From the measured values of distance AB, the following trials were recorded. Trials Distance (1) 120.68 (2) 120.84 (3) 120.75 (4) 120.64 Find the probable error of the mean

To find the probable error of the mean, we can use the following formula:

\[
E_m = 0.6745 \times \frac{\sigma}{\sqrt{n}}
\]

Where:
- \( E_m \) is the probable error of the mean.
- \( \sigma \) is the standard deviation of the measurements.
- \( n \) is the number of trials.

### Step 1: Calculate the Mean (\(\overline{X}\))
The mean (\(\overline{X}\)) of the measurements is calculated as:

\[
\overline{X} = \frac{\sum X_i}{n}
\]

Where \( X_i \) are the individual measurements, and \( n \) is the number of trials.

\[
\overline{X} = \frac{120.68 + 120.84 + 120.75 + 120.64}{4} = \frac{483.91}{4} = 120.7275
\]

### Step 2: Calculate the Standard Deviation (\(\sigma\))
The standard deviation (\(\sigma\)) is calculated as:

\[
\sigma = \sqrt{\frac{\sum (X_i - \overline{X})^2}{n-1}}
\]

Let's calculate the deviations:

\[
(120.68 - 120.7275)^2 = (-0.0475)^2 = 0.00225625
\]
\[
(120.84 - 120.7275)^2 = 0.1125^2 = 0.01265625
\]
\[
(120.75 - 120.7275)^2 = 0.0225^2 = 0.00050625
\]
\[
(120.64 - 120.7275)^2 = (-0.0875)^2 = 0.00765625
\]

Now, sum these squared deviations:

\[
\sum (X_i - \overline{X})^2 = 0.00225625 + 0.01265625 + 0.00050625 + 0.00765625 = 0.023075
\]

Now, calculate the standard deviation:

\[
\sigma = \sqrt{\frac{0.023075}{3}} = \sqrt{0.00769167} \approx 0.0877
\]

### Step 3: Calculate the Probable Error of the Mean (\(E_m\))
Now, use the formula for the probable error of the mean:

\[
E_m = 0.6745 \times \frac{0.0877}{\sqrt{4}} = 0.6745 \times \frac{0.0877}{2} = 0.6745 \times 0.04385 \approx 0.0296
\]

### Final Answer
The probable error of the mean is approximately **0.0296**.

Would you like to discuss this further, or do you have any questions?

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Here are 5 related questions to consider:
1. How does the probable error differ from standard error?
2. What assumptions are made in the calculation of probable error?
3. How would the probable error change with more trials?
4. What is the significance of the constant \(0.6745\) in the probable error formula?
5. How does the magnitude of the probable error affect the reliability of the measurement?

**Tip:** When conducting measurements, it's crucial to repeat trials multiple times to reduce random errors and improve the accuracy of your results.

Learn how to calculate the probable error of the mean using recorded distance trials. Understand the statistical concept of probable error, its formula, and its application in measuring the accuracy of mean values. Suitable for advanced high school students and beyond.

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