Whenever you take a measurement during a data collection, you can assume that there is a "real" value that falls within the range of the measurements taken. To calculate the uncertainty, you will need to find the best estimate of your measure, after which you can consider the results by adding or subtracting the uncertainty measure. If you want to know how to calculate uncertainty, just follow these steps.
Steps
Method 1 of 3: Learn the Basics
Step 1. Express uncertainty in its correct form
Suppose we are measuring a stick that falls 4, 2 cm, centimeter plus, centimeter minus. This means that the stick falls "almost" by 4.2 cm, but, in reality, it could be a slightly smaller or larger value, with the error of one millimeter.
Express the uncertainty like this: 4, 2 cm ± 0, 1 cm. You can also write: 4, 2 cm ± 1 mm, as 0, 1 cm = 1 mm
Step 2. Always round the experimental measurement to the same decimal place as the uncertainty
Measures involving an uncertainty calculation are generally rounded to one or two significant digits. The most important point is that you should round the experimental measurement to the same decimal place as the uncertainty to keep the measurements consistent.
- If the experimental measurement were 60 cm, then the uncertainty should also be rounded to a whole number. For example, the uncertainty for this measurement might be 60cm ± 2cm, but not 60cm ± 2, 2cm.
- If the experimental measurement is 3.4 cm, then the uncertainty calculation should be rounded to 0.1 cm. For example, the uncertainty for this measurement may be 3.4cm ± 0.7cm, but not 3.4cm ± 1cm.
Step 3. Calculate uncertainty from a single measurement
Suppose you are measuring the diameter of a round ball with a ruler. This task is really tough, because it is difficult to tell exactly where the outer edges of the ball are with the ruler, as they are curved, not straight. Let's say that the ruler can find the measurement to the tenth of a centimeter: it doesn't mean that you can measure the diameter with this level of precision.
- Study the edges of the ball and the ruler to make sense of how reliable it is to measure its diameter. In a standard ruler, the 5mm markings are clearly seen, but we assume you can get a better approximation. If you feel like you can go down to an accuracy of 3mm, then the uncertainty is 0.3cm.
- Now, measure the diameter of the sphere. Suppose we get about 7.6 cm. Just state the estimated measure together with the uncertainty. The diameter of the sphere is 7.6cm ± 0.3cm.
Step 4. Calculate the uncertainty of a single measurement of multiple objects
Suppose you are measuring a stack of 10 CD cases, all of which are the same length. You want to find the thickness measurement of a single case. This measure will be so small that your uncertainty percentage will be high enough. But when you measure the ten CDs stacked together, you can only divide the result and the uncertainty by the number of CDs to find the thickness of a single case.
- Let's say you can't go beyond 0.2cm using a ruler. Thus your uncertainty is ± 0.2cm.
- Let's assume that all stacked CDs are 22cm thick.
- Now, just divide the measure and uncertainty by 10, which is the number of CDs. 22 cm / 10 = 2, 2 cm and 0, 2 cm / 10 = 0, 02 cm. This means that the case thickness of a single CD is 2.0 cm ± 0.02 cm.
Step 5. Take your measurements several times
To increase the certainty of your measurements, if you are measuring the length of the object or the amount of time it takes for an object to cover a certain distance, you can increase the chances of getting an accurate measurement if you take different measurements. Finding the average of your multiple measurements will help you get a more accurate picture of the measurement when calculating uncertainty.
Method 2 of 3: Calculate the Uncertainty of Multiple Measurements
Step 1. Take several measurements
Suppose you want to calculate how long it takes for a ball to fall from a table to the ground. For best results, you will need to measure the ball as it falls from the top of the table at least a couple of times… let's say five. Then you'll need to find the average of the five measurements and add or subtract the standard deviation from that number to get the most reliable results.
Let's say you measured the following five times: 0, 43, 0, 52, 0, 35, 0, 29 and 0, 49 s
Step 2. Find the average by adding the five different measurements and dividing the result by 5, the amount of measurements taken
0, 43 + 0, 52 + 0, 35 + 0, 29 + 0, 49 = 2, 08. Now divide 2, 08 by 5. 2, 08/5 = 0, 42. The average time is 0, 42 s.
