4. Consider a sample with data values of \( 27,25,20,15,30 \), 34,28 , and 25 . Using the given data set, perform the following: A. Calculate the mean and the median for the given data set B. Identify the mode and the range for the given observation C. Compute the 2 nd, quartile, 6 th deciles and 20th percentiles D. Compute the variance and standard deviation for the data set

Let's analyze the given data set step by step to address each part of the problem. **Given Data Set:** \[ 27, 25, 20, 15, 30, 34, 28, 25 \] First, let's arrange the data in ascending order for easier calculations: \[ 15, 20, 25, 25, 27, 28, 30, 34 \] There are **8** data points in total. --- ### **A. Calculate the Mean and the Median** **1. Mean:** The mean (average) is calculated by summing all the data points and dividing by the number of observations. \[ \text{Mean} = \frac{\sum x_i}{n} = \frac{15 + 20 + 25 + 25 + 27 + 28 + 30 + 34}{8} \] \[ \text{Mean} = \frac{204}{8} = 25.5 \] **2. Median:** The median is the middle value of the ordered data set. Since there are an even number of observations (8), the median is the average of the 4th and 5th data points. \[ \text{Ordered Data}: 15, 20, 25, 25, 27, 28, 30, 34 \] \[ \text{Median} = \frac{25 + 27}{2} = \frac{52}{2} = 26 \] **Summary:** - **Mean:** 25.5 - **Median:** 26 --- ### **B. Identify the Mode and the Range** **1. Mode:** The mode is the value that appears most frequently in the data set. \[ \text{Data Frequency}: 15(1), 20(1), 25(2), 27(1), 28(1), 30(1), 34(1) \] \[ \text{Mode} = 25 \quad (\text{appears twice}) \] **2. Range:** The range is the difference between the maximum and minimum values. \[ \text{Maximum} = 34 \] \[ \text{Minimum} = 15 \] \[ \text{Range} = 34 - 15 = 19 \] **Summary:** - **Mode:** 25 - **Range:** 19 --- ### **C. Compute the 2nd Percentile, Quartiles, 6th Decile, and 20th Percentile** To compute percentiles and deciles, we'll use the following formula to determine the position: \[ \text{Position} = \left( \frac{P}{100} \right) \times (n + 1) \] Where \( P \) is the desired percentile or decile, and \( n = 8 \). 1. **2nd Percentile (P = 2):** \[ \text{Position} = 0.02 \times 9 = 0.18 \] Since the position is less than 1, the 2nd percentile is the first data point. \[ \text{2nd Percentile} = 15 \] 2. **Quartiles:** - **First Quartile (Q1 - 25th Percentile):** \[ \text{Position} = 0.25 \times 9 = 2.25 \] \[ Q1 = x_2 + 0.25 \times (x_3 - x_2) = 20 + 0.25 \times (25 - 20) = 20 + 1.25 = 21.25 \] - **Second Quartile (Q2 - Median - 50th Percentile):** \[ Q2 = 26 \quad (\text{from Part A}) \] - **Third Quartile (Q3 - 75th Percentile):** \[ \text{Position} = 0.75 \times 9 = 6.75 \] \[ Q3 = x_6 + 0.75 \times (x_7 - x_6) = 28 + 0.75 \times (30 - 28) = 28 + 1.5 = 29.5 \] 3. **6th Decile (D6 - 60th Percentile):** \[ \text{Position} = 0.60 \times 9 = 5.4 \] \[ D6 = x_5 + 0.4 \times (x_6 - x_5) = 27 + 0.4 \times (28 - 27) = 27 + 0.4 = 27.4 \] 4. **20th Percentile (P = 20):** \[ \text{Position} = 0.20 \times 9 = 1.8 \] \[ \text{20th Percentile} = x_1 + 0.8 \times (x_2 - x_1) = 15 + 0.8 \times (20 - 15) = 15 + 4 = 19 \] **Summary:** - **2nd Percentile:** 15 - **First Quartile (Q1):** 21.25 - **Second Quartile (Median, Q2):** 26 - **Third Quartile (Q3):** 29.5 - **6th Decile (D6):** 27.4 - **20th Percentile:** 19 --- ### **D. Compute the Variance and Standard Deviation** **1. Variance:** Variance measures the dispersion of the data points from the mean. \[ \text{Mean} = 25.5 \] Calculate each squared deviation: \[ \begin{align*} (15 - 25.5)^2 &= (-10.5)^2 = 110.25 \\ (20 - 25.5)^2 &= (-5.5)^2 = 30.25 \\ (25 - 25.5)^2 &= (-0.5)^2 = 0.25 \\ (25 - 25.5)^2 &= (-0.5)^2 = 0.25 \\ (27 - 25.5)^2 &= (1.5)^2 = 2.25 \\ (28 - 25.5)^2 &= (2.5)^2 = 6.25 \\ (30 - 25.5)^2 &= (4.5)^2 = 20.25 \\ (34 - 25.5)^2 &= (8.5)^2 = 72.25 \\ \end{align*} \] \[ \sum (x_i - \text{Mean})^2 = 110.25 + 30.25 + 0.25 + 0.25 + 2.25 + 6.25 + 20.25 + 72.25 = 242 \] Assuming **Population Variance**: \[ \text{Variance} = \frac{\sum (x_i - \text{Mean})^2}{n} = \frac{242}{8} = 30.25 \] **2. Standard Deviation:** Standard deviation is the square root of the variance. \[ \text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{30.25} = 5.5 \] **Summary:** - **Variance:** 30.25 - **Standard Deviation:** 5.5 --- ## **Final Summary** **A. Mean and Median** - **Mean:** 25.5 - **Median:** 26 **B. Mode and Range** - **Mode:** 25 - **Range:** 19 **C. Percentiles and Deciles** - **2nd Percentile:** 15 - **First Quartile (Q1):** 21.25 - **Second Quartile (Median, Q2):** 26 - **Third Quartile (Q3):** 29.5 - **6th Decile (D6):** 27.4 - **20th Percentile:** 19 **D. Variance and Standard Deviation** - **Variance:** 30.25 - **Standard Deviation:** 5.5