Errors in Measurement

1. Errors in Measurement – Systematic & Random

• Whenever an experiment is performed, two kinds of errors can appear in the measured quantity : (1) systematic errors and (2) random errors.

Systematic errors :

• Occur due to an error in the procedure or miscalibration of the instrument, etc.
• Have the same size and sign for all the measurements ⇒ they can be determined.
• Examples : zero error, bench error.

Random errors :

• Appear randomly because of the operator, fluctuations in external conditions and variability of measuring instruments.
• Have no fixed sign or size.
• Their effect can be somewhat reduced by taking the average of the measured values.

2. Zero Error

• Zero error (Z.E.) = starting reading of the instrument when nothing is being measured.

• Ruler : object placed from the 1 cm mark to 8.7 cm ⇒ observed reading 8.7 cm, Z.E. = 1 cm ⇒ actual length = 8.7 − 1 = 7.7 cm (same as placing it from 0).

3. Reading with Random Error

x → main value ; Δx → random error ; true value lies anywhere from x − Δx to x + Δx

• e.g. length = (10 ± 1) cm ⇒ anywhere from 9 cm to 11 cm.

• Why a reading has error : car odometer reads whole km, 12382 km at the office and 12387 km at home. True readings may be anywhere in 12382.000–12382.999 and 12387.000–12387.999 ⇒ d_min ≈ 4 km, d_max ≈ 6 km ⇒ d = (5 ± 1) km.

4. Finding Δx & Least Count

Least count :

• Least count = smallest value an instrument can reliably detect and record ; it represents the resolution / precision of the tool.
• Smaller least count ⇒ higher precision.

• Clock with minutes only : 8:50 to 8:56. Each reading may mean any second from :00 to :59 ⇒ time interval = (6 ± 1) min.

Only the reading is given :

• The last written digit tells the least count (trailing zeros count) :

So x = 10 ± 0.1 cm, x = 10.0 cm and "x = 10 cm with a scale of least count 1 mm" all mean x = 10 cm, Δx = 0.1 cm.

5. Mean Deviation Method (Many Readings)

• Best method when several observations x₁, x₂, … x_n are given.

• Deviations of individual readings carry a sign (e.g. −1 s, +1 s) ; the error uses the average of their magnitudes.

Repeated values : multiply each |ΔT| by how often it occurs, e.g. 9.0 (×3), 10.0 (×3), 11.0 (×4) ⇒ T_avg = 10.1 s, |ΔT|_avg = (1.1×3 + 0.1×3 + 0.9×4)/10 = 0.72 s ⇒ T = (10.1 ± 0.72) s.

6. Errors in Addition & Subtraction

A = a ± Δa ; B = b ± Δb

• Absolute errors always add — in subtraction too.
• Cross-check : A = (15 ± 0.1) cm, B = (3 ± 0.1) cm ⇒ A + B = 18 ± 0.2 (max 15.1 + 3.1 = 18.2, min 14.9 + 2.9 = 17.8) ; A − B = 12 ± 0.2 (max 15.1 − 2.9 = 12.2, min 14.9 − 3.1 = 11.8).

Multiplying / dividing by a pure number :

x = (10 ± 0.1) cm ⇒ 4x = (40 ± 0.4) cm ; x/2 = (5 ± 0.05) cm

• Sides a = (10 ± 0.1) cm, b = (6 ± 0.1) cm : 3a = (30 ± 0.3) cm ; a + b = (16 ± 0.2) cm ; a − b = (4 ± 0.2) cm ; 4a − 3b = 22 ± (0.4 + 0.3) = (22 ± 0.7) cm.
• For a − b : absolute error 0.2 cm ; relative error 0.2/4 = 1/20 ; percentage error 5 %.

7. Errors in Multiplication, Division & Powers

Standard method :

• Step 1 : take ln on both sides (use ln(ab/c) = ln a + ln b − ln c ; ln a^x = x ln a).
• Step 2 : differentiate both sides (d(ln x) = dx/x ; constants give 0).
• Step 3 : replace dx → Δx.

Let 7z² = 9π² x y³25 t⁴ ⇒ ln 7 + 2 ln z = ln(9π²/25) + ln x + 3 ln y − 4 ln t

Differentiate : 2zdz = 1xdx + 3ydy − 4tdt

• Each power multiplies the relative error ; relative errors always add, even for quantities in the denominator.
• Pure constants (7, 9π², 25 …) carry no error.
• e.g. 7Z = 9A²B³5C⁴ ⇒ ΔZZ = 2ΔAA + 3ΔBB + 4ΔCC

8. Product & Ratio – Error vs Percentage Change

Product A = ab :

Ratio x = a/b :

a = 10 ± 0.1, b = 5 ± 0.1 ⇒ x = 2 ; Δx = 2(0.1/10 + 0.1/5) = 0.06 ⇒ x = 2 ± 0.06 (cross-check : 10.1/4.9 = 2.061, 9.9/5.1 = 1.941)

Percentage change vs percentage error :

• Change (definite increase / decrease) : keep signs, Δx/x = Δa/a − Δb/b. Error (±, unknown sign) : always add.

9. Power Formulas : % Change vs % Error

• Constants (numbers, π) drop out. e.g. z² = (constant) × x y³t⁴ :

• For error, all terms are added (worst case), even for quantities in the denominator.
• Power n multiplies the relative error by |n|.

