Absolutely, I'd be happy to provide you with a step-by-step solution for finding the variance and standard deviation for Adimas's math test scores!
Firstly, let's clearly restate the scores given:
[tex]\[ 76, 87, 55, 88, 67, 84, 77, 82, 91, 85, 90 \][/tex]
### Step 1: Calculate the Mean
Given that the mean [tex]\( \bar{x} \)[/tex] was calculated as:
[tex]\[
\bar{x} = \frac{76 + 87 + 55 + 88 + 67 + 84 + 77 + 82 + 91 + 85 + 90}{11} = \frac{892}{11} \approx 80.18
\][/tex]
### Step 2: Compute Each Score’s Deviation from the Mean and Square It
We need to find the squared differences from the mean for each score:
[tex]\[
(76 - 80.18)^2, (87 - 80.18)^2, (55 - 80.18)^2, \ldots, (90 - 80.18)^2
\][/tex]
Here are those calculations:
[tex]\[
(76 - 80.18)^2 = (-4.18)^2 = 17.47
\][/tex]
[tex]\[
(87 - 80.18)^2 = (6.82)^2 = 46.51
\][/tex]
[tex]\[
(55 - 80.18)^2 = (-25.18)^2 = 634.01
\][/tex]
[tex]\[
(88 - 80.18)^2 = (7.82)^2 = 61.15
\][/tex]
[tex]\[
(67 - 80.18)^2 = (-13.18)^2 = 173.15
\][/tex]
[tex]\[
(84 - 80.18)^2 = (3.82)^2 = 14.60
\][/tex]
[tex]\[
(77 - 80.18)^2 = (-3.18)^2 = 10.11
\][/tex]
[tex]\[
(82 - 80.18)^2 = (1.82)^2 = 3.31
\][/tex]
[tex]\[
(91 - 80.18)^2 = (10.82)^2 = 117.04
\][/tex]
[tex]\[
(85 - 80.18)^2 = (4.82)^2 = 23.22
\][/tex]
[tex]\[
(90 - 80.18)^2 = (9.82)^2 = 96.43
\][/tex]
### Step 3: Find the Sum of the Squared Deviations
Next, sum all these squared deviations:
[tex]\[
17.47 + 46.51 + 634.01 + 61.15 + 173.15 + 14.60 + 10.11 + 3.31 + 117.04 + 23.22 + 96.43 = 1196
\][/tex]
### Step 4: Calculate the Variance
The variance [tex]\( \sigma^2 \)[/tex] is the sum of squared deviations divided by the number of scores [tex]\( n \)[/tex]:
[tex]\[
\sigma^2 = \frac{1196}{11} \approx 108.88
\][/tex]
So, the variance of Adimas's grades, rounded to the nearest hundredth, is:
[tex]\[
\sigma^2 \approx 108.88
\][/tex]
### Step 5: Calculate the Standard Deviation
The standard deviation [tex]\( \sigma \)[/tex] is the square root of the variance:
[tex]\[
\sigma = \sqrt{108.88} \approx 10.43
\][/tex]
Therefore, the standard deviation of Adimas's grades, rounded to the nearest hundredth, is:
[tex]\[
\sigma \approx 10.43
\][/tex]
### Summary:
- Variance [tex]\( \sigma^2 \approx 108.88 \)[/tex]
- Standard Deviation [tex]\( \sigma \approx 10.43 \)[/tex]
By following these steps, we have found the variance and standard deviation of Adimas's math test scores.
