Find the standard deviation of the nine measurements 4.26, 4.23, 4.27, 4.26, 4.24, 4.24, 4.26, 4.24, 4.25.

Step-by-step solution

1. 1

   Read the data correctly first. Written as "4,26 4,23 …" the commas are decimal separators (European notation), not list separators — otherwise you would have $18$ values of $4$ and $2x$ instead of $9$ measurements clustered near $4.25$. Reading them as separators is the one mistake that changes the answer completely.

2. 2

   Compute the mean.

   $\bar{x} = \frac{38.25}{9} = 4.25$

   The sum $38.25$ comes out exactly, so the mean is exactly $4.25$ and the deviations will all be simple hundredths.

3. 3

   Work with deviations in hundredths. Subtracting $4.25$ from each value gives

   $+0.01,\ -0.02,\ +0.02,\ +0.01,\ -0.01,\ -0.01,\ +0.01,\ -0.01,\ 0$

   They sum to $0$, confirming the mean. Squaring and adding:

   $\sum (x-\bar{x})^2 = 6(0.0001) + 2(0.0004) + 1(0) = 0.0006 + 0.0008 = 0.0014$

4. 4

   Divide by n for a population, by n − 1 for a sample. With $n = 9$:

   $\sigma^2 = \frac{0.0014}{9} = 0.000155\overline{5}, \qquad s^2 = \frac{0.0014}{8} = 0.000175$

5. 5

   Take square roots and interpret the size.

   $\sigma = \sqrt{0.0001556} \approx 0.01247, \qquad s = \sqrt{0.000175} \approx 0.01323$

   Rounded: $\sigma \approx 0.0125$ and $s \approx 0.0132$. Repeated measurements of one quantity are normally treated as a sample, so $s \approx 0.0132$ is the value to quote for measurement precision.