Sphere, R = (5 ± 0.1) cm (2%) :

• K = ½mv² : ΔKK = Δmm + 2Δvv. m = 5.0 kg, v = 10.0 m/s (Δm = 0.1 kg, Δv = 0.1 m/s) ⇒ 2% + 2% = 4%.
• T = 2π√(l/g) : ΔTT = ½Δll + ½Δgg. e.g. 4% in l, 2% in g ⇒ 3% in T.
• g = π²(1 ± 0.02) means Δg/g = 0.02 directly. With l = (4 ± 0.01) m : T = 4 s, ΔT/T = 0.125% + 1% = 1.125% ⇒ T = (4 ± 0.045) s.

10. Time Period from n Oscillations

• Time for n oscillations = t ± (least count of watch). The error is shared by all n oscillations :

e.g. 100 oscillations in 200 s, LC = 1 s ⇒ T = (2 ± 0.01) s ⇒ 0.5% error in T.

• Finding g with a pendulum : g = 4π²lT² ⇒

e.g. l = 50 cm (± 1 cm), 50 oscillations in 100 s (± 1 s) ⇒ T = 2 ± 1/50 s ⇒ Δg/g = 2% + 2 × 1% = 4%.

11. Non-Power Functions : Direct Differentiation

• Used when the formula is not a product of powers : 1/v + 1/u = 1/f, x² + 3y² = 5z², 1/R_eq = 1/R₁ + 1/R₂, Δx = d sinθ, N = N₀e^−λt.
• Method : differentiate directly, replace d → Δ, then make every term positive for error.

(same error result for 1/v + 1/u = 1/f and 1/v − 1/u = 1/f)

• Mirror, 1/f = 1/v + 1/u with u = −(30 ± 1) cm, v = (20 ± 1) cm : f = 60 cm, Δf = 3600(1/400 + 1/900) = 13 cm ⇒ f = (60 ± 13) cm, ≈ 21.67%.
• r = 1 − a1 + a (a appears twice) : differentiate directly ⇒ Δr = 2Δa(1 + a)².

12. Trigonometric & Exponential Formulas

• Log method : take ln on both sides, then differentiate ; for error make all terms positive.
• A quantity that is exactly known has zero error (e.g. Δλ = 0).
• Angle errors must be in radians : 1° = π/180 rad.

e.g. λ = 500 nm (exact), a = (100 ± 1) nm, θ = 30° ± 1° ⇒ φ = π/5, % error = (1 + 5√3π/9)% ≈ 4%.

• 2d sinθ = λ (λ exact, Δθ constant) ⇒ Δd/d = cotθ Δθ ⇒ the fractional error in d decreases as θ increases from 0 to 90°.

• In an exponent, the % error of t gets multiplied by the value of t.

Similarly E = A²e^−αt (α = 0.2 s⁻¹), 1.25% in A, 1.50% in t, at t = 5 s : %E = 2(1.25%) + (0.2)(5)(1.50%) = 4%.

13. Why Relative Errors Add (Series Expansion)

z = x/y : z ± Δz = x ± Δxy ± Δy = xy(1 ± Δxx)(1 ± Δyy)⁻¹

• To first power : (1 ± Δy/y)⁻¹ ≈ 1 ∓ Δy/y. Relative errors in independent variables are always added :

Assumes Δx/x ≪ 1, Δy/y ≪ 1, so higher powers are neglected.

14. Round Off

• Digit to be removed < 5 : simply remove it.
• Digit to be removed > 5 : remove it and add 1 to the digit before it.
• Digit to be removed = 5 : if the digit before it is even, simply remove ; if odd, remove and add 1.
• The even/odd rule is used only when the removed part is exactly 5 (e.g. 185.00). If anything non-zero follows the 5 (e.g. 185.46), round up.

15. Significant Digits : Rules

• Significant digits : the important digits of a number which convey its meaning according to its accuracy.

• 1. All non-zero digits (1–9) are significant.
• 2. Zeros with any non-zero digit anywhere to their LEFT are significant.
• 3. All other zeros (leading zeros) are not significant.
• 4. Countable things have infinite significant digits : 8 apples = 8.000…, 35 students = 35.000…
• The power of 10 is not counted : 5.12 × 10⁷ m → 3 ; 23 × 10⁻⁵ s → 2.

Note : NCERT treats trailing zeros of a whole number without a decimal point as not significant (700 → 1). To avoid this ambiguity write the value in powers of ten : 7 × 10² (1 S.D.) or 7.00 × 10² (3 S.D.).

16. Change of Units & Powers of Ten

• Significant digits do not change with a change of unit. Use powers of 10 to keep the count same.

17. Operations with Significant Digits

Addition & subtraction : first round off, then operate :

• Round every number to the least number of decimal places among the inputs, then add/subtract.

21.735 + 9.46 + 11.0 → 21.7 + 9.5 + 11.0 = 42.2 ;  21.735 + 9.46 + 11 → 22 + 9 + 11 = 42
813 + 34 + 9.6 → 813 + 34 + 10 = 857 ;  813 + 34 − 9.6 → 837
56 + 7.3 + 2.46 → 56 + 7 + 2 = 65 ;  56.0 + 7.3 + 2.46 → 56.0 + 7.3 + 2.5 = 65.8

Multiplication, division & all other functions : first operate, then round off :

• Round the answer to the least number of significant digits among the inputs (use powers of 10 if needed).
• A measured value written without decimals counts by its digits : 5 → 1 S.D., but 5.0 → 2 S.D.

Roots, powers & exact numbers :

• Roots and powers : the answer keeps the same S.D. as the input.
• Exact / countable numbers (2 in d = 2r, 100 rupees, 8 students) have infinite S.D. ⇒ they never limit the answer.

18. Formula Summary