{ "metadata": { "name": "", "signature": "sha256:2ffa422208323473e7fa809c26d007c1a4e73e5340d80167d553ec696ed3e811" }, "nbformat": 3, "nbformat_minor": 0, "worksheets": [ { "cells": [ { "cell_type": "heading", "level": 1, "metadata": {}, "source": [ "Chapter 3 : Errors in Measurements and Their Statistical Analysis" ] }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.1 Page No : 59" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# calculating guarantee value of capacitance\n", "\n", "# Variables\n", "As = 1;\n", "Er = 0.05;\n", "\n", "# Calculations and Results\n", "Aau = As*(1+Er);\n", "print \"Upper limit(micro F) = \", Aau\n", "Aal = As*(1-Er);\n", "print \"Lower limit(micro F) = \", Aal\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Upper limit(micro F) = 1.05\n", "Lower limit(micro F) = 0.95\n" ] } ], "prompt_number": 2 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.2 Page No : 61" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# calculating percentage limiting error\n", "\n", "# Variables\n", "As = 150.;\n", "Er = 0.01;\n", "\n", "# Calculations\n", "dA = As*Er;\n", "As1 = 75.;\n", "Er = (dA/As1)*100;\n", "\n", "# Results\n", "print \"Percentage limiting error = \", Er\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Percentage limiting error = 2.0\n" ] } ], "prompt_number": 3 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.3 Page No : 63" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the range of readings\n", "\n", "# Variables\n", "fsd = 1000.;\n", "TP = 100.;\n", "\n", "# Calculations and Results\n", "Efsd = (1./100)*1000;\n", "print \"magnitude of Error when specified in terms of full scale deflection (w) = \", Efsd\n", "print \"Thus the meter will read between 90W and 110W\"\n", "\n", "Etv = (1./100)*100;\n", "print \"magnitude of Error when specified in terms of true value (w) = \", Etv\n", "print \"Thus the meter will read between 99W and 101W\"\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "magnitude of Error when specified in terms of full scale deflection (w) = 10.0\n", "Thus the meter will read between 90W and 110W\n", "magnitude of Error when specified in terms of true value (w) = 1.0\n", "Thus the meter will read between 99W and 101W\n" ] } ], "prompt_number": 4 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.4 Page No : 65" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the limiting error in percent\n", "\n", "# Variables\n", "dA = 0.05*5*10**-6;\n", "As = 2.5*10**-6;\n", "\n", "# Calculations\n", "Er = (dA/As)*100;\n", "\n", "# Results\n", "print \"percemtage limiting error = +/-\", Er\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "percemtage limiting error = +/- 10.0\n" ] } ], "prompt_number": 5 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.5 Page No : 67" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the range of readings specified interms of f.s.d. and true value\n", "\n", "print \"Range when specified interms of f.s.d.\"\n", "\n", "# Calculations and Results\n", "Error_fsd = 1.*1000./100\n", "Range_lower_value = 100-Error_fsd;\n", "print \"Lower value of range (kN/m2)\", Range_lower_value\n", "Range_upper_value = 100+Error_fsd;\n", "print \"Upper value of range (kN/m2)\", Range_upper_value\n", "print \"Range when specified interms of True value\",\n", "Error_true = 1*100/100.\n", "Range_lower_value = 100-Error_true;\n", "print \"Lower value of range (kN/m2)\", Range_lower_value\n", "Range_upper_value = 100+Error_true;\n", "print \"Upper value of range (kN/m2)\", Range_upper_value\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Range when specified interms of f.s.d.