Experimental errors calculator

Introduction

Sources of error

Error propagation

Experimental values

Error in measurements

Standard Deviation

Error in an average

Error in measurements

Insert the theoretical value

Insert the experimental value

Calculate

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The absolute error is:

The relative error is:

The percentage error is:

Which sources of error do you think were there?

Error in an average

Insert the standard error in each measurement

Insert the number of values included in the average

Calculate

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The average error is:

Is your experimental value within the error in the average?

Which sources of error do you think were there?

Standard Deviation

Insert each value you have measured. Separate the values using a 'space'.

Calculate

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The mean value is:

The standard deviation(σ) is:

Compare the absolute error to the standard deviation. Is the absolute error within the 1σ, 2σ or 3σ of the standard deviation?

Do you regard your measurements to be precise?

WWhich sources of error do you think were there?

Experimental values

Insert each value you have measured. Separate the values using a 'space'.

Insert the systematic error (if any, indicate if it is + or -).

Calculate

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The corrected set of measurements is:

The mean value is:

The maximum probable error is:

What sources of error are there?

There are three general types of errors which may distort experimental outcomes:
- Human error
- Systematic error
- Random error

Systematic errors are related to the precision of the instrument. For example, lets assume that one attempts to measure the length of a straight line with a common ruler which has scaled divisions every 0,1cm. If the actual length of the line is 3,48 cm the experimenter would measure 3,5 cm. So, the precision of the measurement depends on the precision of the instrument. In the case of the line, the correct answer would be (3,5±0,1) cm; meaning that the length of the line is between 3,4 and 3,6 cm. Thus, all measurements should be accompanied by an error factor, also called reading error, which corresponds to the smallest subdivision of the instrument.

Another systematic error stems from the wrong calibration of an instrument. For example, if a speedometer has as initial value 10

Systematic errors however, don't only occur due to the precision of instruments but also due to the experimental set up. If for example one is performing an experiment about the free fall of objects in a non-vacuum environment, then the resistance of air would produce a systematic error to all measurements.

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When conducting an experiment using laboratory equipment and not simulations, the measurements of any physical quantity can never be 100% accurate. The deviation between an experimental value and the actual true value of a quantity can be due to many reasons which we usually refer to as error factors. Such factors can for example be the accuracy of the measuring tool, the experimental set-up used or random mistakes made by the experimenter.Let's have a look at the following example.Say we are asked to measure the diameter of the ball in the image below.

If we use a ruler to measure the diameter, the answer would be 6,2 cm. However, if we measure the ball using a micrometer, which is a more accurate instrument, the answer would be 6,1845 cm. The deviation between these two measurements is due to the presence of error factors. In this case, the error is mainly coming from the accuracy of our instrument. Still, even the measurement with the micrometer, which is a more accurate instrument, could still deviate from the true value. The source of error there could be for example to the calibration of the instrument.

**Accuracy** is a measure of how close a measurement is to the true value of a quantity. This can be determined by calculating the percentage error of a measurement.

** Precision** on the other hand means that a set of independent measurements of the same quantity have low deviation. This can be determined by calculating the standard deviation of our measurements.

The Calculation of experimental error factors tool will help you in calculating different types of errors so you can have a better assessment of your result when performing experiments.

If we use a ruler to measure the diameter, the answer would be 6,2 cm. However, if we measure the ball using a micrometer, which is a more accurate instrument, the answer would be 6,1845 cm. The deviation between these two measurements is due to the presence of error factors. In this case, the error is mainly coming from the accuracy of our instrument. Still, even the measurement with the micrometer, which is a more accurate instrument, could still deviate from the true value. The source of error there could be for example to the calibration of the instrument.

Accuracy and precision of measurements

Although accuracy and precision have similar meanings, in math and science they have different definitions which we need to be aware of.
The Calculation of experimental error factors tool will help you in calculating different types of errors so you can have a better assessment of your result when performing experiments.

Error propagation

When making an experiment, it is often the case that we can't measure directly the quantity we want. Instead, we measure other ones that when combined, provide us with the desired quantity. For example we cannot measure the density of a material directly. To achieve that, we have to take measurements of mass and volume and then use the formula to calculate the density. So, what is the error of a measurement in such a case?

The error of our measurement depends on the combination of the individual measurements, or in other words in the type of formula we are using. Thus we have to use each case separately. In all the cases below we refer to 'A' as the quantity we seek to measure through other quantities; x, y, z as the measured quantities A depends up on; and 'c' is a given constant. So in our previous example A would be the density and x,y would be the mass and volume respectively while we have no z and no c in our equation. Respectively the errors for each quantity are δA, δx, δy and δz.

If A is the product of multiplying a quantity with a constant (called 'c')

Then the error δA for A is

If A is the sum or difference of x, y and z (it makes no difference whether we have '+' or '-')

Then the error δA for A is

If A is the product of x,y and z (it makes no difference if it is multiplication or division) and a constant c (if there is no constant in your equation just set c=1)

Then the error δA for A is

If A is a polynomial function of one variable, x (c again being a constant)

Then the error δA for A is

Before starting your experiment, you need check your instruments for possible zero error. To do that, set your instrument to zero and check its indication. Does it say zero or does it give another value? If so, that value is considered to be a systematic error and you will have to correct all your measurements by subtracting it.

The **theoretical value** of a quantity can refer to different things. If we are trying to verify a given theory then the theoretical value is the value derived from the respective mathematical equation.For example, let's say we are performing an experiment to measure the time in which an object will reach the ground from a given height according to the laws of free fall. Then, the theoretical value will be the one derived from the equationwhereas the experimental value would be the one we actually measured during the experiment using a stop watch.

Another case is when we are measuring quantities whose value is already known from the bibliography. For example, let's say we want to measure the density of glycerin. In that case, in our experiment we would be taking measurements of mass and volume and calculate the density using the equationThe theoretical value would be the one derived from bibliography which is 1.261 g/cm^{3}.

Another case is when we are measuring quantities whose value is already known from the bibliography. For example, let's say we want to measure the density of glycerin. In that case, in our experiment we would be taking measurements of mass and volume and calculate the density using the equationThe theoretical value would be the one derived from bibliography which is 1.261 g/cm

The

When we present a measurement it is also important to present the error of that measurement. We usually accompany our measurement with either the

The

is the relative error expressed in terms of percentage.

However, you must always keep in mind that low standard deviation only indicates that the set of measurements have a good

The formula to calculate the standard deviation is:

The **maximum probable error** is the maximum deviation from the mean.