WHAT IS A DETERMINISTIC EXPERIMENT? - DUDAS - 2025

The deterministic experiment, in statistics, is one that has a predictable and reproducible result as long as the same initial conditions and parameters are maintained. That is, the cause-effect relationship is fully known.

For example, the time it takes for the sand of a clock to move from one compartment to the other is a deterministic experiment, because the result is predictable and reproducible. As long as the conditions are the same it will take the same time to travel from capsule to capsule.

Figure 1. The time it takes for the sand to move from one compartment to the other is a deterministic experiment. Source: Pixabay

Many physical phenomena are deterministic, some examples being the following:

- An object denser than water, such as a stone, will always sink.

- A float, which is less dense than water, will always float up (unless a force is exerted to keep it submerged).

- The boiling temperature of water at sea level is always 100 ºC.

- The time it takes for a die that is dropped from rest to fall, since it is determined by the height from which it was dropped and this time is always the same (when it is dropped from the same height).

Taking advantage of the example of the dice. If it is dropped, even when care is taken to give it the same orientation and always at the same height, it is difficult to predict which side it will show up once it has stopped on the ground. This would be a random experiment.

Theoretically, if data such as: position were known with infinite precision; initial speed and orientation of the die; shape (with rounded or angular edges); and coefficient of restitution of the surface on which it falls, perhaps it would be possible to predict, by complex calculations, which face the die will show up when stopped. But any slight variation in the starting conditions would give a different result.

Such systems are deterministic and at the same time chaotic, since a small alteration of the initial conditions changes the final result in a random way.

Measurement

Deterministic experiments are completely measurable, but even so the measurement of their result is not infinitely precise and has a certain margin of uncertainty.

Take, for example, the following totally deterministic experiment: dropping a toy car down a straight sloping track.

Figure 2. A car descends a rectilinear slope in a deterministic experiment. Source: Pixabay.

It is always released from the same starting point, being careful not to give any impulse. In this case, the time it takes for the car to travel the track must always be the same.

Now a child sets out to measure the time it takes for the cart to travel the track. For this you will use the stopwatch that is built into your mobile phone.

Being an observant boy, the first thing you notice is that your measuring instrument has finite precision, because the smallest time difference that the stopwatch can measure is 1 hundredth of a second.

Then the child proceeds to carry out the experiment and with the mobile stopwatch measures 11 times - let's say to be sure - the time it took for the stroller to travel the inclined plane, obtaining the following results:

The boy is surprised, because at school he had been told that this is a deterministic experiment, but for each measure he obtained a slightly different result.

Variations in measurement

What can be the causes that each measurement has a different result?

One cause may be the precision of the instrument, which as already mentioned is 0.01s. But note that the differences in the measurements are above that value, so other causes must be considered, such as:

- Small variations of the starting point.

- Differences in the start and pause of the stopwatch, due to the child's reaction time.

Regarding the reaction time, there is certainly a delay from when the child sees the cart start to move, until he presses the timer.

Similarly, on arrival there is a delay due to time to reaction time. But the start and arrival delays are compensated, so the time obtained must be very close to the true one.

In any case, the compensation for the reaction delay is not exact, because the reaction times can have small variations in each test, which explains the differences in the results.

What then is the true result of the experiment?

Results of a measurement and error

To report the final result, we must use statistics. Let's first see how often the results are repeated:

- 3.03s (1 time)

- 3.04s (2 times)

- 3.05s (1 time)

- 3.06s (1 time)

- 3.08s (1 time)

- 3.09s 1 time

- 3.10s (2 times)

- 3.11s (1 time)

- 3.12s (1 time)

When ordering the data, we realize that a more repeated mode or result cannot be specified. Then the result to report is the arithmetic mean, which can be calculated like this:

The result of the above calculation is 3.074545455. Logically, it does not make sense to report all these decimals in the result, because each measurement has only 2 decimal places of precision.

Applying the rounding rules, it can be stated that the time it takes for the cart to travel the track is the arithmetic mean rounded to two decimal places.

The result that we can report for our experiment is:

- Measurement error

As we have seen in our example of a deterministic experiment, every measurement has an error, since it cannot be measured with infinite precision.

In any case, the only thing that can be done is to improve the instruments and the measurement methods, in order to obtain a more precise result.

In the previous section, we gave a result for our deterministic experiment of the time it takes for the toy car to travel a sloping track. But this result contains an error. Now we will explain how to calculate that error.

- Calculation of the measurement error

In the measurements for time, a dispersion is noted in the measurements made. Standard deviation is a frequently used form in statistics to report the spread of data.

Variance and standard deviation

The way to calculate the standard deviation is like this: first you find the variance of the data, defined in this way:

If the variance is taken the square root, then the standard deviation is obtained.

Figure 3. Formulas for the mean and standard deviation. Source: Wikimedia Commons.

The standard deviation for the toy car descent time data is:

σ = 0.03

The result was rounded to 2 decimal places, because the precision of each of the data is 2 decimal places. In this case, 0.03s represents the statistical error of each of the data.

However, the average or arithmetic mean of the times obtained has a smaller error. The mean error is calculated by dividing the standard deviation by the square root of the total number of data.

Average error = σ / √N = 0.03 / √11 = 0.01

That is, the statistical error of the time average is 1 hundredth of a second and in this example, it coincides with the appreciation of the stopwatch, but this is not always the case.

As a final result of the measurement, it is reported then:

t = 3.08s ± 0.01s is the time it takes for the toy car to travel the inclined track.

It is concluded that even when it is a deterministic experiment, the result of its measurement does not have infinite precision and always has a margin of error.

And also, to report the final result it is necessary, even when it is a deterministic experiment, to use statistical methods.

References

1. CanalPhi. Deterministic experiment. Recovered from: youtube.com
2. MateMovil. Deterministic experiment. Recovered from: youtube.com
3. Pishro Nick H. Introduction to probability. Recovered from: probabilitycourse.com
4. Ross. Probability and statistics for engineers. Mc-Graw Hill.
5. Statistic how to. Deterministic: Definition and Examples. Recovered from: statisticshowto.datasciencecentral.com
6. Wikipedia. Typical deviation. Recovered from: es.wikipedia.com
7. Wikipedia. Experiment (probability theory). Recovered from: en.wikipedia.com