Question: 4) Find the range, mean, median, mode, and standard deviation of the following data set. 54, 46,…

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

54, 46, 37, 40, 48, 48, 39

\[ \sum (x - \bar{x})^2 = \]

\[ s = \sqrt{\frac{\sum (x - \bar{x})^2}{n - 1}} \]

Solution

To find the range, mean, median, mode, and standard deviation for the data set \(54, 46, 37, 40, 48, 48, 39\), we will solve step by step: 1. Range The range is the difference between the maximum and minimum values. Maximum value = 54 Minimum value = 37 \[ \text{Range} = 54 - 37 = 17 \] 2. Mean The mean is the average of the data. \[ \text{Mean} = \frac{54 + 46 + 37 + 40 + 48 + 48 + 39}{7} = \frac{312}{7} \approx 44.57 \] 3. Median The median is the middle value when the numbers are arranged in order. Ordered data set: 37, 39, 40, 46, 48, 48, 54 Median = 46 4. Mode The mode is the number that appears most frequently. Mode = 48 5. Standard Deviation To find the standard deviation, use the formula provided. First, calculate the deviations from the mean and their squares: \[ \begin{array}{ccc} \text{Data value} & \text{Deviation from Mean} & \text{Deviation squared} \\ 54 & 54 - 44.57 \approx 9.43 & (9.43)^2 \approx 88.87 \\ 46 & 46 - 44.57 \approx 1.43 & (1.43)^2 \approx 2.05 \\ 37 & 37 - 44.57 \approx -7.57 & (-7.57)^2 \approx 57.31 \\ 40 & 40 - 44.57 \approx -4.57 & (-4.57)^2 \approx 20.88 \\ 48 & 48 - 44.57 \approx 3.43 & (3.43)^2 \approx 11.76 \\ 48 & 48 - 44.57 \approx 3.43 & (3.43)^2 \approx 11.76 \\ 39 & 39 - 44.57 \approx -5.57 & (-5.57)^2 \approx 31.02 \\ \end{array} \] Sum of deviation squared: \[ \sum (x - \bar{x})^2 = 88.87 + 2.05 + 57.31 + 20.88 + 11.76 + 11.76 + 31.02 = 223.65 \] Using the formula for standard deviation: \[ s = \sqrt{\frac{223.65}{6}} \approx \sqrt{37.275} \approx 6.10 \] Thus, the range is 17, the mean is approximately 44.57, the median is 46, the mode is 48, and the standard deviation is approximately 6.10.