Question

john measures the length of one side of his room with a tape measure. he repeats his measurements four times and records them in the table below:

trial 1	trial 2	trial 3	trial 4

table 1: recorded distances of one side of johns room.

what is the uncertainty in the average measurement?

note: use the small data set formula for the calculation of the uncertainty in the mean.

multiple choice

• 0.30 m
• 0.080 m
• 0.0750 m
• 0.15 m

Explanation:

Step1: Calculate the mean

First, find the average (mean) of the measurements. The measurements are \( 4.50 \, \text{m} \), \( 4.60 \, \text{m} \), \( 4.30 \, \text{m} \), and \( 4.40 \, \text{m} \). The formula for the mean \( \bar{x} \) is \( \bar{x} = \frac{\sum_{i = 1}^{n} x_{i}}{n} \), where \( n = 4 \) and \( x_{i} \) are the individual measurements.

\[
\bar{x} = \frac{4.50 + 4.60 + 4.30 + 4.40}{4} = \frac{17.80}{4} = 4.45 \, \text{m}
\]

Step2: Calculate the deviations

Next, find the absolute deviations of each measurement from the mean.

• For \( 4.50 \, \text{m} \): \( |4.50 - 4.45| = 0.05 \, \text{m} \)
• For \( 4.60 \, \text{m} \): \( |4.60 - 4.45| = 0.15 \, \text{m} \)
• For \( 4.30 \, \text{m} \): \( |4.30 - 4.45| = 0.15 \, \text{m} \)
• For \( 4.40 \, \text{m} \): \( |4.40 - 4.45| = 0.05 \, \text{m} \)

Step3: Calculate the average deviation (uncertainty)

For a small data set, the uncertainty in the mean (average deviation) is calculated by taking the average of these absolute deviations. The formula is \( \text{Uncertainty} = \frac{\sum_{i = 1}^{n} |x_{i} - \bar{x}|}{n} \).

\[
\text{Uncertainty} = \frac{0.05 + 0.15 + 0.15 + 0.05}{4} = \frac{0.40}{4} = 0.10 \, \text{m}
\]

Wait, there might be a mistake. Wait, the small data set formula for uncertainty in the mean (standard error for small n, sometimes called the average deviation or using the range? Wait, no, maybe the formula is the standard deviation divided by the square root of n, but for small n, sometimes the formula is (maximum - minimum)/2 for range, but no, the problem says "small data set formula for the calculation of the uncertainty in the mean". Wait, maybe I used the wrong formula. Let's check again.

Wait, another approach: The uncertainty in the mean for a small data set can be calculated as the standard deviation divided by the square root of n. Let's calculate the standard deviation first.

First, mean \( \bar{x} = 4.45 \, \text{m} \)

Deviations from mean: \( (4.50 - 4.45)^2 = 0.0025 \), \( (4.60 - 4.45)^2 = 0.0225 \), \( (4.30 - 4.45)^2 = 0.0225 \), \( (4.40 - 4.45)^2 = 0.0025 \)

Sum of squared deviations: \( 0.0025 + 0.0225 + 0.0225 + 0.0025 = 0.05 \)

Variance \( s^2 = \frac{\sum (x_i - \bar{x})^2}{n - 1} = \frac{0.05}{3} \approx 0.0167 \)

Standard deviation \( s = \sqrt{0.0167} \approx 0.129 \, \text{m} \)

Then, the standard error (uncertainty in the mean) is \( \frac{s}{\sqrt{n}} = \frac{0.129}{\sqrt{4}} = \frac{0.129}{2} \approx 0.0645 \, \text{m} \), which is close to 0.0750 m (maybe due to rounding). Wait, maybe the problem uses the average of the absolute deviations. Wait, let's recalculate the absolute deviations:

\( |4.50 - 4.45| = 0.05 \), \( |4.60 - 4.45| = 0.15 \), \( |4.30 - 4.45| = 0.15 \), \( |4.40 - 4.45| = 0.05 \)

Sum of absolute deviations: \( 0.05 + 0.15 + 0.15 + 0.05 = 0.40 \)

Average absolute deviation: \( 0.40 / 4 = 0.10 \). But the options include 0.0750 m. Wait, maybe the formula is (max - min)/2 for the range, then divide by sqrt(n)? Wait, max is 4.60, min is 4.30, range is 0.30, so (0.30)/2 = 0.15, then divide by sqrt(4) = 2, so 0.15 / 2 = 0.075 m. Ah, that's the 0.0750 m option. So maybe the small data set formula is (range / 2) / sqrt(n), where range is max - min.

Let's check:

Range = 4.60 - 4.30 = 0.30 m

Uncertainty = (range / 2) / sqrt(n) = (0.30 / 2) / sqrt(4) = 0.15 / 2 = 0.075 m = 0.0750 m.

Yes, that matches the option 0.0750 m.

Answer:

0.0750 m (corresponding to the option with 0.0750 m)