Datta Meghe Institute of Engineering, Technology &

Research, Sawangi (Meghe), Wardha

Department of Electronics & Telecommunication

Subject: Electronics MeasurementSem/Branch: IIIrd Sem/ ENTC

Subject Teacher: Mr. P.Y.Shende

UNIT 1

Statistical Analysis, Errors, Units And Standards

Syllabus

• Statistical Analysis • Errors• Units• Standards

Introduction

• Measurement can be done with the help of instrument.

• Measurement field is divided into three group: 1) Mechanical Instrument 2) Electrical Instrument 3) Electronic Instrument

Methods Of Measurement:-

• Direct Method• Indirect method

Random vs Systemic Errors• Types of measurement errors: Blunders, Systemic and

Random Errors• Blunders are gross errors such as incorrect instrument setting

and wrong injection of radiopharmaceuticals.• Systemic errors are results differing consistently from the

correct one such as the length measurement by “warped” ruler.

• Random errors are variations in results from one measurement to another (physical limitation or variation of the quantity) such as the rate of the radiation emission.

Accuracy and Precision

• Measurement results having systemic errors are said to be inaccurate

• Measurements that are very reproducible (same result for repeated measurements) is said to be precise.

• It is possible that result is precise but inaccurate and vice versa

Standard Deviation

• Standard deviation is calculated from the series of measurements where the number of measurements, n, and the mean value, are known:

N

n

i

iSD

n

NN

1

_ 2

1

2/1

Standard Deviation, Variance and Nuclear Counting

• Standard Deviation, SD, is an estimation of Variance .

• In nuclear counting, SDN

Applications of Statistical Analysis• Effects of averaging• Counting rates• Significance of difference between counting measurements• Effects of background• Minimum detectable activity (MDA)• Comparing counting systems• Estimating required counting times• Optimal division of counting times• Statistics of ratemeters.

Systematic and Random Errors

• Error: Defined as the difference between a calculated or observed value and the “true” value

– Blunders: Usually apparent either as obviously incorrect data points or results that are not reasonably close to the expected value. Easy to detect.

– Systematic Errors: Errors that occur reproducibly from faulty calibration of equipment or observer bias. Statistical analysis in generally not useful, but rather corrections must be made based on experimental conditions.

– Random Errors: Errors that result from the fluctuations in observations. Requires that experiments be repeated a sufficient number of time to establish the precision of measurement.

Accuracy vs. Precision

• Accuracy: A measure of how close an experimental result is to the true value.

• Precision: A measure of how exactly the result is determined. It is also a measure of how reproducible the result is.

– Absolute precision: indicates the uncertainty in the same units as the observation

– Relative precision: indicates the uncertainty in terms of a fraction of the value of the result

Uncertainties• In most cases, cannot know what the “true” value is

unless there is an independent determination (i.e. different measurement technique).

• Only can consider estimates of the error.

• Discrepancy is the difference between two or more observations. This gives rise to uncertainty.

• Probable Error: Indicates the magnitude of the error we estimate to have made in the measurements. Means that if we make a measurement that we “probably” won’t be wrong by that amount.

variance: average squared deviation of all possible observationsfrom a sample mean (calculated from sum of squares)

standard deviation: positive square root of the variancesmall std dev: observations are clustered tightly

around a central valuelarge std dev: observations are scattered widely

about the mean

2i = lim [1/N S (xi - µ)2]

i=1

n

s2i = S (xi - µ)2

i=1

n

N - 1

where: µ is the mean,xi is observed value, andN is the number of observations

N->∞

Number decreased from N toN - 1for the “sample” varianceas µ is used in the calculation

Fundamental and Derived Units The units of fundamental physical quantities are called fundamental units. They are length, mass and time. These units can neither be derived from one another nor can be resolved into any other units. They are independent of one another.

Units of physical quantities can be expressed in terms of fundamental units and such units are called derived units.Unit of area can be an example for derived unit. If L is the length of square then L x L = L2 is its area. Similarly, the volume of a cube is L x L x L = L3 cubic area. Units of any physical quantity can be derived from its defining equation.