\n", "Lower value of range (kN/m2) 90.0\n", "Upper value of range (kN/m2) 110.0\n", "Range when specified interms of True value Lower value of range (kN/m2) 99.0\n", "Upper value of range (kN/m2) 101.0\n" ] } ], "prompt_number": 7 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.6 Page No : 69" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the magnitude and limiting error in ohm and in percentage of the resistance\n", "\n", "# Variables\n", "R1 = 37.;\n", "R1_le = 5.*R1/100;\n", "R2 = 75.;\n", "R2_le = 5.*R2/100;\n", "R3 = 50.;\n", "R3_le = 5.*R3/100;\n", "\n", "# Calculations and Results\n", "R = R1+R2+R3;\n", "print \"Value of resistance (ohm) = \", R\n", "\n", "R_le = R1_le+R2_le+R3_le;\n", "print \"Limiting Value of resistance (ohm) = \", R_le\n", "\n", "Limiting_error_percentage = R_le*100/R;\n", "print \"Limiting Value of resistance (percentage) = +/-\", Limiting_error_percentage\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Value of resistance (ohm) = 162.0\n", "Limiting Value of resistance (ohm) = 8.1\n", "Limiting Value of resistance (percentage) = +/- 5.0\n" ] } ], "prompt_number": 8 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.7 Page No : 71" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# calculate the value of relative limiting error in resistance\n", "\n", "# Variables\n", "Re_P = 1.5;\n", "Re_I = 1;\n", "\n", "# Calculations\n", "Re_resistance = (Re_P+2*Re_I);\n", "\n", "# Results\n", "print \"the value of relative limiting error of resistance in percentage(+/-) = \", Re_resistance" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "the value of relative limiting error of resistance in percentage(+/-) = 3.5\n" ] } ], "prompt_number": 9 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.8 Page No : 73" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the guaranteed values of the resistance\n", "\n", "# Variables\n", "R1 = 100.;\n", "R1_le_perunit = 0.5; # R1_le_perunit indicates dR1/R1 = 0.5%\n", "R2 = 1000.;\n", "R2_le_perunit = 0.5;\n", "R3 = 842.;\n", "R3_le_perunit = 0.5;\n", "\n", "# Calculations and Results\n", "Rx = R2*R3/R1;\n", "print \"Value of resistance (ohm) = \", Rx\n", "Rx_le_perunit = R1_le_perunit+R2_le_perunit+R3_le_perunit;\n", "\n", "print \"Limiting Value of resistance per unit (dRx/Rx) = \", Rx_le_perunit\n", "Er_Le = Rx_le_perunit*Rx/100;\n", "print \"Limiting Value of resistance (ohm) = +/-\", Er_Le\n", "print \"Guarantee value of the resistance (ohm) = \",\n", "G1 = Rx+Er_Le;\n", "G2 = Rx-Er_Le;\n", "print G1,G2\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Value of resistance (ohm) = 8420.0\n", "Limiting Value of resistance per unit (dRx/Rx) = 1.5\n", "Limiting Value of resistance (ohm) = +/- 126.3\n", "Guarantee value of the resistance (ohm) = 8546.3 8293.7\n" ] } ], "prompt_number": 10 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.9 Page No : 75" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the percentage limiting error and range of resistance values\n", "print \"error in decade a = so, decade a is set at 4000 ohm\",\n", "Er_a = 4000.*0.1/100;\n", "print Er_a\n", "print \"error in decade b = so, decade b is set at 600 ohm\",\n", "Er_b = 600*0.1/100.;\n", "print Er_b\n", "print \"error in decade c = so, decade c is set at 30 ohm\",\n", "Er_c = 30*0.1/100.;\n", "print Er_c\n", "print \"error in decade d = so, decade d is set at 9 ohm\",\n", "Er_d = 9*0.1/100;\n", "print Er_d\n", "Er_total = Er_a+Er_b+Er_c+Er_d;\n", "Re_le_percentage = Er_total*100./4639;\n", "print \"Percentage Relative limiting error = \", Re_le_percentage\n", "Range_lower = 4639-Er_total;\n", "print \"Lower value of range (ohm) = \", Range_lower\n", "Range_upper = 4639+Er_total;\n", "print \"upper