Let's go through the steps Emi took and identify where the first error occurred.
1. Emi correctly calculated the mean of the data set:
[tex]\[
\text{Mean} = \frac{87 + 46 + 90 + 78 + 89}{5} = \frac{390}{5} = 78
\][/tex]
2. Emi needed to compute the squared differences from the mean for each data point.
[tex]\[
(87 - 78)^2 = 9^2 = 81
\][/tex]
[tex]\[
(46 - 78)^2 = (-32)^2 = 1024
\][/tex]
[tex]\[
(90 - 78)^2 = 12^2 = 144
\][/tex]
[tex]\[
(78 - 78)^2 = 0^2 = 0
\][/tex]
[tex]\[
(89 - 78)^2 = 11^2 = 121
\][/tex]
3. Summing these squared differences:
[tex]\[
81 + 1024 + 144 + 0 + 121 = 1370
\][/tex]
4. Emi then needed to compute the variance by dividing the sum of squared differences by the number of data points (since this is population data, we use [tex]\( N \)[/tex], not [tex]\( N-1 \)[/tex]):
[tex]\[
\sigma^2 = \frac{1370}{5} = 274
\][/tex]
Now, let's compare this with what Emi wrote to identify the mistake:
Emi's steps were:
[tex]\[
\sigma^2 = \frac{(87-78)^2 + (46-78)^2 + (90-78)^2 + (78-78)^2 + (89-78)^2}{5}
\][/tex]
[tex]\[
= \frac{81 + 1024 + 144 + 0 + 121}{5}
\][/tex]
[tex]\[
= \frac{1370}{5}
\][/tex]
It seems Emi's written steps were correct up to this point. This means that there is no error in calculus of differences or means or summing up squared differences.
Upon review, the provided erroneous variance calculation:
[tex]\[
=\frac{-878}{5}
\][/tex]
is not correct. This might be a typographical error that Emi made after the summation step or a copy-paste mistake. The actual sum calculation is correct as shown.
Thus, there was no error up to summing up the squared differences, and the first visible error occurs while computing the sum and incorrectly stating:
[tex]\[
\frac{-878}{5}=-175.6
\][/tex]
She should have used the correct sum of 1370. Therefore the identified first error should be: a typing error in summarized squared differences. Other than that all steps and results are internally correct.
This should correct her calculated variance to:
[tex]\(\sigma^2 = 274.0\)[/tex].
There is no error of using [tex]\(N-1\)[/tex] instead of [tex]\(N\)[/tex] as it’s population variance and she used correct number of data points, i.e. [tex]\(N = 5\)[/tex].
