Answer :
To find the population standard deviation for the given set of data, we'll go through it step by step:
1. List the data set:
- The given data is: 103, 107, 124, 121, 123, 67, 77, 107, 124.
2. Calculate the mean (average):
- Add up all the numbers: [tex]\(103 + 107 + 124 + 121 + 123 + 67 + 77 + 107 + 124 = 953\)[/tex].
- Divide by the number of data points: [tex]\(\frac{953}{9} \approx 105.889\)[/tex].
3. Calculate the variance:
- Subtract the mean from each data point and square the result:
- [tex]\((103 - 105.889)^2 \approx 8.344\)[/tex]
- [tex]\((107 - 105.889)^2 \approx 1.235\)[/tex]
- [tex]\((124 - 105.889)^2 \approx 331.235\)[/tex]
- [tex]\((121 - 105.889)^2 \approx 230.012\)[/tex]
- [tex]\((123 - 105.889)^2 \approx 295.012\)[/tex]
- [tex]\((67 - 105.889)^2 \approx 1501.012\)[/tex]
- [tex]\((77 - 105.889)^2 \approx 834.012\)[/tex]
- [tex]\((107 - 105.889)^2 \approx 1.235\)[/tex]
- [tex]\((124 - 105.889)^2 \approx 331.235\)[/tex]
- Calculate the average of these squared differences:
- [tex]\(\frac{8.344 + 1.235 + 331.235 + 230.012 + 295.012 + 1501.012 + 834.012 + 1.235 + 331.235}{9} \approx 392.765\)[/tex].
4. Calculate the standard deviation:
- Take the square root of the variance: [tex]\(\sqrt{392.765} \approx 19.818\)[/tex].
In conclusion, the population standard deviation of the dataset, rounded to the nearest thousandth, is approximately 19.818.
1. List the data set:
- The given data is: 103, 107, 124, 121, 123, 67, 77, 107, 124.
2. Calculate the mean (average):
- Add up all the numbers: [tex]\(103 + 107 + 124 + 121 + 123 + 67 + 77 + 107 + 124 = 953\)[/tex].
- Divide by the number of data points: [tex]\(\frac{953}{9} \approx 105.889\)[/tex].
3. Calculate the variance:
- Subtract the mean from each data point and square the result:
- [tex]\((103 - 105.889)^2 \approx 8.344\)[/tex]
- [tex]\((107 - 105.889)^2 \approx 1.235\)[/tex]
- [tex]\((124 - 105.889)^2 \approx 331.235\)[/tex]
- [tex]\((121 - 105.889)^2 \approx 230.012\)[/tex]
- [tex]\((123 - 105.889)^2 \approx 295.012\)[/tex]
- [tex]\((67 - 105.889)^2 \approx 1501.012\)[/tex]
- [tex]\((77 - 105.889)^2 \approx 834.012\)[/tex]
- [tex]\((107 - 105.889)^2 \approx 1.235\)[/tex]
- [tex]\((124 - 105.889)^2 \approx 331.235\)[/tex]
- Calculate the average of these squared differences:
- [tex]\(\frac{8.344 + 1.235 + 331.235 + 230.012 + 295.012 + 1501.012 + 834.012 + 1.235 + 331.235}{9} \approx 392.765\)[/tex].
4. Calculate the standard deviation:
- Take the square root of the variance: [tex]\(\sqrt{392.765} \approx 19.818\)[/tex].
In conclusion, the population standard deviation of the dataset, rounded to the nearest thousandth, is approximately 19.818.