value of range (ohm) = \", Range_upper\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "error in decade a = so, decade a is set at 4000 ohm 4.0\n", "error in decade b = so, decade b is set at 600 ohm 0.6\n", "error in decade c = so, decade c is set at 30 ohm 0.03\n", "error in decade d = so, decade d is set at 9 ohm 0.009\n", "Percentage Relative limiting error = 0.1\n", "Lower value of range (ohm) = 4634.361\n", "upper value of range (ohm) = 4643.639\n" ] } ], "prompt_number": 11 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.10 Page No : 77" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "# Calculate the magnitude of power and limiting error\n", "\n", "# Variables\n", "F = 4.58;\n", "L = 397.;\n", "R = 1202.*10**-9;\n", "t = 60.;\n", "\n", "# Calculations and Results\n", "P = (2*math.pi*9.81*F*L*R)/(t*10**6);\n", "print \"Magnitude of power (W) = \", P\n", "dF_pu = 0.02/F; # per unit error in force\n", "dL_pu = 1.3/L; # per unit error in Length\n", "dR_pu = 1/R; # per unit error in revolution\n", "dt_pu = 0.5/t; # per unit error in time\n", "dP_pu = dF_pu+dL_pu+dR_pu+dt_pu;\n", "dP_le = dP_pu*P;\n", "print \"Magnitude of limiting error in power (W)\", dP_le\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Magnitude of power (W) = 2.24521573806e-09\n", "Magnitude of limiting error in power (W) 0.00186789998434\n" ] } ], "prompt_number": 12 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.11 Page No : 79" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the magnitude of Force and limiting error\n", "\n", "# Variables\n", "E = 200.*10**9;\n", "L = 25.*10**-3;\n", "b = 4.75*10**-3;\n", "d = 0.9*10**-3;\n", "\n", "# Calculations and Results\n", "I = (b*d**3)/12;\n", "x = 2.5*10**-3;\n", "F = (3*E*I*x)/(L**3);\n", "print \"Magnitude of Force (N) = \", F\n", "\n", "dE_pu = 0./E; # per unit error in E\n", "db_pu = 0.0075/b; \n", "dd_pu = 0.0075/d; \n", "dx_pu = 0.025/x; \n", "dL_pu = 0.025/L;\n", "dF_pu = (dE_pu+db_pu+3*dd_pu+dx_pu+3*dL_pu)*10**-3;\n", "\n", "print \"limiting error in force (N) = +/-\", dF_pu\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Magnitude of Force (N) = 27.702\n", "limiting error in force (N) = +/- 0.0395789473684\n" ] } ], "prompt_number": 13 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.12 Page No : 81" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# calculate the power loss and relative error \n", "\n", "# Variables\n", "I = 64.*10**-3;\n", "R = 3200.;\n", "\n", "# Calculations and Results\n", "P = (I**2)*R;\n", "print \"Power(W) = \", P\n", "Re = 2*0.75+0.2;\n", "print \"Relative error (%) = \", Re\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Power(W) = 13.1072\n", "Relative error (%) = 1.7\n" ] } ], "prompt_number": 14 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.13 Page No : 83" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate the true power as a percentage of measured power\n", "\n", "# Variables\n", "I = 30.4;\n", "R = 0.015;\n", "\n", "# Calculations\n", "I_true = I*(1+0.012);\n", "R_true = R*(1-0.003);\n", "P_true = (I_true**2)*R_true;\n", "P_measured = (I**2)*R;\n", "R = P_true*100/P_measured;\n", "\n", "# Results\n", "print \"true power as a percentage of measured power(%) = \", R\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "true power as a percentage of measured power(%) = 102.1071568\n" ] } ], "prompt_number": 15 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.14 Page No : 85" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#calculate the total resistance, error of each register and fractional error of total resistance\n", "\n", "# Variables\n", "R1 = 250.;\n", "R2 = 500.;\n", "R3 = 