Metric Units of Length, Mass and Capacity

In the International System of Units ( SI ) each physical quantity - length, mass, volume, etc… is represented by a specific SI unit.

basic unit: the metric units of length (meter), mass (gram), liquid volume (liter) and temperature (degree Celsius).

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Converting between Metric units

If converting from a larger unit (e.g. m) to a smaller unit (e.g. cm), check what number of smaller units are needed to make 1 larger unit, then multiply that number with the relevant number of the larger units.

If converting from a smaller unit (e.g. cm) to a larger unit (e.g. m), check what number of smaller units are needed to make 1 larger unit, then divide that number into the relevant number of the larger units.

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kilo hecto deca Basic unit deci centi milli

Non-Metric Units of Length, Mass and Capacity

There are some countries that use non-metric units (imperial units) more than metric-units. Some common imperial units are given below:

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Quantity measured Imperial unit Approximately equals (correct to 2 decimal places)

Length

1 inch 2.54 cm1 foot 30.48 cm1 yard 92.44 cm1 mile 1.61 km

Mass

1 ounce 28.35 g1 pound 453.59 g1 stone 6.35 kg1 ton 1016.05 kg

Capacity

1 ounce 28.41 ml1 pint 0.57 L1 quart 1.14 L1 gallon 4.55 L

Electrical properties.• The electrical conductivity is defined via J = σ E. This is related to the

probably more• familiar Ohm’s Law, I = V/ R, where I [amps, A] is the current resulting from

applied• potential difference V [voltage drop, V] across a sample, and R is the

electrical resistance• [ohms, Ω].• To get to Ohm’s Law, use J = I /A (current density = current per unit area), E

= V/ L and,• finally, R = ρL / A where the resistivity ρ is the inverse of conductivity, ρ = 1/• σ . Resistivity is the intrinsic property, resistance depends on sample

dimensions.

• The mathematical similarity between electrical conduction and fluid permeability is more

• obvious when we recall that electrical fields represent gradients of electric potential, E = -

• ∇φ. Then we have: J = -σ φ for electrical conductivity, and q = ∇-(κ/η) p for fluid flow.∇

• Electrical Conductivity Temperature Dependence• The electrical conductivity of metals decreases with increasing

temperature, but increases• for semi-conductors. Below is a simplified sketch of why. Phonons

again play a role! (A• fuller explanation involves discussion of energy bands in solids

Metals:

• Conduction pictured as the motion of free electrons in an “electron gas”. This is a similar

• picture to the “phonon gas” of heat transport in crystal lattices. In both cases it is phonons

• that are the main scattering agent. Once again, the distance between scattering collisions is

• the mean free path. Or we can think of the collision time, being the time between collisions.

• In metals, electrons are largely responsible for both heat and charge transport, the thermal

• (k) and electrical (σ) conductivities are proportional, as related by the Weidemann Franz

•

• law: k / σ = LT, where T is temperatures [Kelvin] and L = 2.45×10−8 W Ω K−2. Actually,

• this is an empirical relation rather than a ‘law’.• With increasing temperature the number density of phonons

increases linearly, and in• proportion so does the scattering of electrons, so that the

electrical conductivity decreases• as σ ~ 1/T (for T in Kelvin). As T → 0, there are fewer phonons

activated, but scattering by• impurities means that the resistivity doesn’t go to 0 and the

conductivity remains finite.

Semiconductors:

• The conduction picture in semiconductors is more complicated. Rather than a ‘sea of free

• electrons’, we can think of the electrons as being more tightly bound to the atoms. They are

• said to reside in ‘valence bands’, which are separated by an ‘energy gap’ from the

• ‘conduction band’ in which the electrons would be free to move.

• Statistically, individual electrons can ‘borrow’ from the available thermal energy to make

• The jump to the conduction band more readily as the average thermal energy increases – i.e..

as the temperature increases. In particular, this kind of jumping an energy gap e.g. is

mathematically characterized by an exponential factor which is similar in form to other thermally-activated threshold processes