Modern Physics

# Random error: formula and equations, calculation, examples, exercises

The  random error of a physical quantity consists of unpredictable variations in the measurement of that quantity. These variations can be produced by the phenomenon being measured, by the measuring instrument or by the observer itself.

Such an error is not due to something being done wrong during the experiment, but is an error inherent in the measurement process or in the phenomenon being studied. This makes the measured quantity sometimes a little larger and sometimes a little smaller, but it usually fluctuates around a central value.

Unlike random error, systematic error can be caused by a poor calibration or an inadequate scaling factor in the measuring instrument, even a failure of experimental equipment or an inadequate observation, which causes a deviation in the same direction.

Figure 1 illustrates the difference between systematic and random error in the game of throwing darts on a circled target.

In the case of the left, the darts are concentrated around a point far from the center. The thrower of these darts, while good aim, has a systematic failure, perhaps of visual origin or in throwing mode.

On the other hand, the pitcher on the right (in Figure 1) has a large scatter around the center target, so it is a very imprecise pitcher, with little aim, that unintentionally makes a random error.

## Formulas and equations with random error

When a random error is observed in the measurement process, it is necessary to repeat the measurement several times, because, from a statistical point of view, the greater the number of measurements, the smaller the error in the final estimate of the measurement.

Obviously, in each measurement, it must be ensured that the conditions under which they are carried out are always the same.

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Suppose the measurement is repeated n times. As there is a random error in each measurement, it will have a slightly different value. Suppose the set of n measures is:

{x 1 , x 2 , x 3 ,… .., x n }

So what value to inform for the measure?

### Mean value  and standard deviation

The mean or mean value of the set of measurements must be reported, which we denote by <x> and is calculated as follows:

<x> = (x 1 + x 2 + x 3 + …… + x n ) / n

#### Standard deviation

However, this result has a margin of error given by the standard deviation. To define it, you must first know the deviation and then the variation:

-The deviation i  that each measured value xi has from the mean value <x> is:

i = x i – <x>

If the average of the deviations were calculated, it would be systematically obtained <d> = 0 , because:

<d> = (d 1 + d 2 + d 3 + …… + d n ) / n =

= [(x 1 – <x>) + (x 2  – <x>) +… + (x n – <x>)] / n

<d> = (x 1 + x 2 +… + x n ) / n – n <x> / n = <x> – <x> = 0

-The average of the deviations is not useful to know the dispersion of the measurements. On the other hand, the mean squared value of the deviations or variance, denoted by σ 2 , is.

It is calculated according to the following formula:

σ 2 = (d 2 + d 2 +…. + d 2 ) / (n -1)

In statistics, this quantity is called the variance .

And the square root of the variance is known as the standard deviation σ :

σ = √ [(d 2 + d 2 + … + d 2 ) / (n -1)]

The standard deviation σ indicates that:

1.- 68% of the measurements made are included in the interval [<x> – σ, <x> + σ] .

2.- 95% of the measurements are in the interval [<x> – 2σ, <x> + 2σ] .

3.- 99.7% of the measurements performed are in the range [<x> – 3σ, <x> + 3σ] .

## How to calculate random error?

The measurement result is the mean value of the n measurements, indicated by <x> and calculated according to the following formula:

<x> = (∑x i ) / n

However, <x> is not the “exact” value of the measurement as <x> is affected by the random error ε,  calculated as follows:

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ε = σ / √n

Where:

σ = √ [(∑ (xi – <x>) 2 ) / (n -1)]

The final measurement result must be reported in one of the following ways:

1. <x> ± σ / √n = <x> ± ε with a confidence level of 68%.
2. <x> ± 2σ / √n = <x> ± 2ε with a 95% confidence level.
3. <x> ± 3σ / √n = <x> ± 3ε with a confidence level of 99.7%.

Random error affects the last significant number of the measurement, which usually coincides with the measurement instrument’s evaluation. However, if the random error is too large, the last two significant digits can be affected by the variance.

## Random Error Examples

Random errors can appear in several cases where a measurement is taken:

### Measuring a length with a measuring tape or ruler

When a length is measured with a ruler or tape measure and the readings fall between the tick marks, this intermediate value is estimated.

Sometimes the estimate has excess and other defects; therefore, random errors are being introduced into the measurement process.

### wind speed

When measuring wind speed, there may be changes in the reading from one instant to another, due to the changing nature of the phenomenon.

When reading the volume with a graduated cylinder, even trying to minimize the parallax error, each time it is measured, the observation angle of the meniscus changes a little, which is why the measurements are affected by random error.

### When measuring a child’s height

When measuring a child’s height, especially if the child is a little restless, he makes small changes in posture, slightly altering the reading.

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### When using the bathroom scale

When we want to measure our weight with a bathroom scale, a small change in fulcrum, even a change in posture, can randomly affect the measurement.

## Exercise solved

A toy car is allowed to roll down a straight and sloping track and the time taken to cover the entire track is measured with a stopwatch.

The measurement is carried out 11 times, taking care to always release the cart from the same location, without giving any impulse and keeping the inclination fixed.

The set of results obtained is:

{3.12s 3.09s 3.04s 3.04s 3.10s 3.08s 3.05s 3.10s 3.11s 3.06s, 3.03s}

What is the random error of measurements?

### Solution

As you can see, the results obtained are not unique and vary a little.

The first thing is to calculate the average descent time value, getting 3.074545455 seconds.

Keeping the number of decimal places makes no sense, as each measurement has three significant digits and the second decimal of each measurement is uncertain, as it is at the limit of appreciation of the stopwatch, so the result is rounded to two decimal places:

= 3.08 s.

With the calculator in statistical mode, the standard deviation is  σ = 0.03 s and the standard error is σ / √11 = 0.01 s. The final result is expressed like this:

#### descent time

3.08 s ± 0.01s (with a confidence level of 68%)

3.08 s ± 0.02s (with a 95% confidence level)

3.08 s ± 0.03s (with a confidence level of 99.7%)