375.;\n", "\n", "# Calculations and Results\n", "R_true = 1/((1/R1)+(1/R2)+(1/R3));\n", "print \" True value of resistance(ohm) = \", R_true\n", "\n", "dR1 = 0.025*R1;\n", "dR2 = -0.36*R2;\n", "dR3 = 0.014*R3;\n", "R1_effective = R1+dR1;\n", "R2_effective = R2+dR2;\n", "R3_effective = R3+dR3;\n", "R_effective = 1/((1/R1_effective)+(1/R2_effective)+(1/R3_effective));\n", "print \" Effective value of resistance(ohm) = \", R_effective\n", "\n", "Fractional_error = (R_true-R_effective)/R_true;\n", "print \"Fractional_error\", Fractional_error\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " True value of resistance(ohm) = 115.384615385\n", " Effective value of resistance(ohm) = 103.548741621\n", "Fractional_error 0.102577572617\n" ] } ], "prompt_number": 16 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.15 Page No : 87" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#\n", "print \"When all the components have 0% error then resonant frequency (Hz)\",\n", "\n", "# Variables\n", "L = 160.*10**-6;\n", "C = 160.*10**-12;\n", "fr = (1./(2*math.pi))*(1/(L*C))**0.5;\n", "print fr\n", "\n", "# Calculations and Results\n", "print \"When all the components have +10% error then resonant frequency (Hz)\",\n", "L_new = (160.*10**-6)+0.1*L;\n", "C_new = (160.*10**-12)+0.1*C;\n", "fr_new = (1./(2*math.pi))*(1/(L_new*C_new))**0.5;\n", "print fr_new\n", "\n", "error = (fr_new-fr)/fr;\n", "print \"error = \", error\n", "\n", "print \"When all the components have -10% error then resonant frequency (Hz)\",\n", "L_new = (160.*10**-6)-0.1*L;\n", "C_new = (160.*10**-12)-0.1*C;\n", "fr_new = (1./(2*math.pi))*(1/(L_new*C_new))**0.5;\n", "print fr_new\n", "\n", "error = (fr_new-fr)/fr;\n", "print \"error = \", error\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "When all the components have 0% error then resonant frequency (Hz) 994718.394324\n", "When all the components have +10% error then resonant frequency (Hz) 904289.449386\n", "error = -0.0909090909091\n", "When all the components have -10% error then resonant frequency (Hz) 1105242.66036\n", "error = 0.111111111111\n" ] } ], "prompt_number": 17 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.16 Page No : 89" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "# calculate the Volume and relative error \n", "\n", "# Variables\n", "L = 250.;\n", "d = 50.;\n", "\n", "# Calculations and Results\n", "V = ((math.pi/4)*d**2)*L;\n", "print \"Volume(mm3) = \", V\n", "Re = 2*0.2-0.5;\n", "print \"Relative error (%) = \", Re\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Volume(mm3) = 490873.852123\n", "Relative error (%) = -0.1\n" ] } ], "prompt_number": 18 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.17 Page No : 91" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# calculate the per unit change in the value of spring for different temperature ranges\n", "\n", "# Variables\n", "dG_pu = -240*10**-6;\n", "dD_pu = 11.8*10**-6;\n", "\n", "# Calculations and Results\n", "print \"for temperature change of 20 degree C to 50 degree C (%) = \",\n", "d_th = 30;\n", "dK_pu = (dG_pu+dD_pu)*d_th*100;\n", "print dK_pu\n", "\n", "print \"for temperature change of 20 degree C to -50 degree C (%) = \",\n", "d_th = -70;\n", "dK_pu = (dG_pu+dD_pu)*d_th*100;\n", "print dK_pu\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "for temperature change of 20 degree C to 50 degree C (%) = -0.6846\n", "for temperature change of 20 degree C to -50 degree C (%) = 1.5974\n" ] } ], "prompt_number": 19 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.18 Page No : 93" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate apparent resistance, actual resistance and error\n", "\n", "# Variables\n", "Et = 100.;\n", "It = 5.