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Answer :
To find the variance and standard deviation of the highest temperatures recorded in eight specific states, follow these steps:
1. Find the Mean:
- First, we calculate the mean (average) of the temperatures.
- The temperatures are: 112, 100, 127, 120, 134, 118, 105, and 110.
- Add all the temperatures together: [tex]\(112 + 100 + 127 + 120 + 134 + 118 + 105 + 110 = 922\)[/tex].
- Divide the total by the number of temperatures (8): [tex]\(\frac{922}{8} = 115.75\)[/tex].
- So, the mean temperature is 115.75.
2. Calculate the Variance:
- Next, we find how much each temperature deviates from the mean and square that deviation.
- For each temperature, subtract the mean and square the result:
- [tex]\((112 - 115.75)^2 = 14.0625\)[/tex]
- [tex]\((100 - 115.75)^2 = 252.5625\)[/tex]
- [tex]\((127 - 115.75)^2 = 126.5625\)[/tex]
- [tex]\((120 - 115.75)^2 = 18.0625\)[/tex]
- [tex]\((134 - 115.75)^2 = 331.5625\)[/tex]
- [tex]\((118 - 115.75)^2 = 5.0625\)[/tex]
- [tex]\((105 - 115.75)^2 = 115.5625\)[/tex]
- [tex]\((110 - 115.75)^2 = 33.0625\)[/tex]
- Sum these squared deviations:
[tex]\[
14.0625 + 252.5625 + 126.5625 + 18.0625 + 331.5625 + 5.0625 + 115.5625 + 33.0625 = 893.5
\][/tex]
- Divide the sum by the number of data points (8):
[tex]\(\frac{893.5}{8} = 111.6875\)[/tex].
- So, the variance is 111.6875.
3. Calculate the Standard Deviation:
- The standard deviation is the square root of the variance.
- [tex]\(\sqrt{111.6875} \approx 10.57\)[/tex].
Thus, the variance of the temperatures is 111.6875, and the standard deviation is approximately 10.57.
1. Find the Mean:
- First, we calculate the mean (average) of the temperatures.
- The temperatures are: 112, 100, 127, 120, 134, 118, 105, and 110.
- Add all the temperatures together: [tex]\(112 + 100 + 127 + 120 + 134 + 118 + 105 + 110 = 922\)[/tex].
- Divide the total by the number of temperatures (8): [tex]\(\frac{922}{8} = 115.75\)[/tex].
- So, the mean temperature is 115.75.
2. Calculate the Variance:
- Next, we find how much each temperature deviates from the mean and square that deviation.
- For each temperature, subtract the mean and square the result:
- [tex]\((112 - 115.75)^2 = 14.0625\)[/tex]
- [tex]\((100 - 115.75)^2 = 252.5625\)[/tex]
- [tex]\((127 - 115.75)^2 = 126.5625\)[/tex]
- [tex]\((120 - 115.75)^2 = 18.0625\)[/tex]
- [tex]\((134 - 115.75)^2 = 331.5625\)[/tex]
- [tex]\((118 - 115.75)^2 = 5.0625\)[/tex]
- [tex]\((105 - 115.75)^2 = 115.5625\)[/tex]
- [tex]\((110 - 115.75)^2 = 33.0625\)[/tex]
- Sum these squared deviations:
[tex]\[
14.0625 + 252.5625 + 126.5625 + 18.0625 + 331.5625 + 5.0625 + 115.5625 + 33.0625 = 893.5
\][/tex]
- Divide the sum by the number of data points (8):
[tex]\(\frac{893.5}{8} = 111.6875\)[/tex].
- So, the variance is 111.6875.
3. Calculate the Standard Deviation:
- The standard deviation is the square root of the variance.
- [tex]\(\sqrt{111.6875} \approx 10.57\)[/tex].
Thus, the variance of the temperatures is 111.6875, and the standard deviation is approximately 10.57.
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