type definition common units used notes
amount of substance a measure of the number of particles in a substance mol 1 mol = 6.022 × 10²³ particles (Avogadro's number)
concentration the amount of a substance in a given volume mol/dm³, g/dm³ mol × RFM = mass (g)
length the measurement of distance cm, mm 1 m = 100 cm = 1000 mm
mass the amount of matter in a substance g, kg, tonnes 1 kg = 1000 g, , 1 tonne = 1000 kg
pressure the force exerted by particles in a given area Pa, atm 1 atm = 101,000 Pa 
temperature a measure of the average energy due to the motion of the particles (kinetic store) °C, K 0°C = 273 K
volume the amount of space a substance occupies cm³, dm³ 1 dm³ = 1000 cm³, 1 dm³ is also referred to as a litre (L)

In chemistry, we often use units like cubic decimetres (dm³) and grams (g) instead of the SI units cubic metres (m³) and kilograms (kg). This is because the quantities we measure in the lab are usually much smaller.

Worked Example - Converting between units 1

To convert a measurement from nanometres (nm) to metres (m), use the conversion factor 1 nm = 10⁻⁹ m.

For example, converting 500 nm to m:

  1. Write down the conversion factor: 1 nm = 10⁻⁹ m
  2. Multiply the number of nanometres by the conversion factor: 500 nm × 10⁻⁹ m/nm = 500 × 10⁻⁹ m
  3. Simplify the expression to standard form: 500 × 10⁻⁹ m = 5 × 10⁻⁷ m

Worked Example - Converting between units 2

To convert concentration from grams per cubic decimetre (g/dm³) to moles per cubic decimetre (mol/dm³), use the molar mass of the substance.

For example, converting 10 g/dm³ of NaCl (sodium chloride) to mol/dm³:

  1. Calculate the RFM of NaCl: Na (RAM = 23) + Cl (RAM = 35.5) = 58.5
  2. Divide the concentration by the RFM: 10 g/dm³ ÷ 58.5 = 0.171 mol/dm³ (rounded to 3 significant figures)

Uncertainties in Measurements

Whenever a measurement is made in chemistry, there is always some uncertainty in the result obtained. Uncertainty can arise from the resolution of measuring instruments or from the range of a set of repeat measurements. For example, it may be difficult to judge:

  • Whether a thermometer is showing a temperature of 24.0°C, 24.5°C, or 25.0°C.
  • Exactly when a chemical reaction has finished.

Understanding and estimating uncertainty is crucial in chemical experiments to ensure accuracy and reliability of the results.

Absolute uncertainty

  • From measuring instruments:
    • The resolution of a measuring instrument is the smallest change in a quantity that gives a visible change in the reading. For instance, a thermometer with marks every 1.0°C has a resolution of 1.0°C.
    • The uncertainty of a measuring instrument is typically ± half the smallest scale division. For instance, a thermometer with a resolution of 1.0°C would have an uncertainty of ±0.5°C in every reading.
    • Tip: Some measuring instruments will have a known uncertainty stated on the apparatus itself.
  • From repeat measurements:
    • For a set of repeat measurements, the uncertainty is ± half the range.
    • This means that the value can be given as the mean value ± half the range.

Worked Example - Uncertainty for repeat measurements

For a set of repeat measurements, the uncertainty is ± half the range. This means that the value can be given as the mean value ± half the range.

The table shows the results collected for the first temperature of the disappearing cross experiment. 

Time for cross to disappear (s)
Temperature (°C) Run 1 Run 2 Run 3 Mean time (s)
20 125 115 120 120

To estimate the uncertainty for 20°C:

  1. Range = (biggest value - smallest value) = 125 - 115 = 10 s
  2. Uncertainty = ± half the range = ±5.0 s
  3. So, the mean time is 120 cm³ ±5.0 s

Relative uncertainty

Relative uncertainty, also known as percentage uncertainty, expresses the uncertainty as a fraction of the measured value. This is particularly useful for comparing the precision of measurements of different magnitudes.

  • Relative uncertainty is calculated by dividing the absolute uncertainty by the measured value and multiplying by 100 to get a percentage.
    \[ \text{relative uncertainty} = \text{number of measurements} \times \left(\frac{\text{absolute uncertainty}}{\text{measured value}}\right) \times 100 \]

Worked Example - Relative uncertainty (measuring volume)

Suppose we measure the volume of a liquid using a measuring cylinder, and the reading is 50 cm³ with an absolute uncertainty of ±0.5 cm³.

  1. Since we only took one measurement:
    • \[ \text{relative uncertainty} = \text{1} \times \left(\frac{\text{0.5 cm³}}{\text{50.0 cm³}}\right) \times 100 \]
    • relative uncertainty = 0.01 × 100
  2. So, the relative uncertainty in the volume measurement is 1.0% (given to 2 s.f.)

Worked Example - Relative uncertainty (change in temperature)

Suppose we measure the temperature of a solution at two different times during an experiment. The initial temperature is 25.0°C, and the final temperature is 28.0°C. The thermometer has an absolute uncertainty of ±0.5°C.

  1. Calculate the change in temperature: 28.0°C - 25.0°C = 3.0°C
  2. Since we took two measurements:
    • \[ \text{relative uncertainty} = \text{2} \times \left(\frac{\text{0.5°C}}{\text{3.0°C}}\right) \times 100 \]
    • relative uncertainty = 2 × 0.1667 × 100
  3. So, the relative uncertainty in the temperature change measurement is 33.3% (given to 3 s.f.)

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Did you know?

  • The International System of Units (SI) was established to standardise measurements worldwide, reducing uncertainties and errors in scientific research.
  • The Kelvin scale, used for measuring temperature in scientific contexts, is named after Lord Kelvin, a Scottish physicist.
  • The smallest unit of measure ever observed is the yoctometre (ym), which is equal to 10⁻²⁴ metres.
  • The mole, an SI unit for the amount of substance, was first introduced by Wilhelm Ostwald in the late 19th century.

Why do we care?

  • Precision in measurements can make the difference between a perfect cake and a baking disaster.
  • Accurate rounding and measurements are vital in prescribing the correct dosage of medication.
  • Manufacturing smartphones and other gadgets requires precise measurements to ensure all parts fit together and function properly.
  • Accurate data on pollutants can help monitor and protect the environment.

Key information

  • Rounding to decimal places involves looking at the digit immediately after your desired decimal place and rounding up or down accordingly.
  • Significant figures include all meaningful digits in a number, following specific rules for zeros and non-zero digits.
  • SI units provide a standardised system for measuring quantities in chemistry, ensuring consistency and clarity in scientific communication.
  • Uncertainty in measurements can be absolute or relative, and it’s important to understand and report these uncertainties to ensure the reliability of data.