Accuracy: The Quest for the True Value
Accuracy vs. Precision: The Key Distinction
One of the most important lessons in science is understanding the difference between accuracy and precision. They are often confused, but they describe two very different things.
• Accuracy: How close a result is to the true value.
• Precision: How close repeated measurements are to each other (consistency or repeatability).
Think back to the dartboard. If your darts are all clustered together in one spot, that's high precision. But if that cluster is far from the bullseye, it's not accurate. To be truly good, you want both: darts clustered tightly and right on the bullseye. Let's see this with numbers. Suppose the true length of a table is exactly 150.0 cm.
| Scenario | Three Measurements (cm) | Average (cm) | Accuracy | Precision |
|---|---|---|---|---|
| Low Accuracy, Low Precision | 148.2, 151.7, 149.5 | 149.8 | Low (average is off) | Low (numbers are spread out) |
| Low Accuracy, High Precision | 147.8, 147.6, 147.9 | 147.8 | Low (consistently wrong) | High (numbers are very close) |
| High Accuracy, High Precision | 149.9, 150.1, 150.0 | 150.0 | High (perfect average) | High (numbers are very close) |
The third scenario is the goal in any careful experiment or measurement. High precision without accuracy often indicates a systematic error2, like a scale that is not zeroed properly and always adds 2 grams. Low precision usually comes from random errors3, like small, unpredictable changes in how you read a ruler.
Quantifying Accuracy: Error and Uncertainty
Scientists and engineers don't just say "this is accurate." They use numbers to describe how accurate something is. The main way to do this is by calculating error.
The error in a measurement is the difference between the measured value and the true value.
$ \text{Error} = \text{Measured Value} - \text{True Value} $
A smaller absolute error means higher accuracy. Error can be positive (measured too high) or negative (measured too low).
For example, if you measure the boiling point of water as 99.1°C at sea level, and the true value is 100.0°C, your error is 99.1 - 100.0 = -0.9°C. Often, we talk about percent error to understand the size of the error relative to the true value.
$ \text{Percent Error} = \left| \frac{\text{Error}}{\text{True Value}} \right| \times 100\% $
In our boiling point example: Percent Error = |(-0.9) / 100.0| × 100% = 0.9%. A 0.9% error is relatively small, showing good accuracy for many purposes.
Another important concept is uncertainty. While error is the difference from the true value, uncertainty is an estimate of how much the measured value could vary. It is often shown with a "±" symbol. If a scale reads 250 g ± 2 g, it means the true value is believed to be between 248 g and 252 g. Smaller uncertainty means we have more confidence in the measurement's accuracy.
The Sources of Inaccuracy: What Goes Wrong?
Why don't we get the true value every time? Inaccuracy creeps in from different sources, broadly categorized as systematic errors and random errors.
| Error Type | Description | Effect on Accuracy | How to Reduce It |
|---|---|---|---|
| Systematic Error (Bias) | A consistent, repeatable error linked to the system or method. It pushes all measurements in one direction. | Reduces accuracy. Causes low accuracy even with high precision. | Calibrate instruments, use correct technique, control experimental conditions. |
| Random Error | Unpredictable, small variations in measurement due to chance. | Reduces precision, which can make it harder to find the accurate average. | Take more measurements and average them, use more sensitive equipment. |
| Human Error (Mistakes) | Blunders like misreading an instrument, recording data incorrectly, or using the wrong formula. | Can completely destroy both accuracy and precision. | Carefulness, double-checking work, following procedures. |
A classic example of a systematic error is using a metal ruler on a very hot day. The ruler expands, so every centimeter mark is actually slightly longer than a true centimeter. Every measurement you make with it will be less than the true length. Random error could be you slightly misjudging where the end of an object lines up with the ruler's marks each time you measure.
Accuracy in Action: From Labs to Daily Life
Accuracy isn't just for scientists in white coats. It is a practical concern in many fields that affect us directly.
1. Medicine and Pharmacy: When a doctor prescribes medicine, the dosage must be extremely accurate. A 5 mg pill versus a 50 mg pill is a tenfold difference that could be harmless or deadly. Pharmacists use highly accurate scales to measure compounds.
2. Navigation and GPS4: Your phone's GPS determines your location by calculating the time it takes for signals to travel from satellites. If the timing is off by even a thousandth of a second, the calculated position could be off by hundreds of kilometers. Engineers work tirelessly to make these clocks incredibly accurate.
3. Sports and Timing: In Olympic races, where victories are decided by hundredths of a second, timing systems must be extraordinarily accurate. A tiny systematic error could give the gold medal to the wrong athlete.
4. Cooking and Baking: Baking is a science. If a recipe calls for 1 teaspoon of baking powder and you use a tablespoon (which is three teaspoons), your cake will rise too much and collapse—a tasty lesson in the importance of accurate measurement!
5. Climate Science: To understand global warming, scientists need accurate temperature records from around the world over decades. Systematic errors, like a thermometer placed near a heat source, must be identified and corrected to see the true trend.
Improving Accuracy: Tools and Techniques
How do we get closer to the true value? Here are some standard methods used across disciplines:
Calibration: This is the number one tool against systematic error. Calibration means checking an instrument's readings against a known standard. You might place a 100 gram certified weight on a scale to see if it reads 100.0 grams. If it reads 101.5 grams, you adjust (calibrate) it to read correctly.
Using More Precise Instruments: A ruler with millimeter marks is more precise (and can be more accurate) than one with only centimeter marks. A digital thermometer that reads to 0.1°C allows for more accurate readings than one that only shows whole degrees.
Repeating Measurements and Averaging: This helps cancel out random errors. If you measure something multiple times, the random errors will sometimes be positive and sometimes negative. The average of all measurements tends to be closer to the true value than any single measurement.
Controlling Variables: In an experiment, you try to keep everything constant except the one thing you are testing. This reduces unexpected systematic errors. For instance, measuring the growth of plants with the same amount of water, light, and soil type.
Peer Review and Replication: In science, other scientists check each other's work (peer review) and try to repeat experiments (replication). If many independent teams get the same accurate result, confidence in that result grows.
Important Questions
Yes, absolutely. This is a very common situation. A real-world example is a clock that is running 5 minutes fast. Every time you look at it, it consistently gives the same wrong time (high precision), but it is never accurate because it is always 5 minutes ahead of the true time. This is caused by a systematic error—the clock's mechanism is set incorrectly.
Percent error puts the size of the error into context. An error of 1 meter is huge if you're measuring the length of a desk (about 200% error) but very small if you're measuring the distance between two cities (maybe a 0.001% error). Percent error tells you the relative significance of the mistake, helping you decide if the accuracy is good enough for your purpose.
One of the simplest and most effective ways is to calibrate your instruments before you start. For example, make sure a digital balance reads 0.00 when nothing is on it (tare it). Or, check a thermometer in ice water to see if it reads 0°C. This simple step can remove a major source of systematic error right at the beginning.
Footnote
1 Navigation systems: Systems like GPS (Global Positioning System) that use satellites to determine geographic location.
2 Systematic error: An error that is consistently in the same direction (always too high or always too low) due to a flaw in the measurement system.
3 Random error: An error that varies unpredictably in both direction and magnitude from one measurement to the next, caused by uncontrollable factors.
4 GPS: Global Positioning System. A satellite-based navigation system that provides location and time information anywhere on Earth.