*10**-3;\n", "\n", "# Calculations and Results\n", "Rt = Et/It;\n", "print \"apparent value of resistance (ohm) = \", Rt\n", "Rv = 1000.*150;\n", "\n", "Rx = Rt*Rv/(Rv-Rt);\n", "print \"true value of resistance(ohm)\", Rx\n", "\n", "Er_percentage = (Rt-Rx)/Rx*100;\n", "print \"percentage error = \", Er_percentage\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "apparent value of resistance (ohm) = 20000.0\n", "true value of resistance(ohm) 23076.9230769\n", "percentage error = -13.3333333333\n" ] } ], "prompt_number": 20 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.19 Page No : 95" ] }, { "cell_type": "code", "collapsed": false, "input": [ "# Calculate apparent resistance, actual resistance and error\n", "\n", "# Variables\n", "Et = 40.;\n", "It = 800.*10**-3;\n", "\n", "# Calculations and Results\n", "Rt = Et/It;\n", "print \"apparent value of resistance (ohm) = \", Rt\n", "Rv = 1000.*150;\n", "Rx = Rt*Rv/(Rv-Rt);\n", "print \"true value of resistance(ohm)\", Rx\n", "Er_percentage = ((Rt-Rx)/Rx)*100;\n", "print \"percentage error = \", Er_percentage\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "apparent value of resistance (ohm) = 50.0\n", "true value of resistance(ohm) 50.0166722241\n", "percentage error = -0.0333333333333\n" ] } ], "prompt_number": 21 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.20 Page No : 97" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#Calculate the error and percentage error in the measurement of deflection\n", "\n", "# Variables\n", "l = 0.2;\n", "E = 200*10**9;\n", "b = 20.*10**-3;\n", "d = 5.*10**-3;\n", "D = (4.*l**3)/(E*b*d**3);\n", "F = 1*9.81;\n", "\n", "# Calculations and Results\n", "x_true = D*F;\n", "print \"True value of deflection\", x_true\n", "x_indicated = D*10.31/(1+.1*D);\n", "print \"Indicated value of deflection\", x_indicated\n", "Er = x_indicated-x_true;\n", "print \"error = \", Er\n", "Er_percentage = Er*100/x_true;\n", "print \"Percentage error = \", Er_percentage\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "True value of deflection 0.00062784\n", "Indicated value of deflection 0.000659835777051\n", "error = 3.1995777051e-05\n", "Percentage error = 5.09616734375\n" ] } ], "prompt_number": 22 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.21 Page No : 99" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "from numpy import array\n", "#to find the mean,deviations from the mean, Average deviation, standard deviation and variance\n", "\n", "x = [532. ,548, 543, 535, 546, 531, 543, 536];\n", "X = sum(x);\n", "n = 8; \n", "a = 0.;\n", "\n", "# Calculations and Results\n", "Mean = X/n;\n", "print \"mean (kHZ)\", X/n\n", "d = []\n", "for i in range(n):\n", " d.append(x[i]-Mean)\n", " print \"deviations = \", d[i]\n", " a = a+(abs(d[i]))\n", " \n", "d_average = a/n;\n", "print \"Average deviation (kHz) = \", d_average\n", "d_2 = sum(array(d)**2);\n", "s = math.sqrt(d_2/(n-1))\n", "print \"standard deviation(kHz)\", s\n", "V = s**2;\n", "print \"varaince (kHZ)2 = \", V\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "mean (kHZ) 539.25\n", "deviations = -7.25\n", "deviations = 8.75\n", "deviations = 3.75\n", "deviations = -4.25\n", "deviations = 6.75\n", "deviations = -8.25\n", "deviations = 3.75\n", "deviations = -3.25\n", "Average deviation (kHz) = 5.75\n", "standard deviation(kHz) 6.54107898482\n", "varaince (kHZ)2 = 42.7857142857\n" ] } ], "prompt_number": 25 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.22 Page No : 101" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "from numpy import array\n", "#to find the mean,standard deviation, probable error and