Step 3. Find the variance of these measures
To do this, first, find the difference between each of the five measures and the average. To do this, just subtract the measurement from 0.42 s. Here are the five differences:
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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
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Now you need to sum the squares of these differences:
(0.01 s)2 + (0, 1 s)2 + (- 0.07 s)2 + (- 0, 13 s)2 + (0.07 s)2 = 0, 037 s.
- Find the mean of the sum of these squares by dividing the result by 5. 0, 037 s / 5 = 0, 0074 s.
Calculate Uncertainty Step 9 Step 4. Find the standard deviation
To find the standard deviation, simply find the square root of the variance. The square root of 0.0074 is 0.09, so the standard deviation is 0.09s.
Calculate Uncertainty Step 10 Step 5. Write the final measure
To do this, simply combine the mean of the measurements with the standard deviation. Since the mean of the measurements is 0.42 s and the standard deviation is 0.09 s, the final measurement is 0.42 s ± 0.09 s.
Method 3 of 3: Perform Arithmetic Operations with Approximate Measurements
Calculate Uncertainty Step 11 Step 1. Add approximate measurements
To add approximate measures, add the measures themselves and also their uncertainties:
- (5cm ± 0.2cm) + (3cm ± 0.1cm) =
- (5cm + 3cm) ± (0, 2cm + 0, 1cm) =
- 8 cm ± 0.3 cm
Calculate Uncertainty Step 12 Step 2. Subtract approximate measurements
To subtract approximate measurements, subtract them and then add their uncertainties:
- (10cm ± 0, 4cm) - (3cm ± 0, 2cm) =
- (10 cm - 3 cm) ± (0, 4 cm + 0, 2 cm) =
- 7 cm ± 0, 6 cm
Calculate Uncertainty Step 13 Step 3. Multiply approximate measurements
To multiply the uncertain measures, simply multiply them and add theirs relative uncertainties (in the form of a percentage). Calculating uncertainty in multiplications does not work with absolute values, as in addition and subtraction, but with relative ones. Get the relative uncertainty by dividing the absolute uncertainty by a measured value and then multiplying by 100 to get the percentage. For instance:
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(6 cm ± 0, 2 cm) = (0, 2/6) x 100 and added a% sign. The result is 3, 3%
Therefore:
- (6cm ± 0.2cm) x (4cm ± 0.3cm) = (6cm ± 3.3%) x (4cm ± 7.5%)
- (6 cm x 4 cm) ± (3, 3 + 7, 5) =
- 24cm ± 10.8% = 24cm ± 2.6cm
Calculate Uncertainty Step 14 Step 4. Divide approximate measurements
To divide the uncertain measures, simply divide their respective values and add theirs relative uncertainties (the same process seen for multiplications):
- (10 cm ± 0, 6 cm) ÷ (5 cm ± 0, 2 cm) = (10 cm ± 6%) ÷ (5 cm ± 4%)
- (10 cm ÷ 5 cm) ± (6% + 4%) =
- 2 cm ± 10% = 2 cm ± 0, 2 cm
Calculate Uncertainty Step 15 Step 5. Increase an uncertain measure exponentially
To increase an uncertain measure exponentially, simply put the measure at the indicated power and multiply the uncertainty by that power:
- (2.0 cm ± 1.0 cm)3 =
- (2.0 cm)3 ± (1.0 cm) x 3 =
- 8, 0 cm ± 3 cm
Advice
You can report the results and standard uncertainty for all results as a whole or for each result within a dataset. As a general rule, data from multiple measurements is less accurate than data extracted directly from single measurements
Warnings
- Optimal science never discusses "facts" or "truths". While the measurement is very likely to fall within your uncertainty range, there is no guarantee that this is always the case. Scientific measurement implicitly accepts the possibility of being wrong.
- The uncertainty thus described is applicable only in normal statistical cases (Gaussian type, with a bell-shaped trend). Other distributions require different methodologies to describe uncertainties.
