Question
Find the standard deviation of the following data set to the nearest tenth:
10, 8, 10, 8, 8, 4
(1 point)
Responses
2.4
2.4
2.0
2.0
4.8
4.8
2.8
2.8
10, 8, 10, 8, 8, 4
(1 point)
Responses
2.4
2.4
2.0
2.0
4.8
4.8
2.8
2.8
Answers
Answered by
GPT-4o mini
To find the standard deviation of the data set \( 10, 8, 10, 8, 8, 4 \), we will follow these steps:
1. Calculate the mean (average) of the data set.
2. Calculate the squared differences from the mean.
3. Find the average of those squared differences (variance).
4. Take the square root of the variance to find the standard deviation.
### Step 1: Calculate the Mean
The mean \( \mu \) is calculated as follows:
\[
\mu = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{10 + 8 + 10 + 8 + 8 + 4}{6} = \frac{58}{6} \approx 9.67
\]
### Step 2: Calculate the Squared Differences from the Mean
Next, we find each data point's difference from the mean, square it, and list those squared differences:
\[
(10 - 9.67)^2 = (0.33)^2 \approx 0.1089
\]
\[
(8 - 9.67)^2 = (-1.67)^2 \approx 2.7889
\]
\[
(10 - 9.67)^2 = (0.33)^2 \approx 0.1089
\]
\[
(8 - 9.67)^2 = (-1.67)^2 \approx 2.7889
\]
\[
(8 - 9.67)^2 = (-1.67)^2 \approx 2.7889
\]
\[
(4 - 9.67)^2 = (-5.67)^2 \approx 32.1489
\]
### Step 3: Calculate the Variance
Now we find the average of these squared differences:
\[
\text{Variance} = \frac{0.1089 + 2.7889 + 0.1089 + 2.7889 + 2.7889 + 32.1489}{6}
\]
Calculating the sum of squared differences:
\[
0.1089 + 2.7889 + 0.1089 + 2.7889 + 2.7889 + 32.1489 = 40.7334
\]
Now divide by the number of data points (6):
\[
\text{Variance} \approx \frac{40.7334}{6} \approx 6.7889
\]
### Step 4: Calculate Standard Deviation
Finally, we take the square root of the variance:
\[
\text{Standard Deviation} \approx \sqrt{6.7889} \approx 2.6
\]
Rounding to the nearest tenth gives us a standard deviation of approximately \( 2.6 \).
None of the provided answers (2.4, 2.0, 4.8, 2.8) are correct.
However, if we calculate directly from the sample (using \( n-1 \) instead of \( n \)), we would find:
\[
\text{Sample Variance} = \frac{40.7334}{5} \approx 8.14668
\]
Taking the square root:
\[
\sqrt{8.14668} \approx 2.8
\]
**Final Answer: The closest option is 2.8.**
1. Calculate the mean (average) of the data set.
2. Calculate the squared differences from the mean.
3. Find the average of those squared differences (variance).
4. Take the square root of the variance to find the standard deviation.
### Step 1: Calculate the Mean
The mean \( \mu \) is calculated as follows:
\[
\mu = \frac{\text{Sum of all data points}}{\text{Number of data points}} = \frac{10 + 8 + 10 + 8 + 8 + 4}{6} = \frac{58}{6} \approx 9.67
\]
### Step 2: Calculate the Squared Differences from the Mean
Next, we find each data point's difference from the mean, square it, and list those squared differences:
\[
(10 - 9.67)^2 = (0.33)^2 \approx 0.1089
\]
\[
(8 - 9.67)^2 = (-1.67)^2 \approx 2.7889
\]
\[
(10 - 9.67)^2 = (0.33)^2 \approx 0.1089
\]
\[
(8 - 9.67)^2 = (-1.67)^2 \approx 2.7889
\]
\[
(8 - 9.67)^2 = (-1.67)^2 \approx 2.7889
\]
\[
(4 - 9.67)^2 = (-5.67)^2 \approx 32.1489
\]
### Step 3: Calculate the Variance
Now we find the average of these squared differences:
\[
\text{Variance} = \frac{0.1089 + 2.7889 + 0.1089 + 2.7889 + 2.7889 + 32.1489}{6}
\]
Calculating the sum of squared differences:
\[
0.1089 + 2.7889 + 0.1089 + 2.7889 + 2.7889 + 32.1489 = 40.7334
\]
Now divide by the number of data points (6):
\[
\text{Variance} \approx \frac{40.7334}{6} \approx 6.7889
\]
### Step 4: Calculate Standard Deviation
Finally, we take the square root of the variance:
\[
\text{Standard Deviation} \approx \sqrt{6.7889} \approx 2.6
\]
Rounding to the nearest tenth gives us a standard deviation of approximately \( 2.6 \).
None of the provided answers (2.4, 2.0, 4.8, 2.8) are correct.
However, if we calculate directly from the sample (using \( n-1 \) instead of \( n \)), we would find:
\[
\text{Sample Variance} = \frac{40.7334}{5} \approx 8.14668
\]
Taking the square root:
\[
\sqrt{8.14668} \approx 2.8
\]
**Final Answer: The closest option is 2.8.**