Question

The refractive index of glass is found to have the values 1.49, 1.50, 1.52, 1.54 and 1.48. Percentage error in the measurement is?

Answer 1

Mean value = sum of observations /number of observations
= (1.49 + 1.50 + 1.52 + 1.54 + 1.48)/5
= (7.53)/5
= 1.506 ≈ 1.50

Now, absolute error,
y₁ = |1.49 - 1.50| = 0.01
y₂ = |1.50 - 1.50| = 0.00
y₃ = |1.52 - 1.50| = 0.02
y₄ = |1.54 - 1.50| = 0.04
y₅ = |1.48 - 1.50| = 0.02

Now, mean absolute error = (y₁ + y₂ + y₃ + y₄ + y₅)/5
= (0.01 + 0.00 + 0.02 + 0.04 + 0.02)/5
= 0.09/5 = 0.018

∵ relative error = Mean absolute error/mean value
= 0.018/1.50
= 0.012

Now, % error in the measurement = 100 × relative error = 100 × 0.012 = 1.2%

Answer 2

Hello Dear.

Here is the answer---


Given Observations---

1.49,1.50,1.52,1.54, 1.48

No. of Observations = 5

∵ Mean = Sum of all Observations/No. of Observations
= (1.49+1.50+1.52,+1.54+ 1.48)/5
= 7.53/5
= 1.506

Taking an approx value,
Mean = 1.50
[By taking an approximate value answer can be easily calculated]

For Calculating Absolute Error,
x₁ = |1.50 - 1.49|
= 0.01

x₂ = |1.50 - 1.50|
= 0.00

x₃ = |1.50 - 1.52|
= |-0.02|
= 0.02

x₄ = |1.54 - 1.50|
= 0.04

x₅ = |1.48 - 1.50|
= |- 0.02 |
= 0.02

∴  Absolute Error(Δx)  = (x₁ + x₂ + x₃ + x₄ + x₅)/5
  = (0.01 + 0.00 + 0.02 + 0.04 + 0.02)/5
  = 0.09/5
  = 0.018

∵  Relative Error % = (Absolute Error/Mean Value) × 100 %
   = (0.018/1.50) × 100 %
   = 0.012 × 100%
   = 1.2%.

∴ Relative Error % is 1.2 %.


There can be the little difference in the answer, because we have taken the approx mean value. But such difference can be neglected because it will be too hard calculations which will take lot of time. In Examination also such differences are not considered.


Hope it helps.