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

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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
- **Mean:** 25.5 - **Median:** 26 - **Mode:** 25 - **Range:** 19 - **20th Percentile:** 19 - **Quartiles:** - 1st Quartile (25th Percentile): 21.25 - 2nd Quartile (50th Percentile, Median): 26 - 3rd Quartile (75th Percentile): 29.5 - **60th Percentile:** 27.4 - **Variance:** ≈ 34.57 - **Standard Deviation:** ≈ 5.88

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Extra Insights

A. Mean and Median

B. Mode and Range

C. Quartiles, Deciles, and Percentiles

D. Variance and Standard Deviation

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