range\n", "\n", "# Variables\n", "x = [41.7, 42, 41.8, 42, 42.1, 41.9, 42, 41.9, 42.5, 41.8];\n", "X = sum(x) \n", "print X\n", "d = [-.27, .03, -.17, .03, .13, -.07, .03, -.07, .53, -.17];\n", "\n", "# Calculations and Results\n", "d_2 = sum(array(d)**2);\n", "n = 10.;\n", "print \"mean length(deg C)\", X/n\n", "print \"standard deviation(if data is infinite)(deg C)\", math.sqrt(d_2/n)\n", "print \"standard deviation(deg C)\", math.sqrt(d_2/(n-1))\n", "r1 = .6745*math.sqrt(d_2/(n-1));\n", "print \"probable error of 1 reading(deg C)\", r1\n", "print \"probable error of mean(deg C)\", r1/math.sqrt(n-1)\n", "print \"range(deg C)\", max(x)-min(x)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "419.7\n", "mean length(deg C) 41.97\n", "standard deviation(if data is infinite)(deg C) 0.21\n", "standard deviation(deg C) 0.221359436212\n", "probable error of 1 reading(deg C) 0.149306939725\n", "probable error of mean(deg C) 0.0497689799083\n", "range(deg C) 0.8\n" ] } ], "prompt_number": 28 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.23 Page No : 103" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "from numpy import array\n", "#to find the arithematic mean, maen deviation,standard deviation, prpobable error of 1 reading, standard deviation and probable error of mean, standard deviation of standard deviation\n", "\n", "# Variables\n", "T = [397, 398, 399, 400, 401, 402, 403, 404, 405];\n", "f = [1, 3, 12, 23, 37, 16, 4, 2, 2];\n", "\n", "# Calculations and Results\n", "Tf = sum(abs(array(T).transpose()*f));\n", "print \"mean temp(deg C)\", Tf/sum(f)\n", "d = [-3.78, -2.78, -1.78, -.78, .22, 1.22, 2.22, 3.22, 4.22];\n", "d = array(d)\n", "print \"mean deviation(deg C)\", sum(array(f).transpose()*d)/sum(f)\n", "print \"standard deviation(deg C)\", math.sqrt(sum(array(f).transpose()*d**2)/sum(f))\n", "print \"probable error of 1 reading(deg C)\", .6745*math.sqrt(sum(array(f).transpose()*d**2)/sum(f))\n", "print \"probable error of mean(deg C)\", (.6745*math.sqrt(sum(array(f).transpose()*d**2)/sum(f)))/math.sqrt(sum(f))\n", "print \"standard deviation of mean(deg C)\", (math.sqrt(sum(array(f).transpose()*d**2)/sum(f)))/math.sqrt(sum(f))\n", "print \"standard deviation of standard deviation(deg C)\", (math.sqrt(sum(array(f).transpose()*d**2)/sum(f)))/math.sqrt(sum(f)*2)\n", "\n", " " ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ " mean temp(deg C) 400\n", "mean deviation(deg C) -1.7763568394e-17\n", "standard deviation(deg C) 1.38260623462\n", "probable error of 1 reading(deg C) 0.932567905249\n", "probable error of mean(deg C) 0.0932567905249\n", "standard deviation of mean(deg C) 0.138260623462\n", "standard deviation of standard deviation(deg C) 0.0977650244208\n" ] } ], "prompt_number": 31 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.24 Page No : 105" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math\n", "#to find probable no of resistors\n", "\n", "# Variables\n", "x = .15; #deviation \n", "o = .1; #standard deviation\n", "t = x/o;\n", "A = .4432 #area under gaussian curve corresponding to t\n", "\n", "# Calculations\n", "n = 2*A*1000;\n", "\n", "# Results\n", "print \"no of resistors\", math.floor(n)\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "no of resistors 886.0\n" ] } ], "prompt_number": 33 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.25 Page No : 107" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#to find no of 100 rsding exceed 30mm\n", "\n", "# Variables\n", "x = 30-26.3; #mean value 26.3\n", "r = 2.5;\n", "o = r/.6745;\n", "t = x/o;\n", "A = .3413;#area under gaussian curve corresponding to t\n", "\n", "# Calculations\n", "n = 2*A*100;\n", "nn = 