Probable Error of Single Observation

The probable error is a measure of the precision of observations. It defines a range around the mean (or most probable value) within which there is a 50% probability that any single observation will fall. For a set of observations, the probable error of a single observation ($Es$) is commonly calculated using the residuals ($v$) of the observations.

A residual ($v$) is the difference between an individual observed value and the most probable value (usually the mean) of the series of observations.

The standard formula for calculating the probable error of a single observation from a series of $n$ observations with residuals $v$ is:

\( \rm Es = \pm 0.6745 \sqrt{\frac{\Sigma v^2}{n-1}} \)

Let's break down the components of this formula:

• $Es$: Represents the probable error of a single observation.
• $\pm$: Indicates that the error can be positive or negative relative to the mean.
• $0.6745$: This is a constant derived from the properties of the normal distribution. It signifies that there is a 50% chance that the actual error of a single observation will be within $\pm 0.6745$ times the standard error of a single observation.
• $\Sigma v^2$: This is the sum of the squares of the residuals ($v$) for all observations. Squaring the residuals makes all values positive and gives more weight to larger errors.
• $n$: Represents the total number of observations in the series.
• $n-1$: This is the degrees of freedom. Using $n-1$ in the denominator provides a better estimate of the population variance when the mean is calculated from the sample data.
• $\sqrt{\frac{\Sigma v^2}{n-1}}$: This term represents the standard error of a single observation.

Comparing this standard formula with the given options:

• Option 1: \( \rm Es=\pm0.729\sqrt{\Sigma r^2}/n-1 \) - Uses a different constant (0.729) and potentially different notation for residuals ($r$ instead of $v$). The constant 0.729 is not standard for probable error.
• Option 2: \( \rm Es=\pm0.6745\sqrt{{\Sigma v^2}/n-1} \) - Matches the standard formula exactly, using the correct constant (0.6745), sum of squared residuals ($\Sigma v^2$), and degrees of freedom ($n-1$).
• Option 3: \( \rm Es=\pm0.6745\sqrt{\Sigma v^2}/r-1 \) - Uses the correct constant and sum of squared residuals, but the denominator is incorrect ($r-1$ instead of $n-1$). 'r' is often used for residuals, not the number of observations.
• Option 4: \( \rm Es=\pm0.943\sqrt{\Sigma v^2}/n-1 \) - Uses a different constant (0.943). The constant 0.943 is related to average error, not probable error.

Based on the standard formula for the probable error of a single observation, Option 2 is the correct representation.