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 20 th percentiles D. Compute the variance and standard deviation for the data set

Let's analyze the given data set step by step as per the requirements: **Given Data Set:** \[ 27, \ 25, \ 20, \ 15, \ 30, \ 34, \ 28, \ 25 \] Let's first arrange the data in ascending order for easier computation: \[ 15, \ 20, \ 25, \ 25, \ 27, \ 28, \ 30, \ 34 \] There are **8 data points** in total. --- ### **A. Calculating the Mean and the Median** #### **Mean** The **mean** is the average of all data points. \[ \text{Mean} = \frac{\sum \text{Data Points}}{\text{Number of Data Points}} = \frac{15 + 20 + 25 + 25 + 27 + 28 + 30 + 34}{8} \] \[ \text{Mean} = \frac{204}{8} = 25.5 \] #### **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. \[ \text{Median} = \frac{\text{4th term} + \text{5th term}}{2} = \frac{25 + 27}{2} = 26 \] --- ### **B. Identifying the Mode and the Range** #### **Mode** The **mode** is the value(s) that appear most frequently in the data set. From the ordered data: \[ 15, \ 20, \ 25, \ 25, \ 27, \ 28, \ 30, \ 34 \] - The number **25** appears **twice**, which is more frequent than any other number. \[ \text{Mode} = 25 \] #### **Range** The **range** is the difference between the maximum and minimum values. \[ \text{Range} = \text{Maximum} - \text{Minimum} = 34 - 15 = 19 \] --- ### **C. Computing the 2nd Decile, Quartiles, 6th Decile, and 20th Percentile** To compute percentiles and deciles, we'll use the **Position Formula**: \[ P = \frac{n + 1}{100} \times \text{Percentile} \] Where \( n \) is the number of data points. For our data set, \( n = 8 \). #### **1. 2nd Decile (20th Percentile)** \[ P_{20} = \frac{8 + 1}{100} \times 20 = \frac{9}{100} \times 20 = 1.8 \] - **Position 1.8**: This is between the 1st and 2nd data points. \[ \text{Value} = \text{1st term} + 0.8 \times (\text{2nd term} - \text{1st term}) = 15 + 0.8 \times (20 - 15) = 15 + 4 = 19 \] \[ P_{20} = 19 \] #### **2. Quartiles** - **First Quartile (Q1 - 25th Percentile):** \[ P_{25} = \frac{9}{100} \times 25 = 2.25 \] - **Position 2.25**: Between the 2nd and 3rd data points. \[ Q_1 = 20 + 0.25 \times (25 - 20) = 20 + 1.25 = 21.25 \] - **Second Quartile (Q2 - 50th Percentile, Median):** \[ Q_2 = \text{Median} = 26 \quad \text{(As calculated earlier)} \] - **Third Quartile (Q3 - 75th Percentile):** \[ P_{75} = \frac{9}{100} \times 75 = 6.75 \] - **Position 6.75**: Between the 6th and 7th data points. \[ Q_3 = 28 + 0.75 \times (30 - 28) = 28 + 1.5 = 29.5 \] #### **3. 6th Decile (60th Percentile)** \[ P_{60} = \frac{9}{100} \times 60 = 5.4 \] - **Position 5.4**: Between the 5th and 6th data points. \[ \text{Value} = 27 + 0.4 \times (28 - 27) = 27 + 0.4 = 27.4 \] \[ P_{60} = 27.4 \] #### **4. 20th Percentile** This is the same as the 2nd decile calculated above. \[ P_{20} = 19 \] **Summary of Percentiles and Deciles:** - **2nd Decile (20th Percentile):** 19 - **Quartiles:** - \( Q_1 = 21.25 \) - \( Q_2 = 26 \) (Median) - \( Q_3 = 29.5 \) - **6th Decile (60th Percentile):** 27.4 - **20th Percentile:** 19 --- ### **D. Computing the Variance and Standard Deviation** #### **Variance** The **variance** measures the dispersion of the data points from the mean. \[ \text{Variance} (s^2) = \frac{\sum (x_i - \bar{x})^2}{n - 1} \] Where: - \( x_i \) = each data point - \( \bar{x} \) = mean - \( n \) = number of data points Given: - Mean (\( \bar{x} \)) = 25.5 - \( n = 8 \) Let's compute each \( (x_i - \bar{x})^2 \): \[ \begin{align*} (15 - 25.5)^2 &= (-10.5)^2 = 110.25 \\ (20 - 25.5)^2 &= (-5.5)^2 = 30.25 \\ (25 - 25.5)^2 &= (-0.5)^2 = 0.25 \\ (25 - 25.5)^2 &= (-0.5)^2 = 0.25 \\ (27 - 25.5)^2 &= (1.5)^2 = 2.25 \\ (28 - 25.5)^2 &= (2.5)^2 = 6.25 \\ (30 - 25.5)^2 &= (4.5)^2 = 20.25 \\ (34 - 25.5)^2 &= (8.5)^2 = 72.25 \\ \end{align*} \] \[ \sum (x_i - \bar{x})^2 = 110.25 + 30.25 + 0.25 + 0.25 + 2.25 + 6.25 + 20.25 + 72.25 = 242 \] \[ \text{Variance} = \frac{242}{8 - 1} = \frac{242}{7} \approx 34.57 \] #### **Standard Deviation** The **standard deviation** is the square root of the variance. \[ \text{Standard Deviation} (s) = \sqrt{\text{Variance}} = \sqrt{34.57} \approx 5.88 \] --- ### **Summary** - **Mean:** 25.5 - **Median:** 26 - **Mode:** 25 - **Range:** 19 - **Percentiles and Deciles:** - **2nd Decile (20th Percentile):** 19 - **Quartiles:** - \( Q_1 = 21.25 \) - \( Q_2 = 26 \) (Median) - \( Q_3 = 29.5 \) - **6th Decile (60th Percentile):** 27.4 - **20th Percentile:** 19 - **Variance:** ≈ 34.57 - **Standard Deviation:** ≈ 5.88