1.1.5
Har measurement ek andaza hota hai ek confidence range ke saath . Jab tum kehte ho ki ek rod 5.2 5.2 error hai. Physics tumse perfect hone ko nahi kehti — woh kehti hai ki tum honestly bolo kitne unsure ho . Error analysis usi honesty ki grammar hai.
Kisi bhi instrument ki resolution infinite nahi hoti, koi bhi insaan scale bilkul perfectly nahi padh sakta, aur duniya dholti rehti hai (temperature drift, vibrations, tumhara kaanpta haath). Isliye true value janno hi nahi ja sakti; hume sirf measured value milti hai. Unke beech ka gap error hai. Hum us gap ko do flavours mein split karte hain kyunki dono alag behave karte hain aur hum dono se alag tarike se ladte hain.
Definition Systematic errorWoh error jo har reading ko ek hi direction mein consistent amount se push karta hai . Iska ek pattern hota hai. Causes: zero error (instrument 0 par 0 nahi dikhata), galat calibrated scale, instrumental defect, fixed experimental bias.
Key trait: measurement repeat karne se yeh cancel nahi hota. Average phir bhi galat hota hai.
Woh error jo readings ko true value ke upar aur neeche unpredictably scatter karta hai . Causes: tiny fluctuations, last digit ka judgement, vibrations, noise.
Key trait: repeat karne aur average karne se yeh reduce hota hai, kyunki highs aur lows cancel hone lagte hain.
Intuition Averaging random ko kyun help karta hai par systematic ko nahiRandom errors coin-flip nudges jaisi hain: kabhi + − har baar same nudge hai (jaise hamesha + 0.3 + 0.3
Common mistake "Zyada readings hamesha answer ko accurate banati hain."Kyun sahi lagta hai: zyada data usually = better, aur yeh actually random scatter ko shrink karta hai (precision improve hoti hai).
Fix: averaging sirf random error ko khatam karta hai. + 0.3 Accuracy ≠ precision. Precise = readings tightly clustered; accurate = true value ke aas-paas clustered. Tum precisely galat bhi ho sakte ho.
Maan lo tum koi quantity n a 1 , a 2 , … , a n
Random errors symmetrically scatter hote hain. Woh value jo total spread (sum of squared deviations) ko minimise karti hai woh arithmetic mean hai. Isliye mean true value ka sabse achha single guess hai.
a mean = n 1 ∑ i = 1 n a i
Definition Absolute errorEk reading best estimate se kitni door hai:
Δ a i = ∣ a mean − a i ∣ WHAT: miss ki size, quantity ke same units mein. WHY absolute value: hume deviation ki magnitude se matlab hai, sign se nahi.
A R P → A bsolute (units hain) → R elative (divide karo, units nahi) → P ercent (×100).
"A Rabbit Percent." Har step hai "pichla wala aur zyada comparable."
Worked example Example 1 — poora pipelineEk pendulum ka measured period (s): 2.63 , 2.56 , 2.42 , 2.71 , 2.80
Mean: 5 2.63 + 2.56 + 2.42 + 2.71 + 2.80 = 5 13.12 = 2.624 Yeh step kyun? Best estimate = mean (random scatter ko sabse achhe se khatam karta hai).
Absolute errors: ∣2.624 − 2.63∣ = 0.006 0.064 0.204 0.086 0.176 Kyun? Har reading ki best guess se doori.
Mean absolute error: 5 0.006 + 0.064 + 0.204 + 0.086 + 0.176 = 5 0.536 = 0.107 ≈ 0.11 Kyun? Typical deviation = hamari uncertainty.
Result: T = 2.62 ± 0.11 (error ko decimals match karne ke liye round karo)
Relative: 2.62 0.11 = 0.042 Percentage: 4.2% Kyun? Taaki hum ise, say, kisi mass measurement ke 4.2%
Worked example Example 2 — flavour pehchannaEk balance bina kisi object ke 0.20 5.00 5.01 4.99
Scatter ± 0.01 random hai (last-digit judgement).
