Chapter 1 - Philosophy of Measurement

Example 1 - pg 33

In [3]:
#Example 1.1:#ARITHEMATIC MEAN,AVERAGE DEVIATION ,STANDARD DEVIATION AND PROBABLE ERROR
#calculate the ARITHEMATIC MEAN,AVERAGE DEVIATION ,STANDARD DEVIATION AND PROBABLE ERROR
#given
import math, numpy
T=numpy.array([99.7,99.8,99.9,100,100.1,100.2,100.3]);#VOLTS
f=numpy.array([2,8,20,40,21,6,3]);#frequency of occurence
q=numpy.array([T[0]*f[0],T[1]*f[1],T[2]*f[2],T[3]*f[3],T[4]*f[4],T[5]*f[5],T[6]*f[6]]);#
#calculations
AM=(q[0]+q[1]+q[2]+q[3]+q[4]+q[5]+q[6])/100;#arithematic mean in mm
qb=numpy.zeros(7)
for i in range(0,7):
    qb[i]= T[i]-AM

Q= numpy.array([qb[0],qb[1],qb[2],qb[3],qb[4],qb[5],qb[6]]);#
AV=(-qb[0]*f[0]-qb[1]*f[1]-qb[2]*f[2]-qb[3]*f[3]+qb[4]*f[4]+qb[5]*f[5]+qb[6]*f[6])/100;#
SD=math.sqrt(((qb[0]**2*f[0])+(qb[1]**2*f[1])+(qb[2]**2*f[2])+(qb[3]**2*f[3])+(qb[4]**2*f[4])+(qb[5]**2*f[5])+(qb[6]**2*f[6]))/100);#standard deviation
r1= 0.6745*SD;#PROBABLE ERROR OF ONE READING
#results
print"arithmetic mean is (V)=",AM
print"average deviation is (V)=",AV
print"standard deviation is (V)=",round(SD,4)
print"probable error is (V)=",round(r1,4)

arithmetic mean is (V)= 100.0
average deviation is (V)= 0.084
standard deviation is (V)= 0.1192
probable error is (V)= 0.0804

Example 2 - pg 34

In [4]:
#Example 1.2:#APPRABET RESISTANCE AND PERCENTAGE GROSS ERROR
#calculate the APPARENT RESISTANCE AND PERCENTAGE GROSS ERROR
#given
print"when current reading is 5mA"
vr=100.;#voltmeter reading
ir=5.;#mA
#calculations and results
rt=vr/(ir);#in kilo ohms
print"apparent resistance in kilo ohms is",rt
vm=150.;#range of voltmeter
s=1.;#kilo ohms per volts sensivity
rv=s*vm;#kilo ohms
rx=((rt*rv)/(rv-rt));#kilo ohms
ge=((rx-rt)/rx)*100;#percentage gross error
print"percentage gross error is",round(ge,2)
print"when current reading is 50mA"
vr=100.;#voltmeter reading
ir1=50.;#mA
rt1=vr/(ir1);#in kilo ohms
print"apparent resistance in kilo ohms is",rt1
vm=150.;#range of voltmeter
s=1;#kilo ohms per volts sensivity
rv=s*vm;#kilo ohms
rx1=((rt1*rv)/(rv-rt1));#kilo ohms
ge1=((rx1-rt1)/rx1)*100;#percentage gross error
print"percentage gross error is",round(ge1,2)

when current reading is 5mA
apparent resistance in kilo ohms is 20.0
percentage gross error is 13.33
when current reading is 50mA
apparent resistance in kilo ohms is 2.0
percentage gross error is 1.33

