Content

• Steps
• Method 1 of 3: The Basics
• Method 2 of 3: Calculating Multiple Measurement Uncertainty
• Method 3 of 3: Arithmetic Operations with Errors
• Tips
• Warnings

When measuring something, you can assume that there is some "true value" that lies within the range of values ​​that you find. To calculate a more accurate value, you need to take the measurement result and evaluate it when adding or subtracting an error. If you want to learn how to find such an error, follow these steps.

Steps

Method 1 of 3: The Basics

1. 1 Express the error correctly. Let's say when measuring a stick, its length is 4.2 cm, plus or minus one millimeter. This means that the stick is approximately 4.2 cm, but in fact it can be slightly less or more than this value - with an error of up to one millimeter.
   • Write the error as: 4.2 cm ± 0.1 cm. You can also rewrite this as 4.2 cm ± 1 mm, since 0.1 cm = 1 mm.
2. 2 Always round off measurement values ​​to the same decimal place as the uncertainty. Measurement results that take into account uncertainty are usually rounded off to one or two significant figures. The most important point is that you need to round the results to the same decimal place as in the error in order to maintain consistency.
   • If the measurement result is 60 cm, then the error should be rounded to the nearest whole number. For example, the error of this measurement may be 60 cm ± 2 cm, but not 60 cm ± 2.2 cm.
   • If the measurement result is 3.4 cm, then the error is rounded to 0.1 cm.For example, the error of this measurement may be 3.4 cm ± 0.7 cm, but not 3.4 cm ± 1 cm.
3. 3 Find the error. Let's say you measure the diameter of a round ball with a ruler. This is difficult because the curvature of the ball will make it difficult to measure the distance between two opposite points on its surface. Let's say a ruler can give a result with an accuracy of 0.1 cm, but this does not mean that you can measure the diameter with the same accuracy.
   • Examine the ball and ruler to get an idea of ​​how accurately you can measure the diameter. The standard ruler has a clear 0.5 cm mark, but you may be able to measure the diameter with greater accuracy than this. If you think you can measure the diameter with an accuracy of 0.3 cm, then the error in this case is 0.3 cm.
   • Let's measure the diameter of the ball. Let's say you got a reading of about 7.6 cm. Just indicate the measurement result along with the error. The ball diameter is 7.6 cm ± 0.3 cm.
4. 4 Calculate the error in measuring one item out of several. Let's say you are given 10 compact discs (CDs), each the same size. Let's say you want to find the thickness of just one CD. This value is so small that the error is almost impossible to calculate.However, to calculate the thickness (and its uncertainty) of one CD, you can simply divide the measurement (and its uncertainty) of the thickness of all 10 CDs stacked together (one on top of the other) by the total number of CDs.
   • Let's say the accuracy of measuring a stack of CDs using a ruler is 0.2 cm. So your error is ± 0.2 cm.
   • Let's say the thickness of all CDs is 22 cm.
   • Now divide the measurement result and the error by 10 (the number of all CDs). 22 cm / 10 = 2.2 cm and 0.2 cm / 10 = 0.02 cm. This means that the thickness of one CD is 2.20 cm ± 0.02 cm.
5. 5 Measure several times. To improve the accuracy of measurements, whether it is measuring length or time, measure the desired value several times. Calculation of the average value from the obtained values ​​will increase the measurement accuracy and the calculation of the error.

