# Sources of Uncertainty ## Understanding Uncertainty - Measurements of quantities are made with the aim of finding the true value of that quantity - In reality, it is impossible to obtain the true value of any quantity as there will always be a degree of uncertainty - The uncertainty is an estimate of the difference between a measurement reading and the true value - Random and systematic errors are two types of measurement errors that lead to uncertainty

Uncertainty is an estimate of the difference between a measured value and the true value

## Random Error - Random errors cause unpredictable fluctuations in an instrument's readings as a result of uncontrollable factors, such as environmental conditions - For example, air currents affecting a mass balance, or variations in starting a stopwatch - This affects the **precision** of the measurements taken, causing a wider spread of results about the mean value - Random errors cannot be predicted or completely eliminated ### Reducing Random Error - **Repeat** measurements several times and calculate an average from them - This reduces the impact of unpredictable variations - The more readings taken, the more reliable the mean value becomes

Random errors affect precision - they cause scatter in repeated measurements

## Systematic Error - Systematic errors arise from the use of faulty instruments or from flaws in the experimental method - For example, a thermometer that consistently reads 2°C too high, or always measuring from the wrong point on a ruler - This type of error is repeated consistently every time the instrument is used or the method is followed - This affects the **accuracy** of all readings obtained - all measurements are shifted by the same amount in the same direction - Systematic errors cannot be reduced by taking repeat readings ### Reducing Systematic Errors - Instruments should be **recalibrated**, or different instruments should be used - Corrections or adjustments should be made to the technique - For example, ensuring measuring equipment is properly zeroed before use - Checking calibration against known standards

Systematic errors affect accuracy - repeating measurements will not reduce them

## Zero Error - This is a type of systematic error which occurs when an instrument gives a reading when the true reading is zero - For example, a mass balance that starts at 3 g instead of 0 g - Or a voltmeter that shows 0.1 V when not connected to a circuit ### Correcting Zero Errors - Zero errors can be removed by taking the difference of the offset from each value - If a scale starts at 3 g instead of 0 g, a measurement of 75 g would actually be $75 - 3 = 72$ g - The offset could be positive or negative - Positive zero error: instrument reads above zero when it should read zero - Negative zero error: instrument reads below zero when it should read zero ### Zero Errors on Graphs - Systematic errors on graphs are shown by the offset of the line from the origin - If a graph should pass through the origin but doesn't, this indicates a systematic error in the measurements - The y-intercept represents the magnitude of the systematic error

Always check instruments read zero before taking measurements to avoid zero errors

## Precision & Accuracy These are two distinct concepts that are often confused. ### Precision - Precise measurements are ones in which there is very little spread about the mean value - In other words, how close the measured values are to each other - If a measurement is repeated several times, it can be described as precise when the values are very similar to, or the same as, each other - For example, measuring the diameter of a ball five times and getting: 5.23 cm, 5.24 cm, 5.23 cm, 5.22 cm, 5.23 cm would be precise - The precision of a measurement is reflected in the values recorded - Measurements to a greater number of decimal places are said to be more **precise** than those to a whole number - For example, 3.456 s is more precise than 3.5 s

Precision refers to how close repeated measurements are to each other

### Accuracy - A measurement is considered accurate if it is close to the true value - For example, if the true diameter of a ball is 5.24 cm, and measurements give an average of 5.23 cm, this is accurate - The accuracy can be increased by: - Repeating measurements and finding a mean of the results - Identifying and eliminating systematic errors - Using more precise instruments

Accuracy refers to how close a measurement is to the true value

### Distinguishing Precision and Accuracy - Repeating measurements also helps to identify anomalies that can be omitted from the final results - Measurements can be: - **Accurate and precise**: clustered closely together near the true value - **Accurate but not precise**: scattered measurements that average to the true value - **Precise but not accurate**: clustered closely together but far from the true value - **Neither accurate nor precise**: scattered measurements far from the true value The following charts illustrate these four scenarios for repeated measurements where the true value is 10.0 cm: **Accurate and Precise:**

**Accurate but Not Precise:**

**Precise but Not Accurate:**

**Neither Accurate nor Precise:**

## Key Comparison | Property | What it measures | Affected by | How to improve | |----------|-----------------|-------------|----------------| | **Precision** | How close measurements are to each other | Random errors | Take multiple readings and calculate mean | | **Accuracy** | How close measurements are to true value | Systematic errors | Calibrate instruments, improve technique |

Measurements can be precise but not accurate if each reading has the same systematic error

## Common Misconceptions - It is a very common mistake to confuse precision with accuracy - Measurements can be precise but **not** accurate if each measurement reading has the same error - For example, a ruler with a worn end might give measurements of 10.2 cm, 10.3 cm, 10.2 cm (precise) when the true value is 11.0 cm (not accurate) - Precision refers to the ability to take multiple readings with an instrument that are close to each other - Accuracy is the closeness of those measurements to the true value - Taking more readings improves precision but does not necessarily improve accuracy unless systematic errors are addressed