Example 3 - pg 35

In [5]:
#Example 1.3:#limiting error
#calculate the limiting error
#given
fs=1.5;#full scale in percentage
vr=500.;#voltmeter reading
#calculations
ea=(fs/100)*vr;#volts
le=150;#limiting error voltage
lep=((ea/le)*100);#limiting error
#results
print "limiting error at 150 V is (%)=",lep

limiting error at 150 V is (%)= 5.0

Example 4 - pg 35

In [7]:
#Example 1.4:#ARITHMETIC MEAN,AVERAGE DEVIATION ,STANDARD DEVIATION AND PROBABLE ERROR
#calculate the ARITHMETIC MEAN,AVERAGE DEVIATION ,STANDARD DEVIATION AND PROBABLE ERROR
#given
import math,numpy
T=numpy.array([99.7,99.8,100,100.2,100.3]);#VOLTS
q=numpy.array([T[0],T[1],T[2],T[3],T[4]]);#
#calculations
AM=(q[0]+q[1]+q[2]+q[3]+q[4])/5;#arithematic mean in mm
qb=numpy.zeros(6)
for i in range(0,5):
    qb[i]= T[i]-AM;

Q= numpy.array([qb[0],qb[1],qb[2],qb[3],qb[4]]);#
AV=(-qb[0]-qb[1]-qb[2]+qb[3]+qb[4])/5;#
SD=math.sqrt(((qb[0]**2)+(qb[1]**2)+(qb[2]**2)+(qb[3]**2)+(qb[4]**2))/5);#standard deviation
V=SD**2;#
#results
print"arithmetic mean is, (V)=",AM
print"average deviation is, (V)=",AV
print"standard deviation is, (V)=",round(SD,3)
print"Variance is ,(V)=",V

arithmetic mean is, (V)= 100.0
average deviation is, (V)= 0.2
standard deviation is, (V)= 0.228
Variance is ,(V)= 0.052

Example 5 - pg 36

In [8]:
#Example 1.5:#error
#calculate the error in power
#given
i=8.2;#in amperes
r=20.;#ohms
nd=100.;#divisions
ra=10.;#range in amperes
d=0.5;#divisions
amcr=1.;#ammemeter constant error
crr=-0.2;#construction error
#calculations
p=i**2*r;#watts
rd1=ra/nd;#reading of one division
per=((d*rd1)/i)*100;#possible ameter rwading error
ter=amcr+per;#total ammeter error
ep=(((2*ter)+crr)/nd)*p;#
#results
print "power is,(W)=",p
print "error in power is,(W)=",round(ep,3)

power is,(W)= 1344.8
error in power is,(W)= 40.606

Example 6 - pg 37

In [9]:
#Example 1.6:#ARITHMETIC MEAN
#calculate the arithmetic mean
#given
import numpy
T=numpy.array([99.7,99.8,99.9,100,100.1,100.2,100.3]);#VOLTS
f=numpy.array([2,8,20,40,21,6,3]);#frequency of occurence
#calculations
qb=numpy.zeros(8)
for i in range (0,7):
    qb[i]=T[i]-T[5];

prdtc=(qb[0]*f[0])+(qb[1]*f[1])+(qb[2]*f[2])+(qb[3]*f[3])+(qb[4]*f[4])+(qb[5]*f[5])+(qb[6]*f[6])
am=T[5]+(prdtc/100);#arithemetic mean
#results
print "arithmetic mean is (V)=",am

arithmetic mean is (V)= 100.0

Example 7 - pg 37

In [11]:
#Example 1.7:#limiting error
#calculate the limiting error for power
#given
fse=1.;#full scale deflection
vr=150.;#range in volts
v1=100.;#volts
ve=100.;#range in mA
#calculations
ev=(fse/100)*vr;#voltas
le100=((ev)/v1)*100;#in percentage
ee=(fse/100)*ve;#mA
e1=55;#mA
le50=((ee/e1)*100);#in percentage
ler=le100+le50;#
#results
print "limiting error for power is, (%)=",round(ler,1)

limiting error for power is, (%)= 3.3

Example 8 - pg 37

In [12]:
#pg 37
#Example 1.8:# error
#calculate the volume, percentage error
#given
fse=1.;#full scale deflection
e=0.60;#meters
#calculations
v=(e)**3;#volume in m^3
ev=3*fse;#error in volume
evv=(ev/100)*v;#
#results
print "volume is,(m^3)=",v
print "percentage error in volume is,(%)=",ev
print "error in volume is,(m^3)=",evv

volume is,(m^3)= 0.216
percentage error in volume is,(%)= 3.0
error in volume is,(m^3)= 0.00648

