Answer :
To find the standard deviation of the given data set {4, 7, 8, 9, 12, 12}, we will follow these steps:
1. Calculate the Mean:
- First, find the mean (average) of the data set.
- Add all the values together: 4 + 7 + 8 + 9 + 12 + 12 = 52
- Divide by the number of values: [tex]\( \frac{52}{6} \approx 8.67 \)[/tex]
2. Find the Squared Differences from the Mean:
- Subtract the mean from each data point and then square the result:
- [tex]\( (4 - 8.67)^2 \approx (-4.67)^2 = 21.8 \)[/tex]
- [tex]\( (7 - 8.67)^2 \approx (-1.67)^2 = 2.79 \)[/tex]
- [tex]\( (8 - 8.67)^2 \approx (-0.67)^2 = 0.45 \)[/tex]
- [tex]\( (9 - 8.67)^2 \approx 0.33^2 = 0.11 \)[/tex]
- [tex]\( (12 - 8.67)^2 \approx 3.33^2 = 11.09 \)[/tex]
- [tex]\( (12 - 8.67)^2 \approx 3.33^2 = 11.09 \)[/tex]
3. Average the Squared Differences:
- Add up all the squared differences: [tex]\( 21.8 + 2.79 + 0.45 + 0.11 + 11.09 + 11.09 = 47.33 \)[/tex]
- Divide by the number of values: [tex]\( \frac{47.33}{6} = 7.89 \)[/tex]
4. Calculate the Standard Deviation:
- Take the square root of the average squared differences: [tex]\( \sqrt{7.89} \approx 2.81 \)[/tex]
From the calculations above, the standard deviation of the data set {4, 7, 8, 9, 12, 12} is approximately 2.81.
Therefore, the correct choice from the given options is not listed, but the closest to 2.81 would be closest if rounded. In this context, the correct understanding concludes that the exact answer is 2.8087 which can be confirmed on further precision requests.
1. Calculate the Mean:
- First, find the mean (average) of the data set.
- Add all the values together: 4 + 7 + 8 + 9 + 12 + 12 = 52
- Divide by the number of values: [tex]\( \frac{52}{6} \approx 8.67 \)[/tex]
2. Find the Squared Differences from the Mean:
- Subtract the mean from each data point and then square the result:
- [tex]\( (4 - 8.67)^2 \approx (-4.67)^2 = 21.8 \)[/tex]
- [tex]\( (7 - 8.67)^2 \approx (-1.67)^2 = 2.79 \)[/tex]
- [tex]\( (8 - 8.67)^2 \approx (-0.67)^2 = 0.45 \)[/tex]
- [tex]\( (9 - 8.67)^2 \approx 0.33^2 = 0.11 \)[/tex]
- [tex]\( (12 - 8.67)^2 \approx 3.33^2 = 11.09 \)[/tex]
- [tex]\( (12 - 8.67)^2 \approx 3.33^2 = 11.09 \)[/tex]
3. Average the Squared Differences:
- Add up all the squared differences: [tex]\( 21.8 + 2.79 + 0.45 + 0.11 + 11.09 + 11.09 = 47.33 \)[/tex]
- Divide by the number of values: [tex]\( \frac{47.33}{6} = 7.89 \)[/tex]
4. Calculate the Standard Deviation:
- Take the square root of the average squared differences: [tex]\( \sqrt{7.89} \approx 2.81 \)[/tex]
From the calculations above, the standard deviation of the data set {4, 7, 8, 9, 12, 12} is approximately 2.81.
Therefore, the correct choice from the given options is not listed, but the closest to 2.81 would be closest if rounded. In this context, the correct understanding concludes that the exact answer is 2.8087 which can be confirmed on further precision requests.
