{"id":115,"date":"2026-06-17T18:21:31","date_gmt":"2026-06-17T18:21:31","guid":{"rendered":"https:\/\/flowxiom.com\/?p=115"},"modified":"2026-06-17T18:21:34","modified_gmt":"2026-06-17T18:21:34","slug":"a-level-ial-unit-1-03-errors-and-uncertainties","status":"publish","type":"post","link":"https:\/\/flowxiom.com\/?p=115","title":{"rendered":"A-Level IAL Unit 1 03 Errors and Uncertainties"},"content":{"rendered":"\n

Unit 1 03 Errors and Uncertainties<\/h2>
<\/div><\/div>
<\/div>

1. Learning Objectives<\/h3><\/div>
    \n
  • <\/span>Subject Content:<\/strong>\n
      \n
    • <\/span>Distinguish between Precision<\/strong> and Accuracy<\/strong>.
      \n<\/li>\n
    • <\/span>Identify Random<\/strong> and Systematic<\/strong> errors.
      \n<\/li>\n
    • <\/span>Calculate absolute, fractional, and percentage Uncertainties<\/strong>.
      \n<\/li>\n
    • <\/span>Combine uncertainties in calculations (addition, multiplication, powers).
      \n<\/li>\n<\/ul>\n<\/li>\n
    • <\/span>Language Goals:<\/strong>\n
        \n
      • <\/span>Describe data sets using “precise but not accurate” etc.
        \n<\/li>\n
      • <\/span>Explain the “source of error” in experiments.
        \n<\/li>\n<\/ul>\n<\/li>\n<\/ul><\/div>

        2. Key Terminology<\/h3><\/div>
        \n\n\n\n\n\n\n\n\n\n
        Term<\/strong><\/th>\nDefinition<\/strong><\/th>\n<\/tr>\n<\/thead>\n
        Accuracy<\/strong><\/td>\nHow close a reading is to its true value<\/strong>.<\/td>\n<\/tr>\n
        Precision<\/strong><\/td>\nHow close the readings are to each other<\/strong>.<\/td>\n<\/tr>\n
        Random Error<\/strong><\/td>\nUnpredictable variation between measurements.<\/td>\n<\/tr>\n
        Systematic Error<\/strong><\/td>\nConstant bias in one direction (e.g., zero error).<\/td>\n<\/tr>\n
        Absolute Uncertainty<\/strong><\/td>\nThe range (<\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span>) of a measurement.<\/td>\n<\/tr>\n
        Percentage Uncertainty<\/strong><\/td>\n<\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table><\/div>
        <\/div>

        3. Accuracy vs. Precision<\/h3><\/div>
          \n
        • <\/span>Accuracy:<\/strong> How close the measured value is to the “true value”. Affected by systematic errors.
          \n<\/li>\n
        • <\/span>Precision:<\/strong> The “concentration” (spread) between multiple measured values. Affected by random errors.
          \n<\/li>\n<\/ul><\/div>
          \n

          Exam Tip:<\/strong> A measurement can be very precise<\/strong> (consistent) but inaccurate<\/strong> (wrong) if there is a zero error in the instrument.<\/p>\n<\/blockquote><\/div>

          <\/div>

          4. Types of Errors<\/h3><\/div>
          \n\n\n\n\n\n
          Type<\/strong><\/th>\nCause<\/strong><\/th>\nHow to Reduce<\/strong><\/th>\n<\/tr>\n<\/thead>\n
          Random Error<\/strong><\/td>\nFluctuations in conditions, human reaction time, parallax error (variable).<\/td>\nRepeat<\/strong> and Average<\/strong> the readings.<\/td>\n<\/tr>\n
          Systematic Error<\/strong><\/td>\nPoorly calibrated instruments, zero error, background interference.<\/td>\nRecalibrate<\/strong> or check against a known standard.<\/td>\n<\/tr>\n<\/tbody>\n<\/table><\/div>
          <\/path><\/path><\/path><\/svg><\/div>
          Caution<\/div><\/div>
          \n

          Repeating measurements does NOT reduce systematic errors.<\/p>\n<\/div><\/div><\/div>

          This is a very important principle in experimental physics. Simply put: Repeated measurements can only “dilute” random fluctuations, but cannot eliminate a fixed bias.<\/strong><\/p><\/div>

          1. The Core Reason: Directionality of Error<\/h3><\/div>

          Systematic errors are consistent and unidirectional. This means that no matter how many times you measure, the result will always deviate in the same direction (e.g., always larger than the true value, or always smaller).<\/p><\/div>

          When we take the average of repeated measurements, we are trying to cancel out random errors that fluctuate up and down. However, since a systematic error is “wrong in the same way” every time, averaging them preserves the error rather than eliminating it.<\/p><\/div>

