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To determine which city had the highest variability in temperatures over the course of one week, we can calculate the standard deviation of the temperatures for each city. The standard deviation provides a measure of how spread out the temperatures are around the mean (average) temperature.
Let's summarize the given temperatures for each city:
- Salem: 56, 61, 55, 62, 58, 61, 60
- Oxford: 71, 52, 58, 50, 89, 60, 61
### Step-by-Step Solution
1. Calculate Mean (Average) Temperature for Each City:
- For Salem:
[tex]\( \text{Mean}_{\text{Salem}} = \frac{56 + 61 + 55 + 62 + 58 + 61 + 60}{7} \)[/tex]
- For Oxford:
[tex]\( \text{Mean}_{\text{Oxford}} = \frac{71 + 52 + 58 + 50 + 89 + 60 + 61}{7} \)[/tex]
2. Calculate the Variance for Each City:
[tex]\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \][/tex]
Where [tex]\(x_i\)[/tex] represents each temperature and [tex]\(N\)[/tex] is the number of days (7 days).
- For Salem:
[tex]\[ \text{Variance}_{\text{Salem}} = \frac{(56 - \text{Mean}_{\text{Salem}})^2 + (61 - \text{Mean}_{\text{Salem}})^2 + ... + (60 - \text{Mean}_{\text{Salem}})^2}{7} \][/tex]
- For Oxford:
[tex]\[ \text{Variance}_{\text{Oxford}} = \frac{(71 - \text{Mean}_{\text{Oxford}})^2 + (52 - \text{Mean}_{\text{Oxford}})^2 + ... + (61 - \text{Mean}_{\text{Oxford}})^2}{7} \][/tex]
3. Calculate the Standard Deviation:
The standard deviation is simply the square root of the variance:
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} \][/tex]
4. Comparison of Standard Deviations:
With the calculated standard deviations, we compare them to determine which city has higher variability.
### Results
After performing these calculations, we get the following standard deviations:
- The standard deviation of temperatures for Salem is approximately 2.507.
- The standard deviation of temperatures for Oxford is approximately 12.352.
### Conclusion
Since the standard deviation for Oxford (12.352) is greater than that for Salem (2.507), we conclude that Oxford had the highest variability in temperatures over the week.
Let's summarize the given temperatures for each city:
- Salem: 56, 61, 55, 62, 58, 61, 60
- Oxford: 71, 52, 58, 50, 89, 60, 61
### Step-by-Step Solution
1. Calculate Mean (Average) Temperature for Each City:
- For Salem:
[tex]\( \text{Mean}_{\text{Salem}} = \frac{56 + 61 + 55 + 62 + 58 + 61 + 60}{7} \)[/tex]
- For Oxford:
[tex]\( \text{Mean}_{\text{Oxford}} = \frac{71 + 52 + 58 + 50 + 89 + 60 + 61}{7} \)[/tex]
2. Calculate the Variance for Each City:
[tex]\[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{N} \][/tex]
Where [tex]\(x_i\)[/tex] represents each temperature and [tex]\(N\)[/tex] is the number of days (7 days).
- For Salem:
[tex]\[ \text{Variance}_{\text{Salem}} = \frac{(56 - \text{Mean}_{\text{Salem}})^2 + (61 - \text{Mean}_{\text{Salem}})^2 + ... + (60 - \text{Mean}_{\text{Salem}})^2}{7} \][/tex]
- For Oxford:
[tex]\[ \text{Variance}_{\text{Oxford}} = \frac{(71 - \text{Mean}_{\text{Oxford}})^2 + (52 - \text{Mean}_{\text{Oxford}})^2 + ... + (61 - \text{Mean}_{\text{Oxford}})^2}{7} \][/tex]
3. Calculate the Standard Deviation:
The standard deviation is simply the square root of the variance:
[tex]\[ \text{Standard Deviation} = \sqrt{\text{Variance}} \][/tex]
4. Comparison of Standard Deviations:
With the calculated standard deviations, we compare them to determine which city has higher variability.
### Results
After performing these calculations, we get the following standard deviations:
- The standard deviation of temperatures for Salem is approximately 2.507.
- The standard deviation of temperatures for Oxford is approximately 12.352.
### Conclusion
Since the standard deviation for Oxford (12.352) is greater than that for Salem (2.507), we conclude that Oxford had the highest variability in temperatures over the week.
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