100-math.floor(n);\n", "\n", "# Results\n", "print \"no of readings exceed\", nn/2\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "no of readings exceed 16.0\n" ] } ], "prompt_number": 34 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.26 Page No : 109" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#to find no of rods of desired length\n", "\n", "# Variables\n", "n = 25000.; #no of rods\n", "n1 = 12500.; #length>10mm\n", "n2 = 2000.; #length>10.25\n", "a = n1-n2; #101.96:\n", " print \"rejected value\", x[i]\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "[ 0.49953805 0.18672828 1.84653522 0.56337678 2.04762722 0.26014282\n", " 0.76127684 0.04309114 0.31440574 0.21864765]\n", "rejected value 0.00433\n" ] } ], "prompt_number": 41 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.32 Page No : 121" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "from numpy import array\n", "#calculate standard deviation\n", "\n", "# Variables\n", "x = array([.9 ,2.3, 3.3, 4.5, 5.7, 6.7]);\n", "y = array([1.1, 1.6, 2.6, 3.2, 4 ,5]);\n", "n = 6.;\n", "\n", "# Calculations\n", "a = ((n*sum(x*y)-(sum(x)*sum(y)))/((sum(x**2)*n)-sum(x)**2));\n", "b = ((sum(y)*sum(x**2)-(sum(x)*sum(x*y)))/((sum(x**2)*n)-sum(x)**2));\n", "\n", "sdy = math.sqrt((1/n)*sum((a*x+b-y)**2));\n", "sdx = sdy/a;\n", "\n", "sa = math.sqrt(n/(n*sum(x**2)-sum(x)**2))*sdy;\n", "sb = math.sqrt(sum(x**2)/(n*sum(x**2)-sum(x)**2))*sdy;\n", "\n", "# Results\n", "print \"s_a\", sa\n", "print \"s_b\", sb\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "s_a 0.0354898238647\n", "s_b 0.155116651415\n" ] } ], "prompt_number": 44 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.34 Page No : 123" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#determine value of total current considering errors as limiting errors ans as smath.radians(numpy.arcmath.tan(rd deviations\n", "\n", "# Variables\n", "I1 = 200.;\n", "I2 = 100.;\n", "dI1 = 2.;\n", "dI2 = 5.;\n", "\n", "# Calculations and Results\n", "I = I1+I2;\n", "dI = ((I1/I)*(dI1/I1)+(I2/I)*(dI2/I2));\n", "print \"error considered as limiting errors\"\n", "print \"I\", I\n", "print \"dI\", dI*I\n", "sdI = math.sqrt(dI1**2+dI2**2);\n", "print \"error considered as standard deviations\"\n", "print \"I\", I\n", "print \"sdI\", sdI\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "error considered as limiting errors\n", "I 300.0\n", "dI 7.0\n", "error considered as standard deviations\n", "I 300.0\n", "sdI 5.38516480713\n" ] } ], "prompt_number": 45 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.35 Page No : 125" ] }, { "cell_type": "code", "collapsed": false, "input": [ "#determine probable error in the computed value of resistnce\n", "\n", "# Variables\n", "r_V = 12.;\n", "I = 10.;\n", "r_Rv = r_V/I;\n", "V = 100.;\n", "r1 = 2.;\n", "\n", "# Calculations\n", "r_Ri = V*r1/I**2;\n", "r_R = math.sqrt(r_Rv**2+r_Ri**2);\n", "\n", "# Results\n", "print \"r_R\", r_R\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "r_R 2.33238075794\n" ] } ], "prompt_number": 46 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.37 Page No : 127" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#to find Cq and its possible errors\n", "\n", "# Variables\n", "d = 12.5;\n", "A = (math.pi/4)*d**2*10**-6;\n", "W = 392.;\n", "t = 600.;\n", "p = 1000.;\n", "g = 9.81;\n", "h = 3.66;\n", "\n", "# Calculations and Results\n", "Cq = W/(t*p*A*math.sqrt(2*g*h));\n", "print \"Cq\", Cq\n", "dW = .23/W;\n", "dt = 2/t;\n", "dp = .1/100;\n", "dA = 2*.002;\n", "dg = .1/100;\n", "dh = .003/h;\n", "dd = .002;\n", "dCq = Cq*(dW+dt+dp+dA+dg/2+dh/2);\n", "print \"%age absolute error\", dCq*100/Cq\n", "\n", "sdCq = Cq*math.sqrt(dW**2+dt**2+dp**2+4*dd**2+.25*(dg**2+dh**2));\n", "print \"%age standard deviation error\", sdCq*100/Cq\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "Cq 0.628253425224\n", "%age absolute error 0.982990409278\n", "%age standard deviation error 0.537339132325\n" ] } ], "prompt_number": 48 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.38 Page No : 129" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "\n", "# Variables\n", "V = 110.2;\n", "I = 5.3;\n", "\n", "# Calculations and Results\n", "P = V*I\n", "print \"power(W) dissipated\", P\n", "w_v = .2;\n", "w_i = 0.06;\n", "dp = math.sqrt((w_v*I)**2+(w_i*V)**2);\n", "print \"uncertainity in power(%)\", dp*100/P\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "power(W) dissipated 584.06\n", "uncertainity in power(%) 1.14653078526\n" ] } ], "prompt_number": 49 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.39 Page No : 131" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#to find uncertainity in combined resistance in both series and in parrallel\n", "\n", "# Variables\n", "R1 = 100.;\n", "R2 = 50.;\n", "wR1 = .1;\n", "wR2 = 0.03;\n", "\n", "# Calculations and Results\n", "print \"series conn\"\n", "R = R1+R2\n", "print \"resistance(ohm)\", R\n", "dR1 = 1;\n", "dR2 = 1;\n", "wR = math.sqrt((dR1*wR1)**2+(dR2*wR2)**2)\n", "print \"uncertainity in resistance(ohm)\", wR\n", "\n", "print \"parrallel conn\"\n", "R = R1*R2*(R1+R2)**-1\n", "print \"resistance(ohm)\", R\n", "dR1 = (R2/(R1+R2))-((R1*R2)/(R1+R2)**2);\n", "dR2 = (R1/(R1+R2))-((R1*R2)/(R1+R2)**2);\n", "wR = math.sqrt((dR1*wR1)**2+(dR2*wR2)**2)\n", "print \"uncertainity in resistance(ohm)\", wR\n", "\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "series conn\n", "resistance(ohm) 150.0\n", "uncertainity in resistance(ohm) 0.104403065089\n", "parrallel conn\n", "resistance(ohm) 33.3333333333\n", "uncertainity in resistance(ohm) 0.0173561103909\n" ] } ], "prompt_number": 50 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.40 Page No : 133" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#to calculate uncertainity in measurement\n", "\n", "# Variables\n", "l = 150;\n", "dl = 0.01;\n", "b = 50;\n", "\n", "# Calculations and Results\n", "wA = l*dl;\n", "print \"when no uncertainity in measurement of length\"\n", "print \"uncertainity in measurement of area(m*m)\", wA\n", "\n", "print \"when no certainity in measurement of length\"\n", "wA = 1.5*1.5;\n", "wB = 0.01;\n", "wL = math.sqrt((wA**2-(l*wB)**2)/b**2);\n", "print \"uncertainity in measurement of length(m)\", wL\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "when no uncertainity in measurement of length\n", "uncertainity in measurement of area(m*m) 1.5\n", "when no certainity in measurement of length\n", "uncertainity in measurement of length(m) 0.0335410196625\n" ] } ], "prompt_number": 51 }, { "cell_type": "heading", "level": 2, "metadata": {}, "source": [ "Example 3.41 Page No : 135" ] }, { "cell_type": "code", "collapsed": false, "input": [ "import math \n", "#to calculate uncertainity in power\n", "\n", "# Variables\n", "E = 100.;\n", "dE = .01;\n", "I = 10.;\n", "dI = 0.01;\n", "R = 10.;\n", "dR = .01;\n", "\n", "# Calculations and Results\n", "dP = math.sqrt(4*dE**2+dR**2)*100; #P = E**2/R\n", "print \"%age uncertainity in power measurement\", dP\n", "\n", "dP = math.sqrt(dE**2+dI**2)*100; #P = E*I\n", "print \"%age uncertainity in power measurement\", dP\n" ], "language": "python", "metadata": {}, "outputs": [ { "output_type": "stream", "stream": "stdout", "text": [ "%age uncertainity in power measurement 2.2360679775\n", "%age uncertainity in power measurement 1.41421356237\n" ] } ], "prompt_number": 53 } ], "metadata": {} } ] }