          <\/div>

          2. An Intuitive Example: The Uncalibrated Scale<\/h3><\/div>

          Imagine you have an electronic scale that is not zeroed; it reads 0.1 kg even when nothing is on it (this is a systematic error).<\/p><\/div>

            \n
          • <\/span>True Weight:<\/strong> An apple weighs 0.5 kg.
            \n<\/li>\n
          • <\/span>1st Measurement:<\/strong> Reads 0.6 kg (0.5 + 0.1).
            \n<\/li>\n
          • <\/span>2nd Measurement:<\/strong> Reads 0.6 kg.
            \n<\/li>\n
          • <\/span>100th Measurement:<\/strong> Still reads 0.6 kg.
            \n<\/li>\n<\/ul><\/div>

            If you add up these 100 results and calculate the average, the result is still 0.6 kg<\/strong>. Repeating the measurement did not bring you closer to the true 0.5 kg. You have simply measured a wrong<\/strong> (inaccurate) value very precisely<\/strong>.<\/p><\/div>

            <\/div>

            3. Mathematical Representation<\/h3><\/div>

            Suppose the true value is <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-math><\/mjx-container><\/span>, and the systematic error is <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-math><\/mjx-container><\/span> (a constant).<\/p><\/div>

            Each measured result <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-math><\/mjx-container><\/span> is:<\/p><\/div>

            <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-math><\/mjx-container><\/span><\/p><\/div>

            If you measure <\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> times and take the average (<\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-math><\/mjx-container><\/span>):<\/p><\/div>

            <\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-math><\/mjx-container><\/span><\/p><\/div>

            <\/mjx-c><\/mjx-mo><\/mjx-over><\/mjx-c><\/mjx-mi><\/mjx-base><\/mjx-mover><\/mjx-texatom><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-texatom><\/mjx-math><\/mjx-container><\/span><\/p><\/div>

            Conclusion:<\/strong><\/p><\/div>

            The average still contains the full systematic error <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-texatom><\/mjx-script><\/mjx-msub><\/mjx-math><\/mjx-container><\/span>.<\/p><\/div>

            <\/div>

            4. Only Random Errors Can Be Reduced by Repetition<\/h3><\/div>

            Unlike systematic errors, Random Errors are unpredictable; sometimes they are high, sometimes low.<\/p><\/div>

              \n
            • <\/span>The first time might measure 0.52 kg (+0.02).
              \n<\/li>\n
            • <\/span>The second time might measure 0.48 kg (-0.02).\n After averaging many measurements, the +0.02 and -0.02 tend to cancel each other out, bringing the result closer to the true value.
              \n<\/li>\n<\/ul><\/div>

              Summary<\/h4><\/div>
                \n
              • <\/span>Systematic Error:<\/strong> Repeated measurements do not help<\/strong> because the error is constant. Solution: Calibrate instruments or improve experimental design.
                \n<\/li>\n
              • <\/span>Random Error:<\/strong> Repeated measurements help<\/strong> because the errors cancel out.<\/li>\n<\/ul><\/div>
                <\/div>

                5. Combining Uncertainties<\/h3><\/div>

                In A-level exams, you need to calculate the uncertainty of final results based on formulas.<\/p><\/div>

                Rule 1: Addition and Subtraction<\/h4><\/div>
                  \n
                • <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> or <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span><\/strong>
                  \n<\/li>\n
                • <\/span>Rule:<\/strong> Add the Absolute Uncertainties<\/strong>.
                  \n<\/li>\n
                • <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span>
                  \n<\/li>\n<\/ul><\/div>

                  Rule 2: Multiplication and Division<\/h4><\/div>
                    \n
                  • <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> or <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-math><\/mjx-container><\/span><\/strong>
                    \n<\/li>\n
                  • <\/span>Rule:<\/strong> Add the Percentage Uncertainties<\/strong>.
                    \n<\/li>\n
                  • <\/span><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-math><\/mjx-container><\/span>
                    \n<\/li>\n<\/ul><\/div>

                    Rule 3: Powers<\/h4><\/div>
                      \n
                    • <\/span><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-script><\/mjx-msup><\/mjx-math><\/mjx-container><\/span><\/strong>
                      \n<\/li>\n
                    • <\/span>Rule:<\/strong> Multiply percentage uncertainty by the power index<\/strong>.
                      \n<\/li>\n
                    • <\/span><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mi><\/mjx-mrow><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-math><\/mjx-container><\/span>
                      \n<\/li>\n<\/ul><\/div>
                      <\/div>

                      6. Rules for Significant Figures (s.f.)<\/h3><\/div>

                      In A-level Physics, the use of significant figures is not a suggestion but a grading standard<\/strong>.<\/p><\/div>

