How do you find the precision of each measurement?

Precision, in the context of measurements, refers to the repeatability or reproducibility of a measurement. It describes how closely individual measurements of the same quantity agree with each other. Finding the precision of each individual measurement isn't usually the goal. Instead, you assess the overall precision of a set of measurements. Here's how you typically assess the precision, and how it relates to individual measurements:

1. Gather Multiple Measurements: The foundation of determining precision is having a set of measurements for the same quantity. The more measurements you have, the better the estimate of precision will be.

2. Calculate the Average (Mean): Compute the average of all your measurements. This is often considered the best estimate of the "true" value based on your data.

   • Formula: Average = (Measurement1 + Measurement2 + ... + MeasurementN) / N where N is the number of measurements.

3. Calculate the Deviation: For each individual measurement, calculate its deviation from the average. The deviation is the absolute value of the difference between the measurement and the average.

   • Formula: Deviation = |Measurement - Average|

4. Calculate the Standard Deviation (a common measure of precision): Standard deviation quantifies the spread of the data around the mean. A smaller standard deviation indicates higher precision (less spread).

   • Formula:

     • First, calculate the variance: Variance = Σ (Deviation)^2 / (N-1)
     • Then, take the square root of the variance: Standard Deviation = √Variance

     Here, Σ means the sum of all the squared deviations, and N is the number of measurements. Using (N-1) instead of N in the variance calculation gives the sample standard deviation, which is usually what's wanted in experimental work.

5. Express Precision: Precision is often expressed using statistical measures like standard deviation, standard error, or confidence intervals. You can also express precision as a percentage of the mean (relative standard deviation or coefficient of variation).

   • Standard Deviation: A direct measure of the spread of the data.
   • Relative Standard Deviation (RSD) or Coefficient of Variation (CV): RSD = (Standard Deviation / Average) * 100%. This expresses the standard deviation as a percentage of the mean, making it easier to compare the precision of measurements with different magnitudes.

How Individual Deviations Relate to Overall Precision

While you calculate the deviation for each individual measurement, remember that you don't report the "precision of each measurement" individually. Instead, the individual deviations are used to calculate a single value (like standard deviation) that characterizes the precision of the entire set of measurements. Large deviations in individual measurements will contribute to a larger standard deviation, indicating lower overall precision. Smaller deviations contribute to a smaller standard deviation, indicating higher precision.

Example:

Let's say you measure the length of an object three times and get the following results: 10.2 cm, 10.1 cm, and 10.3 cm.

1. Average: (10.2 + 10.1 + 10.3) / 3 = 10.2 cm

2. Deviations:

   • |10.2 - 10.2| = 0 cm
   • |10.1 - 10.2| = 0.1 cm
   • |10.3 - 10.2| = 0.1 cm

3. Standard Deviation:

   • Variance: ((0)^2 + (0.1)^2 + (0.1)^2) / (3-1) = 0.02 / 2 = 0.01
   • Standard Deviation: √0.01 = 0.1 cm

4. Relative Standard Deviation: (0.1 cm / 10.2 cm) * 100% ≈ 0.98%

In this example, the standard deviation of 0.1 cm (or RSD of about 1%) tells you about the overall precision of your length measurements. It reflects how much the individual measurements varied from each other. You don't say "the precision of the first measurement was 0 cm".