Calculate the standard deviation, Error And Probable Error in Screw Gauge experiment

APPARATUS REQUIRED

i) Steel ball

ii) Micrometer Screw gauge

THEORY

The precision with which a physical quantity is measured depends inversely upon the deviation or dispersion of the set of measured values x₁ about their mean value x̄. If the values are widly  dispersed the precision is said to below.

The deviation δ1 = x₁ – x̄

Average deviation: 

The average value of the deviation of all the individual measurements from the arithmetic mean is known as average deviation and is denoted by d ¯

Therefore, d ¯ = (x₁-x̄) + (x₂-x̄)…………+ (x_n-x̄) / n

        = δ₁ + δ₂ +………… δ_n / n

        = Σ δi / n

Standard deviation:

The square root of the mean square deviations of an infinite set of measurements is known as standard deviation and is donated by σ. The number of observations is sufficiently large.

For a fairly large sample of measurements which have reasonable normal distribution about the mean value x̄.

ஃ Average deviation/standard deviation = d ¯/σ   = 0.80.

Standard error:

The quantity of σ/√n is known as standard error and denoted by σ_M

from a finite number of observations n may be the internal μ ± . σ_M is 0.68 that it may lie between 0.95 and for 0.99.

Probable error:

The probable error is a quantity such that there is an even chance whether the true value for the quantity measured differs from the mean value by an amount greater or less there.

Probable error = ± 0.6745 √s / n(n-1)

= ± 0.6745 standar error

OBSERVATION TABLE:

no. of obs.	Radius of curvature	δ	δ2
1	15.25	0.07	0.0049
2	15.42	+0.10	0.0100
3	15.30	-0.02	0.0004
4	15.20	-0.12	0.0144
5	15.35	+0.03	0.0009
6	15.40	+0.08	0.0064

Mean radius of curvature = 15.32

S = δ²₁ +δ²₂ +δ²₃ +δ²₄ +δ²₅ +δ²₆

   = 0.049+0·0100 +0.0004 +0.0144 +0.0009+0.0064

  = 0.0370

a) Average deviation d ¯ = δ₁+δ₂+δ₃+δ₄+δ₅+δ₆                                

            = 0.07 +0.10 -0.02 +0.03 +0.08 -0·12 / 6

           = 0.14/6  

           = 0.02333

b) Standard deviation σ = √S/n-1

    = √0.0370/6-1

    = 0.0860

c) Standard error σ_M   = S/ n(n-1)     

= √0.0370/6(6-1)

= √0.0370/30

= 0.03512 

d)Probable error = ± 0.6745× standard error

                           = ± 0.6745×0.03512

                           = ± 0.0237

Radius of curvature = 15.32 ± 0.0237

RESULT:

Average deviation of steel ball = 0.02333

Standard deviation of steel ball = 0.0860

Standard error of steel ball        = 0.03572

finally probable error of steel ball 12 ± 0.0237