Yuri's first error occurred when he divided by the total number of data points [tex]\( n \)[/tex] instead of [tex]\( n - 1 \)[/tex]. This is a crucial step when calculating the sample standard deviation. For a sample, the denominator should be [tex]\( n - 1 \)[/tex] to account for the degrees of freedom in the estimation of the population standard deviation from the sample data.
Let's go through the steps correctly:
1. Calculate the mean:
[tex]\[
\text{mean} = \frac{12 + 14 + 9 + 21}{4} = \frac{56}{4} = 14
\][/tex]
2. Find the squared differences from the mean:
[tex]\[
(12 - 14)^2 = (-2)^2 = 4
\][/tex]
[tex]\[
(14 - 14)^2 = 0^2 = 0
\][/tex]
[tex]\[
(9 - 14)^2 = (-5)^2 = 25
\][/tex]
[tex]\[
(21 - 14)^2 = 7^2 = 49
\][/tex]
3. Sum the squared differences:
[tex]\[
4 + 0 + 25 + 49 = 78
\][/tex]
4. Divide by [tex]\( n - 1 \)[/tex] (since [tex]\( n = 4 \)[/tex]):
[tex]\[
\frac{78}{4 - 1} = \frac{78}{3} = 26
\][/tex]
5. Take the square root to find the sample standard deviation:
[tex]\[
s = \sqrt{26}
\][/tex]
Therefore, the correct sample standard deviation is:
[tex]\[
s = \sqrt{26}
\][/tex]
The first error Yuri made was not dividing by [tex]\( n - 1 \)[/tex] (which is 3 in this case) when calculating the sample standard deviation. Instead, he incorrectly divided by [tex]\( n \)[/tex].