Method 2 of 3: Calculating Multiple Measurement Uncertainty

1. 1 Take a few measurements. Let's say you want to find how long it takes for the ball to fall from the height of the table. For best results, measure the fall time a number of times, for example, five. Then you need to find the average of the five obtained time measurements, and then add or subtract the standard deviation for the best result.
   • Let's say that as a result of five measurements, the results are obtained: 0.43 s, 0.52 s, 0.35 s, 0.29 s and 0.49 s.
2. 2 Find the arithmetic mean. Now find the arithmetic mean by adding up five different measurements and dividing the result by 5 (the number of measurements). 0.43 + 0.52 + 0.35 + 0.29 + 0.49 = 2.08 s. 2.08 / 5 = 0.42 s. Average time 0.42 s.
3. 3 Find the variance of the obtained values. To do this, first, find the difference between each of the five values ​​and the arithmetic mean. To do this, subtract 0.42 s from each result.
   • • 0.43 s - 0.42 s = 0.01 s
     • 0.52 s - 0.42 s = 0.1 s
     • 0.35 s - 0.42 s = -0.07 s
     • 0.29 s - 0.42 s = -0.13 s
     • 0.49 s - 0.42 s = 0.07 s
     • Now add the squares of these differences: (0.01) + (0.1) + (-0.07) + (-0.13) + (0.07) = 0.037 s.
     • You can find the arithmetic mean of this sum by dividing it by 5: 0.037 / 5 = 0.0074 s.
4. 4 Find the standard deviation. To find the standard deviation, simply take the square root of the arithmetic mean of the sum of squares. The square root of 0.0074 = 0.09 s, so the standard deviation is 0.09 s.
5. 5 Write down your final answer. To do this, record the mean of all measurements plus or minus standard deviation. Since the mean of all measurements is 0.42 s and the standard deviation is 0.09 s, the final answer is 0.42 s ± 0.09 s.

Method 3 of 3: Arithmetic Operations with Errors

1. 1 Addition. To add the values ​​with errors, add separately the values ​​and separately the errors.
   • (5cm ± 0.2cm) + (3cm ± 0.1cm) =
   • (5cm + 3cm) ± (0.2cm + 0.1cm) =
   • 8cm ± 0.3cm
2. 2 Subtraction. To subtract values ​​with uncertainties, subtract values ​​and add up uncertainties.
   • (10cm ± 0.4cm) - (3cm ± 0.2cm) =
   • (10 cm - 3 cm) ± (0.4 cm + 0.2 cm) =
   • 7cm ± 0.6cm
3. 3 Multiplication. To multiply the values ​​with errors, multiply the values ​​and add the RELATIVE errors (in percent). Only the relative error can be calculated, not the absolute one, as is the case with addition and subtraction. To find the relative error, divide the absolute error by the measured value, then multiply by 100 to express the result as a percentage. For example:
   • (6 cm ± 0.2 cm) = (0.2 / 6) x 100 - adding a percent sign gives 3.3%.
     Consequently:
   • (6 cm ± 0.2 cm) x (4 cm ± 0.3 cm) = (6 cm ± 3.3%) x (4 cm ± 7.5%)
   • (6cm x 4cm) ± (3.3 + 7.5) =
   • 24 cm ± 10.8% = 24 cm ± 2.6 cm
4. 4 Division. To divide the values ​​with uncertainties, divide the values ​​and add the RELATIVE uncertainties.
   • (10 cm ± 0.6 cm) ÷ (5 cm ± 0.2 cm) = (10 cm ± 6%) ÷ (5 cm ± 4%)
   • (10 cm ÷ 5 cm) ± (6% + 4%) =
   • 2cm ± 10% = 2cm ± 0.2cm
5. 5 Exponentiation. To raise a value with an error to a power, raise the value to a power, and multiply the relative error by a power.
   • (2.0cm ± 1.0cm) =
   • (2.0 cm) ± (50%) x 3 =
   • 8.0 cm ± 150% or 8.0 cm ± 12 cm

Tips

• You can give an error both for the overall result of all measurements, and for each result of one measurement separately.Typically, data obtained from multiple measurements is less reliable than data obtained directly from individual measurements.

Warnings

• The exact sciences never work with "true" values. While a correct measurement is likely to give a value within the margin of error, there is no guarantee that this will be the case. Scientific measurements allow for error.
• The uncertainties described here are only applicable for normal distribution cases (Gaussian distribution). Other probability distributions require different solutions.