Error flash cards

1. 1.What is meant by the error in a measurement?

   The error is the difference between the measured value and the true (or accepted) value of a quantity. Every measurement has some uncertainty, so no measurement is perfectly exact.

   Hint: Measured value vs true value.

2. 2.Name the three broad categories of errors in measurement.

   (i) Systematic errors, (ii) Random errors, and (iii) Gross errors (mistakes/blunders).

   Hint: One is reproducible, one is scattered, one is a blunder.

3. 3.What is a systematic error?

   An error that tends to occur in one direction (always positive or always negative) due to a definite cause. It follows a rule/pattern and can, in principle, be corrected.

   Hint: Consistent bias, same sign every time.

4. 4.List the main sub-types of systematic errors.

   (i) Instrumental errors, (ii) errors due to imperfect experimental technique/procedure, and (iii) personal errors.

   Hint: Instrument, method, observer.

5. 5.Give an example of an instrumental error.

   Zero error of vernier callipers or a screw gauge, a wrongly calibrated scale, or a thermometer whose zero is shifted. These arise from imperfect design/calibration of the instrument.

   Hint: Zero error, faulty calibration.

6. 6.What is a personal error?

   A systematic error arising from an individual observer's bias or carelessness, e.g. parallax due to not keeping the eye correctly aligned, or a habitual delay in starting a stopwatch.

   Hint: The observer's own habit/bias.

7. 7.How can systematic errors generally be minimised?

   By identifying the cause and correcting for it: recalibrating instruments, applying zero-error corrections, improving experimental technique, and removing personal bias (e.g. avoiding parallax).

   Hint: Correct the known cause.

8. 8.What is a random error?

   An error that occurs irregularly in both magnitude and sign due to unpredictable fluctuations in conditions (temperature, voltage, observer judgement). It cannot be eliminated but its effect is reduced by averaging.

   Hint: Irregular, both signs, scattered results.

9. 9.How is the effect of random errors reduced?

   By taking a large number of readings and computing their arithmetic mean; positive and negative deviations tend to cancel, so the mean is closer to the true value.

   Hint: Repeat and average.

10. 10.What is a gross error (blunder)?

    A large error caused by carelessness or a mistake by the observer, e.g. reading the wrong scale division, recording a wrong value, or an arithmetic slip. It has no fixed pattern.

    Hint: A human blunder, not a subtle effect.

11. 11.How are gross errors avoided?

    By taking readings carefully, repeating measurements, and cross-checking; a blundered reading is simply discarded when spotted.

    Hint: Care and repetition; discard the outlier.

12. 12.Define accuracy of a measurement.

    Accuracy is how close a measured value is to the true value of the quantity. High accuracy means small systematic error.

    Hint: Closeness to the truth.

13. 13.Define precision of a measurement.

    Precision is how closely repeated measurements agree with one another (reproducibility), regardless of the true value. High precision means small random error/scatter.

    Hint: Repeatability, small scatter.

14. 14.Can a measurement be precise but not accurate? Explain.

    Yes. If readings cluster tightly but around a wrong value (e.g. due to a zero error), they are precise but not accurate. Precision concerns scatter; accuracy concerns closeness to the true value.

    Hint: Tight cluster around the wrong spot.

15. 15.Which type of error mainly limits accuracy, and which limits precision?

    Systematic errors mainly limit accuracy; random errors mainly limit precision.

    Hint: Systematic→accuracy, random→precision.

16. 16.How does the least count of an instrument relate to precision?

    A smaller least count allows finer readings, giving greater precision. The precision of a single measurement is limited by the instrument's least count.

    Hint: Finer least count → more precise.

17. 17.Define least count of a measuring instrument.

    The smallest value that can be measured/read directly with the instrument. For a metre scale it is $1\text{ mm}$; for a good stopwatch it may be $0.1\text{ s}$.

    Hint: Smallest directly readable value.

18. 18.What is least count error?

    The uncertainty in a reading equal to the least count of the instrument (often taken as $\pm$ least count, sometimes half the least count). It is a form of random error inherent to the instrument's resolution.

    Hint: Uncertainty of one least count.

19. 19.If the true value of a quantity is $a$ and $n$ readings $a_1,a_2,\dots,a_n$ are taken, how is the best (most probable) value found?

    By the arithmetic mean: $a_{mean}=\dfrac{a_1+a_2+\cdots+a_n}{n}=\dfrac{1}{n}\sum_{i=1}^{n}a_i$. This is taken as the true value in the absence of the actual true value.

    Hint: Arithmetic mean of readings.

20. 20.Define the absolute error of the $i$-th reading.

    The absolute error is $\Delta a_i=|a_{mean}-a_i|$, the magnitude of the difference between the mean value and that reading.

    Hint: Magnitude of (mean − reading).

21. 21.Define the mean absolute error.

    $\Delta a_{mean}=\dfrac{1}{n}\sum_{i=1}^{n}|\Delta a_i|=\dfrac{|\Delta a_1|+|\Delta a_2|+\cdots+|\Delta a_n|}{n}$. It gives the average magnitude of the deviations.

    Hint: Average of the absolute errors.

22. 22.How is a measured result written using the mean and mean absolute error?

    $a=a_{mean}\pm\Delta a_{mean}$, meaning the true value lies (very likely) between $a_{mean}-\Delta a_{mean}$ and $a_{mean}+\Delta a_{mean}$.

    Hint: value $\pm$ mean absolute error.

23. 23.Define relative (fractional) error.

    Relative error $=\dfrac{\Delta a_{mean}}{a_{mean}}$, the ratio of the mean absolute error to the mean value. It is dimensionless.

    Hint: Mean absolute error ÷ mean value.

24. 24.Define percentage error.

    Percentage error $=\dfrac{\Delta a_{mean}}{a_{mean}}\times 100\%$, i.e. the relative error expressed as a percentage.

    Hint: Relative error × 100%.

25. 25.Why is relative (or percentage) error often more meaningful than absolute error?

    Because it compares the error to the size of the quantity. An absolute error of $1\text{ mm}$ is negligible on $1\text{ m}$ but huge on $2\text{ mm}$; the relative error captures this.

    Hint: Error relative to the size being measured.

26. 26.State the rule for propagation of error in a sum, $Z=A+B$.

    The absolute errors add: $\Delta Z=\Delta A+\Delta B$ (maximum possible error). The percentage error must be computed from this $\Delta Z$ and $Z$.

    Hint: Absolute errors add for a sum.

27. 27.State the rule for propagation of error in a difference, $Z=A-B$.

    The absolute errors still add: $\Delta Z=\Delta A+\Delta B$. We always take the worst case, so errors add even for subtraction.

    Hint: Absolute errors add even when you subtract.

28. 28.Why is measuring a small difference of two large quantities error-prone?

    The absolute error $\Delta A+\Delta B$ may become comparable to the small difference $Z=A-B$, so the relative/percentage error blows up. Such measurements should be avoided.

    Hint: Small result, but errors still add.

29. 29.State the rule for propagation of error in a product, $Z=AB$.

    The relative errors add: $\dfrac{\Delta Z}{Z}=\dfrac{\Delta A}{A}+\dfrac{\Delta B}{B}$. Equivalently, percentage errors add.

    Hint: Relative errors add for a product.

30. 30.State the rule for propagation of error in a quotient, $Z=A/B$.

    The relative errors add: $\dfrac{\Delta Z}{Z}=\dfrac{\Delta A}{A}+\dfrac{\Delta B}{B}$. Same rule as for a product.

    Hint: Divide → relative errors still add.

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