In the solution provided, Yuri did not actually make a mistake in calculating the standard deviation. Let's carefully review each step in detail to ensure the accuracy of these calculations:
1. Yuri calculated the differences between each data point and the mean:
- [tex]\( 12 - 14 = -2 \)[/tex]
- [tex]\( 14 - 14 = 0 \)[/tex]
- [tex]\( 9 - 14 = -5 \)[/tex]
- [tex]\( 21 - 14 = 7 \)[/tex]
These differences are correctly calculated as:
-2, 0, -5, and 7.
2. Yuri then squared each of these differences:
- [tex]\( (-2)^2 = 4 \)[/tex]
- [tex]\( 0^2 = 0 \)[/tex]
- [tex]\( (-5)^2 = 25 \)[/tex]
- [tex]\( 7^2 = 49 \)[/tex]
These squared differences are:
4, 0, 25, and 49.
3. Next, he calculated the sum of the squared differences:
- [tex]\( 4 + 0 + 25 + 49 = 78 \)[/tex]
The sum of the squared differences is correctly computed as 78.
4. He then divided this sum by the number of data points (4):
- [tex]\( \frac{78}{4} = 19.5 \)[/tex]
The variance is correctly computed as 19.5.
5. Finally, Yuri found the square root of the variance to get the standard deviation:
- [tex]\( \sqrt{19.5} \approx 4.416 \)[/tex]
The standard deviation is approximately [tex]\( 4.416 \)[/tex].
Each step is meticulously correct as per the detailed calculations provided. Yuri did not make an error in any step of computing the standard deviation. Hence, there is no error in Yuri's calculation. The detailed and correctly performed steps attest to the accuracy of his result.
