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# How to find the average value: what random errors are and what physicists need the average value for

Random errors are inaccuracies, which change randomly at repetition of equally accurate measurements. Quite often, an experimenter obtains somewhat different results with no regard for how many repeated measurements there are; the size of the difference depends on the total sum of instrumental errors and errors of calculation. This situation is caused by the influence of random factors that cannot be eliminated during experiments or courses of natural phenomena (e.g., registration of space radiation by a particle counter). Suppose, we are to define a range of flight of a rubber ball shot from a ballistic gun in a horizontal direction. Even if the ball is under constant experimental conditions it will not be falling onto one and the same place on a table surface – and this is connected with the fact that the ball does not have an ideal shape, and also friction force impacts the striking mechanism so the striker acts differently influencing the position of the gun in the space, etc. This distribution of collected results happens almost always during the performance of a series of experiments, and evidence for this you can find in almost every Turabian paper written on a mathematical topic. The causes of random errors lie in the imperfection of our senses and in many other circumstances, which accompany the process of measurement and cannot be taken into account in advance.

Direct measurements are those that are made depending directly on instrument readings. The accuracy of measurement of physical quantities is determined, first of all, by precision of the instrument selected for measurement. The choice of a measuring tool is a very important step in the preparation for measurements. Before proceeding to the measurement, you must first determine limits of devices’ and instruments’ accuracy (that is, instrumental errors). Consequently, to reduce random errors an experimenter needs to increase the number of tests and try to benefit from using the average value. Thus, there is a partial compensation of random deviations in measurement results and the re-balancing of data that tend to deviate towards overstatement or understatement. Calculation of random errors is produced by methods of probability theory and mathematical statistics, and is defined by the choice of a distribution function of random variables. Such techniques are widely put in practice and one is not able to accomplish an accounting paper without a proper statistical design. For all distribution functions, the basic distribution is Gauss, which is valid for a large number of equally accurate measurements. Finally, before recording the measurement result you must do a correct assessment and rounding of absolute error, and then remember properly how to find the average value. When recording the result, be sure to indicate relative error and confidence probability p with which confidence interval Δx has been calculated.

## How to find the average of a group of numbers: the essential terminology of the science of measurement

It has to be said that a physical quantity and its dimensionality is not the same thing. Hence, physical quantities completely different in its nature can share an equal dimensionality (for example, such quantities as work and torque). Dimensionality does not contain information on how to find the average value or about the mathematical nature of a quantity, that is, whether it is a scalar, vector or tensor quantity. Nevertheless, the dimensionality of a quantity is important to verify the relationship between different physical quantities. For correct substitution of numerical values in formulas, all the values should be expressed in derivative or basic units of the same system; such a formality is very important because only on this condition coefficients depending on the choice of units will not appear in the formulas. To find out how the instruments of measurement work let us analyze a few case study examples and look out on some basic conceptions and terms, namely:

• Measurements can be direct or indirect, single and multiple, equally accurate and unequal; the results of the measurement of physical quantities are always approximate numbers. Multiple measurements are recorded in a table with obligatory indication of units of measure.
• The accuracy of measurements may not exceed the accuracy of the measuring instrument and is characterized by two main parameters: absolute and relative errors. Absolute error indicates how much the true value of an unknown quantity differs from the value that an experimenter has received as the result of the measurement. At the same time, relative error characterizes the quality of the measurement, i.e. how strongly match the measured value and the absolute measurement error.
• all digits of the approximate numbers are called significant digits, except for zeros standing to the left of the first non-zero digit. Approximate numbers are usually written in the standard format: b × 10n, where b is the floating-point number of a decimal containing as many digits as many there are significant digits in the approximate number, and n is a positive or negative integer called number exponent. To understand how to find the average value you must know that if absolute error of an approximate number does not exceed a unit in the last category, then all significant digits of the approximate number should be called true digits.
• A Slip (or blunder) is an error significantly larger than expected at specified conditions. Slips occur due to lack of care and accuracy, due to malfunction of measuring instrument, improper recordings of the results of measurements and so on. It is almost always possible to identify these errors by repeating measurements with other equipment, by using a different method or involving another observer into the procedure of measurement. If the results of measurements are corresponding to gross errors they must be rejected and substituted by results of the new measurements – this is no less important than to know how to write a coursework or compose a physics term paper. In order to eliminate any measurement blunders, all operations must be repeated at least twice or thrice.
• Instrumental errors are caused by imperfection of the instruments’ design and inaccuracies in the instruments’ calibration (e.g. little differences in the length of scale arms, a mismatch between the scale center with the rotation axis in a pointer instrument, a change in the course of a hand stopwatch due to temperature changes, etc.). The reduction of instrumental errors might be achieved by employing more sophisticated and precise instruments. However, it is impossible to completely eliminate instrument errors.
• Methodical errors are caused by shortcomings of the method used for measurements, imperfections of the theory of a physical phenomenon and inaccuracies made during calculation of a formula for finding the measured quantity. This class of errors also may include faults associated with an incomplete view of the experimental conditions. For example, the methodical systematic error will be made in the process of weighing of a body on an analytical balance, unless an experimenter amends the difference between ejection forces acting from the side of the air and having influence both on the body and weights.

