1.1.5 · D1Errors — absolute, relative, percentage; systematic vs rando
Har measurement asliyat mein ek chhota sa interval hota hai — ek best guess jisme "give or take" laga hota hai — aur error analysis bas us interval ki width ko track karne ka bookkeeping hai. Ye bookkeeping karne se pehle, tumhe kuch symbols (sums, means, absolute value, fractions) mein fluent hona chahiye, toh is page pe har ek symbol scratch se earn kiya gaya hai.
Ye parent topic on errors ke liye toolbox page hai. Parent topic mein ∑ ∣ ∣
Kisi bhi symbol se pehle, ye picture apne dimag mein rakh lo: numbers ek seedhi ruler line pe rehte hain, aur ek measurement us line pe ek dot hai. True value ek aur dot hai jo hum dekh nahi sakte. "Error" bas do dots ke beech ki distance hai.
Definition Value, true value, measured value
Ek value bas ek unit ke saath ek number hai, jaise 5.2
True value woh exact answer hai jo nature ke paas hai — ek invisible dot jise hum dhundte hain par kabhi poori tarah touch nahi kar paate.Measured value woh dot hai jahan hamara instrument actually land karta hai.
Picture: line pe do dots. Unke beech ka gap — yahi poora topic iske baare mein hai.
Parent note mein a 1 , a 2 , … , a n
Definition Subscript notation
Ek ==a bina specific numbers ke .
==Subscript i label hai, ek seat number. a 1 a 2 nahi hai — ye bas batata hai kaunsi reading hai.
Letter ==n kitni readings tumne total lii hain. Agar tumne pendulum 5 n = 5 a 1 a 5
Intuition Labels ki zaroorat kyun haiAgar tum kuch ek baar measure karo, toh ek number milta hai aur average karne ko kuch nahi hota. Errors tabhi kabu mein aati hain jab tum repeat karo — aur "har repeat" ke baare mein baat karne ke liye tumhare paas n i
Question: a 3 3 Ye ek label hai jo teesri reading ko naam deta hai — na "a a 3
Parent mein pehla formula hai a mean = n 1 ∑ a i ∑ S um) is show ka star hai.
+ ⋯ + n … ∑ precisely batata hai kahan se shuru karna hai, kahan rokna hai, aur kya add karna hai — koi ambiguity nahi.
Worked example Warm-up sumReadings 2.63 , 2.56 , 2.42 , 2.71 , 2.80 ∑ i = 1 5 a i = 2.63 + 2.56 + 2.42 + 2.71 + 2.80 = 13.12 5 13.12
Definition Arithmetic mean (average)a mean = n 1 ∑ i = 1 n a i = n a 1 + a 2 + ⋯ + a n KYA: saari readings add karo, phir total ko n Picture: agar har reading number line pe ek stack ki gayi coin hoti, toh mean woh point hai jahan stacks perfectly balance karti hain — see-saw ka pivot.
n 13.12 typical reading describe nahi karta. n
Recall Balance idea check karo2.63 , 2.56 , 2.42 , 2.71 , 2.80 13.12/5 = ? Reveal: 2.624 2.80 2.71 mean hai jisse tum phir miloge jab hum scatter quantify karenge.
Parent likhta hai Δ a i = ∣ a mean − a i ∣ Δ
Definition Absolute valueBars ∣ x ∣ x x 0 ∣ + 0.086 ∣ = 0.086 , ∣ − 0.064 ∣ = 0.064 Picture: number line ko 0 0 distance wahi bachti hai.
Intuition Yahan sign kyun hatate hainHum jaanna chahte hain ek reading mean se kitni door miss hui, na kis side miss hui. 0.06 0.06
∣ a − b ∣ a b Kyun sahi lagta hai: pehle a − b Fix: agar woh negative aaye, toh sign flip karo. ∣2.42 − 2.624∣ = ∣ − 0.204 ∣ = 0.204 − 0.204
Δ Δ D for D ifference) kisi quantity ke saath lagaya jaye toh matlab hai "us quantity mein chhota change ya gap ." Toh Δ a a a Picture: measured dot ke around ek chhota bracket ⊢ ⊣
Intuition "Wobble" ke liye ek dedicated symbol kyunErrors gaps aur chhoti amounts ke baare mein hain. Har baar "a Δ Δ Δ jab errors calculations mein combine hoti hain .
Question: Δ a mean Wobble ka typical size — ek reading usually best guess se kitna stray karti hai.
Parent ka relative error hai δ r = a mean Δ a mean δ idea .
Definition Ratio / fraction as "per unit"Ek fraction q p q har ek unit ke liye kitna p error khud cheez ki tulna mein kitni badi hai?
2.62 s 0.11 s = 0.042 cancel ho jaate hain — top aur bottom mein same unit hai — toh jawaab ek bare number hai. Woh "koi units nahi bache" hi exactly reason hai ki relative error tumhe length measurement ko mass measurement se compare karne deta hai.
1 2 1 aadhi wire hai — ek disaster. Wahi 1 2 1
Definition Lower-case deltaδ δ relative/fractional kism ki error ke liye use hota hai (ek plain ratio), taaki visually Δ δ r δ %
Definition Per-cent = "per hundred"Word percent ka literally matlab hai "100 mein se." Ek fraction ko 100 % 1 100 0.042 × 100 = 4.2 , written 4.2% Picture: pie ka wahi piece, lekin ab pie 1 100
Mnemonic A Rabbit PercentA bsolute (units hain) → R elative (divide karo, units cancel) → P ercent (×100). Har step number ko pehle se zyada comparable banata hai.
Relative error small delta
Top se bottom padho: number line aur subscripts bedrock hain; woh sum aur mean build karte hain; mean plus absolute value plus Δ 100 hi errors topic hai.
Right side cover karo aur dekho kya tum reveal karne se pehle har ek ka jawaab de sakte ho.
∑ i = 1 n a i Readings a 1 a n 1 n
a 3 Teesri reading — ye ek seat number hai, power ya multiplication nahi.
n Sabko ∑ n
Sum ko n "Kitni readings hain" strip karne ke liye aur ek typical reading ka size paane ke liye — balance point.
∣ − 0.2 ∣ 0.2
Reading miss hone ki size nikalte waqt absolute value kyun use karte hain? Hum miss ka size chahte hain; signs rakhne se plus aur minus misses cancel ho jaati aur error hide ho jaati.
Kisi quantity pe Δ Ek chhota change ya uncertainty — wobble ka size, koi value nahi.
Relative error dimensionless kyun hai? Ye ek error ko same unit wali value se divide karta hai, toh units cancel ho jaati hain, ek pure number bachta hai.
0.042 100 4.2%
Unit-carrying absolute error ko kaunsa symbol (Δ δ Δ δ