Constant 0.20 systematic zero error hai — har reading 0.20 Fix: 0.20 ≈ 4.80 Kyun? Ek baar identify ho jaane par tum known systematic error ko correct kar sakte ho; random error ko sirf averaging se reduce kar sakte ho.
Worked example Example 3 — accuracy vs precision numbers meinTarget true length = 10.00 10.41 , 10.39 , 10.40 precise (tight) lekin inaccurate (+ 0.40 9.7 , 10.3 , 9.9 , 10.1 accurate (mean = 10.0 Yeh kyun matter karta hai: A ki averaging help nahi karegi; B ki averaging truth par land karti hai.
Common mistake Error ko bahut zyada digits mein report karnaT = 2.624 ± 0.107 Kyun sahi lagta hai: zyada decimals "scientific" lagte hain.
Fix: error khud uncertain hota hai, isliye ise 1 (kabhi kabhi 2) significant figures tak rakhо, aur value ko same decimal place par round karo: 2.62 ± 0.11
Recall Compute karne se pehle predict karoEk wire ka diameter 4 baar read kiya jaata hai: 0.42 , 0.44 , 0.43 , 0.43 Forecast: kya % error 1% 10%
Reveal: mean = 0.43 0.01 , 0.01 , 0.00 , 0.00 = 0.005 = 0.005/0.43 = 0.012 = 1.2% 1% fraction ki jagah digits ki size par zyada dhyan diya.
Systematic aur random error mein ek line mein difference Systematic = ek direction mein consistent bias (averaging se nahi hata); random = dono taraf unpredictable scatter (averaging se reduce hota hai).
Averaging random ko kyun reduce karta hai lekin systematic ko nahi? Random errors ±
Ek single reading ke absolute error ka formula Δ a i = ∣ a mean − a i ∣
Mean absolute error ka formula Δ a mean = n 1 ∑ ∣ a mean − a i ∣
Relative error formula aur kya yeh unitless hai? δ r = Δ a mean / a mean
Percentage error formula δ % = ( Δ a mean / a mean ) × 100%
Accuracy vs precision Accuracy = true value ke kareeb hona; precision = repeated readings ka ek doosre ke kareeb hona.
Known systematic zero error ko kaise fix karte hain? Har reading se constant offset subtract (ya add) karo.
Uncertainty ko sirf 1–2 significant figures mein kyun report karte hain? Error khud uncertain hota hai; extra digits meaningless precision hain.
n readings se true value ka sabse achha single estimate Arithmetic mean.
Recall Feynman: ek 12-saal ke bachche ko explain karoSocho tum apne dost ki height ek tedi ruler se measure kar rahe ho. Agar ruler ka pehla centimetre missing hai, toh har koi jo tum measure karte ho 1 cm chhota aata hai — yeh ek systematic galti hai, har baar same goof, aur zyada dost measure karne se yeh theek nahi hogi. Lekin agar tumhara haath thoda kaanta hai, to kabhi thoda zyada padhte ho, kabhi thoda kam — yeh random hai. Kai baar measure karo aur zyada-readings aur kam-readings cancel ho jaati hain, to average achha hota hai. "Error" bas kitna-tum-off-ho-sakte-ho hai; "relative error" woh miss hai actual size se compare karke — 1 cm off hona ek chiti ke liye bahut matter karta hai lekin haathi ke liye nahi.
Systematic = saari shots same galat jagah lagi hain (tight but off-centre).
Random = shots bullseye ke aas-paas spray hue hain.
Achhi science dono ko theek karna chahti hai: shots tight BHI hon aur centred BHI.
Significant Figures and Rounding — tumhara error kitne digits rakhne deta hai.Combination of Errors — in errors ko + , − , × , ÷ , Least Count and Vernier Calipers — systematic error ka instrumental source.Mean and Standard Deviation — mean absolute error ka statistical cousin.Dimensional Analysis — results check karna jab errors stated ho jaayein.
Zero error, bad calibration
Fluctuations, last-digit judgement
