Question
Christine is currently taking a college astronomy class and the instructor often goves quizzes. On the past seven quizzes, Christine got the scores shown below. Find the standard deviation, rounding to one more decimal place is present in the original data
50 15 31 27 11 42 71
1)20.9
2)437.6
3)31
4)8715.6
50 15 31 27 11 42 71
1)20.9
2)437.6
3)31
4)8715.6
Answers
GPT-4o mini
To calculate the standard deviation, we'll follow these steps:
1. Calculate the mean (average) of the scores.
2. Subtract the mean from each score and square the result.
3. Calculate the average of those squared differences.
4. Take the square root of that average.
### Step 1: Calculate the Mean
The scores are: 50, 15, 31, 27, 11, 42, 71.
\[
\text{Mean} = \frac{\text{Sum of scores}}{\text{Number of scores}} = \frac{50 + 15 + 31 + 27 + 11 + 42 + 71}{7} = \frac{ 50 + 15 + 31 + 27 + 11 + 42 + 71 }{ 7 } = \frac{ 257 }{ 7 } \approx 36.7
\]
### Step 2: Calculate Squared Differences from the Mean
Now we subtract the mean from each score and square the result:
\[
\begin{align*}
(50 - 36.7)^2 & = (13.3)^2 = 176.89 \\
(15 - 36.7)^2 & = (-21.7)^2 = 471.69 \\
(31 - 36.7)^2 & = (-5.7)^2 = 32.49 \\
(27 - 36.7)^2 & = (-9.7)^2 = 94.09 \\
(11 - 36.7)^2 & = (-25.7)^2 = 660.49 \\
(42 - 36.7)^2 & = (5.3)^2 = 28.09 \\
(71 - 36.7)^2 & = (34.3)^2 = 1177.69 \\
\end{align*}
\]
Now we can sum these squared differences:
\[
\text{Sum of squared differences} = 176.89 + 471.69 + 32.49 + 94.09 + 660.49 + 28.09 + 1177.69 \approx 2541.5
\]
### Step 3: Calculate the Variance
We find the variance by taking the average of the squared differences. Since we have a sample (rather than the entire population), we'll divide by \( n-1 \) (where \( n \) is the number of scores, which is 7 in this case):
\[
\text{Variance} = \frac{2541.5}{7-1} = \frac{2541.5}{6} \approx 423.5833
\]
### Step 4: Calculate the Standard Deviation
Finally, we take the square root of the variance to get the standard deviation:
\[
\text{Standard Deviation} = \sqrt{423.5833} \approx 20.6
\]
Rounding to one more decimal place from the original data results in:
\[
\text{Standard Deviation} \approx 20.6
\]
Thus, the closest answer to the standard deviation calculated is:
**1) 20.9**
1. Calculate the mean (average) of the scores.
2. Subtract the mean from each score and square the result.
3. Calculate the average of those squared differences.
4. Take the square root of that average.
### Step 1: Calculate the Mean
The scores are: 50, 15, 31, 27, 11, 42, 71.
\[
\text{Mean} = \frac{\text{Sum of scores}}{\text{Number of scores}} = \frac{50 + 15 + 31 + 27 + 11 + 42 + 71}{7} = \frac{ 50 + 15 + 31 + 27 + 11 + 42 + 71 }{ 7 } = \frac{ 257 }{ 7 } \approx 36.7
\]
### Step 2: Calculate Squared Differences from the Mean
Now we subtract the mean from each score and square the result:
\[
\begin{align*}
(50 - 36.7)^2 & = (13.3)^2 = 176.89 \\
(15 - 36.7)^2 & = (-21.7)^2 = 471.69 \\
(31 - 36.7)^2 & = (-5.7)^2 = 32.49 \\
(27 - 36.7)^2 & = (-9.7)^2 = 94.09 \\
(11 - 36.7)^2 & = (-25.7)^2 = 660.49 \\
(42 - 36.7)^2 & = (5.3)^2 = 28.09 \\
(71 - 36.7)^2 & = (34.3)^2 = 1177.69 \\
\end{align*}
\]
Now we can sum these squared differences:
\[
\text{Sum of squared differences} = 176.89 + 471.69 + 32.49 + 94.09 + 660.49 + 28.09 + 1177.69 \approx 2541.5
\]
### Step 3: Calculate the Variance
We find the variance by taking the average of the squared differences. Since we have a sample (rather than the entire population), we'll divide by \( n-1 \) (where \( n \) is the number of scores, which is 7 in this case):
\[
\text{Variance} = \frac{2541.5}{7-1} = \frac{2541.5}{6} \approx 423.5833
\]
### Step 4: Calculate the Standard Deviation
Finally, we take the square root of the variance to get the standard deviation:
\[
\text{Standard Deviation} = \sqrt{423.5833} \approx 20.6
\]
Rounding to one more decimal place from the original data results in:
\[
\text{Standard Deviation} \approx 20.6
\]
Thus, the closest answer to the standard deviation calculated is:
**1) 20.9**