Question: Find the range, mean, median, mode, and standard deviation of the following data set. 129, 131,…

Find the range, mean, median, mode, and standard deviation of the following data set.

129, 131, 136, 124, 132, 124, 129, 127

Solution

To solve the problem, we will find the range, mean, median, mode, and standard deviation of the data set: 129, 131, 136, 124, 132, 124, 129, 127. First, arrange the data in ascending order: 124, 124, 127, 129, 129, 131, 132, 136 Range: The range is the difference between the largest and smallest values. \[ \text{Range} = 136 - 124 = 12 \] Mean: The mean is the average of all the numbers. \[ \text{Mean} = \frac{124 + 124 + 127 + 129 + 129 + 131 + 132 + 136}{8} = \frac{1032}{8} = 129 \] Median: The median is the middle number in a sorted list. Since there are 8 numbers, the median will be the average of the 4th and 5th numbers. \[ \text{Median} = \frac{129 + 129}{2} = 129 \] Mode: The mode is the number that appears most frequently. Here, both 124 and 129 appear twice, but 124 appears first in the ordered list. So, the mode is 124. Standard Deviation: First, we find the variance. The variance is the average of the squared differences from the mean. The squared differences from the mean (129) are: \[ \begin{align*} (124 - 129)^2 &= 25 \\ (124 - 129)^2 &= 25 \\ (127 - 129)^2 &= 4 \\ (129 - 129)^2 &= 0 \\ (129 - 129)^2 &= 0 \\ (131 - 129)^2 &= 4 \\ (132 - 129)^2 &= 9 \\ (136 - 129)^2 &= 49 \\ \end{align*} \] Sum of squared differences: \[ 25 + 25 + 4 + 0 + 0 + 4 + 9 + 49 = 116 \] Variance: \[ \text{Variance} = \frac{116}{8} = 14.5 \] Standard deviation is the square root of the variance. \[ \text{Standard Deviation} = \sqrt{14.5} \approx 3.81 \] Final summary: - Range: 12 - Mean: 129 - Median: 129 - Mode: 124 - Standard Deviation: 3.81