Question

The errors of an experiment are of random nature and follows a normal distribution. If standard deviation is '$\sigma$', the probable error is:

• A. $\frac{3}{4} \sigma$
• B. $\frac{2}{3} \sigma$
• C. $\frac{1}{2} \sigma$
• D. $\sigma$

Hint:

Probable Error is essentially the Median Absolute Deviation for a normal distribution. Think of it as the "50/50" boundary for your data.

Answer 1

The Correct Option is B

Concept: Probable Error ($P.E.$) is a value that defines a range around the mean such that there is a 50% chance that any single observation will fall within that range. It is a specific multiple of the standard deviation.

Step 1: Define the mathematical relationship.
In a normal distribution, the probable error is calculated using the formula: \[ P.E. = 0.6745 \times \sigma \] This value comes from the Z-score table where the area between $-Z$ and $+Z$ equals 0.50.

Step 2: Convert to a fraction.
The value $0.6745$ is approximately equal to the fraction $\frac{2}{3}$ (which is $\approx 0.6667$).

Step 3: Conclusion.
Among the given options, $\frac{2}{3}\sigma$ is the standard approximation used to define the probable error in terms of standard deviation.