### How to find the average of a set of values: a historical introduction into the world of experimental physics

It is widely known that physics is purely an experimental science. The aim of a physical experiment is to find such parameters of physical phenomena that can be measured, and after the obtaining of numerical values the scientist needs to compare them with the predictions of a testable theory or hypothesis. The parameters of physical objects and processes that can be directly or indirectly measured are called physical quantities. All physical quantities can be divided into two main categories: 1) quantities characterizing the properties and states of bodies (for instance, mass, volume, density, electric resistance, pressure, etc.); 2) quantities characterizing phenomena and processes occurring in time (linear velocity, current strength, work, etc.). To measure a value means to compare it with the uniform value taken as unit. For example, when measuring objects (linear dimensions of length, width, height, diameter, etc.), scientists compare them with a meter; to measure body weight one needs to compare it to a kilogram and so on. By measuring a physical quantity we mean finding the value of a physical quantity by using special technical means, whereby it is possible to determine in how many times the measured value is greater (or less) than the corresponding value adopted as a standard. The result of measurement must be expressed by a certain number. As a rule, all measurements are recorded in a table, with obligatory indication of units of measure (this also concerns writing a case study research ). The system of units is a collection of units of main and derived quantities. Units of derived quantities are formed on the basis of equations connecting these quantities with basic ones. The International System of Units (SI) has been accepted by 21 countries in October 1960 as a result of the General XI international conference on weights and measures; sufficient is to say that it is based on six core units and two additional units. The first three basic units (a meter, a kilogram, a second) permit us to form derived units for all variables with a purely mechanical nature, and three remaining core units (an ampere, a kelvin, a candela) enable us to form derived units for the quantities that cannot be reduced to mechanical: thus, an ampere serves for magnetic and electric quantities, a kelvin is used for thermal quantities, whereas a candela is useful for the quantities in photometry. Angular units (such as radian and steradian) cannot be added to the number of the major units, as this would cause difficulties in the interpretation of the values of dimensions related to rotation (circular arcs, the area of a circle, work of a couple of forces and so on). In essence, these units are derived units, although with that peculiarity that they have the same dimensionality in different systems of units, which helps us not with comprehension of how to find the average value. Dimensionality of base units is set by definitions. This means that the dimensionality of derivative units is defined by equations calculating relationships between quantities taken in the simplest form. For example, the speed unit is formed by replacing quantities with units of these quantities in the equation for the velocity of uniform motion: [V] = (1m) : (1c). This expression is a shorthand of the verbal definition of the speed unit, i.e. the speed of uniform motion at which a distance equal to 1 m can be overcome in a time of 1 sec. In order to form derived units instead of equations for relationships between quantities on can use the formulas of dimensionality; however, this method is very formal and uncommon, as in complex cases it does not allow a physicist to trace the mechanism of unit formation and formulate its simplified, verbal definition. In addition to basic, additional and derived units it is allowable to use multiples and submultiples of units derived from the SI units by using decimal prefixes. Names of multiples and units are formed by adding corresponding prefixes to the original names of units. If a name of unit already comprises a prefix, such as a kilogram, then a new prefix should join the initial name - in this case, the name gram, so that it produces milligram, megagram, etc.

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