- In all measurements, there is always some uncertainty in the measured value.
- We cannot be sure that we have made a completely accurate measurement.
- This can be due to different causes, generally called experimental errors.
Experimental errors
There are several causes of errors when performing an experiment. For example:
- The reading on an ammeter may fluctuate.
- The pointer of an analogue voltmeter may not be exactly on one of the scale divisions.
- Room temperature may fluctuate. Experimental errors lead to measurements that are different from the true values. Experimental errors can be random or systematic.
Random Errors
- Random errors refer to unpredictable variations
- that occur during measurement that leads to inconsistencies in the data.
- Random errors arise from unpredictable random changes in the experiment
- which may occur in the measuring instruments or environmental conditions
Common Sources of Random Errors
- Environmental Variations
- Fluctuations in room temperature
- Fluctuations in humidity
- Unpredictable wind conditions (for outdoor experiments)
- Measuring and other instruments
- Fluctuations in meter readings
- Variation of output from power supply
- Human Factors
- Reaction time
- Parallax error when reading a scale
How to Reduce Random Errors ?
- The effects of random errors can be reduced by taking multiple measurements and calculating an average.
- There may still be some random error but it will be minimised.
Why does Averaging Multiple Measurements Reduce Random Error ?
- Being random in nature, random errors are equally likely to cause the measurement to deviate above or below the true value.
- For example, if the true value of current in a circuit is 0.10 A, ammeter fluctuations are equally likely to give values below 0.10 A and above 0.10 A.
- When multiple measurements are taken, added and averaged, the positive and negative errors cancel out, bringing the mean closer to the true value.
Example: Random errors in ammeter readings
Let’s say the current in a circuit is 0.10 A, if we take 6 readings for current, we would expect 3 of them to be below 0.10 A and 3 of them to be above 0.10 A.
| SN | I/A | Deviance from True Value |
|---|---|---|
| 1 | 0.12 | + 0.02 |
| 2 | 0.09 | - 0.01 |
| 3 | 0.11 | + 0.01 |
| 4 | 0.07 | - 0.03 |
| 5 | 0.08 | - 0.02 |
| 6 | 0.13 | + 0.03 |
Total positive error: 0.06 Total negative error: 0.06 Net error: 0.00 Now if we add our readings and do an average:
\text{Average Current} &= \frac{{0.12+0.09+0.11+0.07+0.08+0.13}}{6} \ &= \frac{0.60}{6} \ &= 0.10 \space A \end{align}$$
- As you can see, when we take in this case 6 measurements, each of them will have some degree of either positive or negative error.
- When we add up the readings and do an average, the positive and negative errors cancel out, giving us a more precise result.
Key Characteristics of Random Errors
- Unpredictability: They cause measurements to fluctuate in an unpredictable manner due to their random nature.
- Impact on precision: Random errors affect the precision of measurements, leading to a spread of the values around the true value.
- Reduction through repetition: Taking multiple measurements and averaging them minimises the impact of random errors as positive and negative errors cancel out.
Systematic Errors
- Systematic errors are consistent inaccuracies
- that cause all measurements to deviate from the true value by a fixed amount or proportion.
- These errors arise from flaws in the experimental setup, measuring instruments, or procedures, leading to results that are consistently too high or too low.
Common Sources of Systematic Errors
- Instrumental errors: Faulty or miscalibrated measuring instruments can introduce consistent deviations.
- A scale that is not zeroed correctly will consistently give readings that are offset by a certain amount.
- Using a metre rule with a damaged end will consistently give readings that are offset by a certain amount
- Observational errors: Consistently misreading instruments or incorrect usage can lead to systematic errors.
- A vertical scale always being read from a position that is too low or too high will result in systematic parallax error.
How to Reduce Systematic Errors ?
- Regularly calibrating measuring instruments against known standards to ensure their accuracy.
- Carefully examine experimental procedures to identify and correct potential sources of bias.
Effect of Averaging on Systematic Errors
- Taking multiple readings and doing an average will not reduce the effect of systematic errors.
- This is because systematic errors consistently offset all measurements in the same direction.
- If an average is calculated, the average will also be offset in the same direction by the same amount of the individual readings. Example: Let’s say the mass of a ball is 100 g. If we use a balance having a positive zero error of + 0.5 g, all readings will be offset by + 0.5 g from the true value. If we take 6 readings: 100.5 g, 100.5 g, 100.5 g, 100.5 g, 100.5 g, 100.5 g and do an average:
\text{Average mass of ball} &= \frac{{100.5 + 100.5+100.5+100.5+100.5+100.5}}{6} \ &= \frac{{6(100.5)}}{6} \ &= 100.5 \space g \end{align}$$ As you can see, even by taking multiple readings and doing an average, the average is still offset by 0.5 g from the true value.
Key Characteristics of Systematic Errors
- Consistency: Systematic errors affect all measurements in the same way, leading to uniform deviation from the true value.
- Predictability: Since these errors are consistent, their effect on the measurement is predictable.
- Impact on accuracy: Systematic errors compromise the accuracy of measurements causing a consistent bias in the results.
Differences between Random and Systematic Errors
| Random Errors | Systematic Errors |
|---|---|
| Cause measurements to fluctuate unpredictably around the true value | Cause consistent and predictable deviations, offsetting the measurements by a fixed value |
| Can be reduced by taking multiple readings and calculating an average since positive and negative errors cancel out | Cannot be reduced by averaging, as their consistent nature means they will bias the average as well |
| Primarily impact precision | Primarily impact accuracy |
Note: Nature of Parallax Errors
Depending on the context, parallax error can be both systematic and random.
Parallax Error as a Systematic Error
Occurs when the observer consistently views the scale from the same incorrect angle:
- Happens if the measurement is always taken from above, below, left or right of the proper eye level.
- This leads to a consistent offset in the readings.
- Affects accuracy more than precision.
- Example: A student always reads a liquid’s meniscus from above proper eye level, so all volume readings are consistently lower than the true value.
Parallax Error as a Random Error
Occurs when the observer’s eye position varies between measurements:
- Eye position changes unpredictable between different measurements.
- Leads to a random variation in readings, increasing uncertainty.
- Affects precision more than accuracy. Example: In multiple trials, a student reads the scale from slightly different angles each time, causing values to fluctuate around the actual reading.