Answer :
To find the variance of the given data set, follow these steps:
1. List the Data: The data set provided is: 198, 190, 245, 211, 193, and 193.
2. Given Mean: The mean ([tex]\(\bar{x}\)[/tex]) of the data set is given as 205.
3. Calculate the Deviations: Find the deviation of each data point from the mean. This means subtracting the mean from each data point.
- Deviation for 198: [tex]\(198 - 205 = -7\)[/tex]
- Deviation for 190: [tex]\(190 - 205 = -15\)[/tex]
- Deviation for 245: [tex]\(245 - 205 = 40\)[/tex]
- Deviation for 211: [tex]\(211 - 205 = 6\)[/tex]
- Deviation for 193: [tex]\(193 - 205 = -12\)[/tex] (for both instances of 193)
4. Square the Deviations: Square each of the deviations to eliminate any negative signs (because variance involves squared values):
- Squared deviation for -7: [tex]\((-7)^2 = 49\)[/tex]
- Squared deviation for -15: [tex]\((-15)^2 = 225\)[/tex]
- Squared deviation for 40: [tex]\((40)^2 = 1600\)[/tex]
- Squared deviation for 6: [tex]\((6)^2 = 36\)[/tex]
- Squared deviations for both -12: [tex]\((-12)^2 = 144\)[/tex] and [tex]\((-12)^2 = 144\)[/tex]
5. Find the Mean of the Squared Deviations: Add all the squared deviations and then divide by the number of data points to find the variance.
[tex]\[
\text{Variance} = \frac{49 + 225 + 1600 + 36 + 144 + 144}{6} = \frac{2198}{6} \approx 366.33
\][/tex]
Thus, the variance of the data set is approximately [tex]\(366.33\)[/tex].
1. List the Data: The data set provided is: 198, 190, 245, 211, 193, and 193.
2. Given Mean: The mean ([tex]\(\bar{x}\)[/tex]) of the data set is given as 205.
3. Calculate the Deviations: Find the deviation of each data point from the mean. This means subtracting the mean from each data point.
- Deviation for 198: [tex]\(198 - 205 = -7\)[/tex]
- Deviation for 190: [tex]\(190 - 205 = -15\)[/tex]
- Deviation for 245: [tex]\(245 - 205 = 40\)[/tex]
- Deviation for 211: [tex]\(211 - 205 = 6\)[/tex]
- Deviation for 193: [tex]\(193 - 205 = -12\)[/tex] (for both instances of 193)
4. Square the Deviations: Square each of the deviations to eliminate any negative signs (because variance involves squared values):
- Squared deviation for -7: [tex]\((-7)^2 = 49\)[/tex]
- Squared deviation for -15: [tex]\((-15)^2 = 225\)[/tex]
- Squared deviation for 40: [tex]\((40)^2 = 1600\)[/tex]
- Squared deviation for 6: [tex]\((6)^2 = 36\)[/tex]
- Squared deviations for both -12: [tex]\((-12)^2 = 144\)[/tex] and [tex]\((-12)^2 = 144\)[/tex]
5. Find the Mean of the Squared Deviations: Add all the squared deviations and then divide by the number of data points to find the variance.
[tex]\[
\text{Variance} = \frac{49 + 225 + 1600 + 36 + 144 + 144}{6} = \frac{2198}{6} \approx 366.33
\][/tex]
Thus, the variance of the data set is approximately [tex]\(366.33\)[/tex].