Example 9 - pg 38

In [14]:
#Example 1.9:#error
#calculate the error in circuit
#given
v=95.;#volts
r=40.;#ohms
evv=0.95;#error in voltage
err=-0.2;#error in resistance
#calculations
i=v/r;#amperes
err1=(err/r)*100;#percentage error 
evv1=(evv/v)*100;#percentage error
x=evv1-err1;#
ei=(x/100)*i;#
#results
print "current in the circuit is,(A)=",i
print "error in current is,(A)=",round(ei,4)
print "percentage error in current is,(%)=",x

current in the circuit is,(A)= 2.375
error in current is,(A)= 0.0356
percentage error in current is,(%)= 1.5

Example 10 - pg 38

In [15]:
#Example 1.10:#MAGNITUDE AND LIMITING ERROR
#calculate the magnitude of resistance and limiting error
#given
print "parralel resistance case"
r1=40.;#ohms
er1=5.;#percentage error
r2=80.;#ohms
er2=5.;#percentage error
r3=50.;#ohms
er3=5.;#percentage error
#calculations
rp=((r1*r2*r3)/(r1*r2+r2*r3+r3*r1));#ohms
Y=(r1*r2+r2*r3+r3*r1);#ohms
ex=er1+er2+er3;#percentage error
ey1=er1+er2;#
ey2=er2+er3;#
ey3=er3+er1;#
y=(((r1*r2*ey1)/Y)+((r2*r3*ey2)/Y)+((r3*r1*ey3)/Y));#error
mer=(y+ex)*rp;#
#results
print "magnitude of resistance in ohms is = ",round(rp,3)
print "limiting error in ohms is = ",round(mer/100,3)
print "percentage error (%) = ",(y+ex)
print "series resistance case"
rs=r1+r2+r3;#ohms
er=(((r1/rs)*er1)+((r2/rs)*er2)+((r3/rs)*er3));#
mer1=(er/100)*rs;#ohms
print "magnitude of resistance in ohms is = ",rs
print "percentage error (%) = ",er
print "limiting error in ohms is = ",mer1

parralel resistance case
magnitude of resistance in ohms is =  17.391
limiting error in ohms is =  4.348
percentage error (%) =  25.0
series resistance case
magnitude of resistance in ohms is =  170.0
percentage error (%) =  5.0
limiting error in ohms is =  8.5

Example 11 - pg 39

In [17]:
#Example 1.11:#MAGNITUDE AND LIMITING ERROR
#calculate the magintude of resistance in ohm and error
#given
r1=50;#ohms
er1=0.5;#percentage error
r2=100;#ohms
er2=0.5;#percentage error
r3=75.5;#ohms
er3=0.5;#percentage error
#calculations
x=((r2/r1)*r3);#ohms
eps=er1+er2;#	
erpsq=eps-er3;#when error in both (PS) and (Q) is positive
erpsq1=eps+er3;#when error in (PS) is positive and (Q) is negetive
oer1=(erpsq/100)*x;#ohms
oer2=(erpsq1/100)*x;#ohms
#results
print "magnitude in ohm = ",x
print "percentage error when error in both (PS) and (Q) is positive (%) = ",erpsq
print "error in ohms when error in both (PS) and (Q) is positive (ohms) = ",oer1
print "percentage error when error in (PS) is positive and (Q) is negative (%) = ",erpsq1
print "error in ohms when error in (PS) is positive and (Q) is negative (ohms) = ",oer2

magnitude in ohm =  151.0
percentage error when error in both (PS) and (Q) is positive (%) =  0.5
error in ohms when error in both (PS) and (Q) is positive (ohms) =  0.755
percentage error when error in (PS) is positive and (Q) is negative (%) =  1.5
error in ohms when error in (PS) is positive and (Q) is negative (ohms) =  2.265