                      A. Uncertainty and s.f.<\/h4><\/div>
                        \n
                      1. \n

                        Uncertainty is usually 1 s.f.<\/strong><\/p>\n

                          \n
                        • <\/span>Example:<\/em> Write <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-math><\/mjx-container><\/span>, instead of <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-math><\/mjx-container><\/span>.
                          \n<\/li>\n<\/ul>\n<\/li>\n
                        • \n

                          The value must match the precision of the uncertainty.<\/strong> (The decimal places of the measured value must align with the decimal places of the uncertainty).<\/p>\n

                            \n
                          • <\/span>Correct:<\/em> <\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-math><\/mjx-container><\/span> (Two decimal places aligned)
                            \n<\/li>\n
                          • <\/span>Incorrect:<\/em> <\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-math><\/mjx-container><\/span> (Not aligned)
                            \n<\/li>\n<\/ul>\n<\/li>\n<\/ol><\/div>

                            B. Calculation Rules<\/h4><\/div>
                              \n
                            1. Multiplication\/Division:<\/strong> The significant figures of the result should match the data with the least significant figures<\/strong> in the question (usually 2 or 3 digits).<\/li>\n
                            2. Intermediate Steps:<\/strong> During intermediate calculations, keep one extra significant figure (e.g., keep 4 s.f.) to avoid rounding errors, and only round off in the final answer.<\/li>\n<\/ol><\/div>
                              \n\n\n\n\n\n\n\n
                              Value<\/strong><\/th>\nNumber of s.f.<\/strong><\/th>\nReason<\/strong><\/th>\n<\/tr>\n<\/thead>\n
                              0.0052<\/strong><\/td>\n2<\/td>\nLeading zeros don’t count<\/td>\n<\/tr>\n
                              5.200<\/strong><\/td>\n4<\/td>\nTrailing zeros after decimal count<\/td>\n<\/tr>\n
                              502<\/strong><\/td>\n3<\/td>\nZeros between digits count<\/td>\n<\/tr>\n
                              <\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table><\/div>
                              <\/div>

                              7. Practice Exercises<\/h3><\/div>

                              Q1.<\/strong> A voltmeter has a zero error of <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mtext><\/mjx-c><\/mjx-mtext><\/mjx-math><\/mjx-container><\/span>. This error is:<\/p><\/div>

                              A. Random and affects precision.<\/p><\/div>

                              B. Random and affects accuracy.<\/p><\/div>

                              C. Systematic and affects precision.<\/p><\/div>

                              D. Systematic and affects accuracy.<\/p><\/div>

                              Q2.<\/strong> The length of a table is measured as <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mtext><\/mjx-c><\/mjx-mtext><\/mjx-math><\/mjx-container><\/span> and its width is <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-mtext><\/mjx-c><\/mjx-mtext><\/mjx-math><\/mjx-container><\/span>.<\/p><\/div>

                              Calculate the percentage uncertainty in the area of the table.<\/p><\/div>

                              Q3.<\/strong> The density <\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> of a metal cube is calculated using the mass <\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> and the length of its side <\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span>:<\/p><\/div>

                              <\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mo><\/mjx-nstrut><\/mjx-c><\/mjx-mi><\/mjx-num><\/mjx-line><\/mjx-dstrut><\/mjx-c><\/mjx-mi><\/mjx-c><\/mjx-mn><\/mjx-script><\/mjx-msup><\/mjx-den><\/mjx-row><\/mjx-dtable><\/mjx-dbox><\/mjx-frac><\/mjx-mfrac><\/mjx-math><\/mjx-container><\/span><\/p><\/div>

                              The percentage uncertainty in <\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> is <\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span>, and the percentage uncertainty in <\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> is <\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span>.<\/p><\/div>

                              What is the percentage uncertainty in the calculated density?<\/p><\/div>

                              Q4.<\/strong> A student measures the time for 20 oscillations of a pendulum to be <\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mtext><\/mjx-c><\/mjx-mtext><\/mjx-math><\/mjx-container><\/span> with a stopwatch. The uncertainty in the timing is <\/mjx-c><\/mjx-mo><\/mjx-c><\/mjx-c><\/mjx-c><\/mjx-mn><\/mjx-c><\/mjx-mtext><\/mjx-c><\/mjx-mtext><\/mjx-math><\/mjx-container><\/span>.<\/p><\/div>

                              What is the absolute uncertainty in the period <\/mjx-c><\/mjx-mi><\/mjx-math><\/mjx-container><\/span> (the time for one oscillation)?<\/p><\/div>

                              <\/div>

                              Practice Exercises Analysis<\/h3><\/div>

                              Q1. Voltmeter Zero Error<\/h4><\/div>