**The following numbers represents number of  students registered in English course.**  

               **8 , 9 , 11 , 12 , 7 , 16 , 23 ,13 , 18 , 14 , 5 , 8**

 **Calculate the following ( given the standard deviation S = 5)**

 **1- Mean**

 **2- Median** 

 **3-Mode** 

 **4- Range**

 **5- coefficient of variation (c.v)** 

 **6- Z18**

Let's go through each of the calculations step by step.

**Given data**:  
\( 8, 9, 11, 12, 7, 16, 23, 13, 18, 14, 5, 8 \)  
Standard deviation \( S = 5 \).

### 1. **Mean**  
The mean \( \bar{x} \) is calculated as the sum of all values divided by the number of values:
\[
\bar{x} = \frac{8 + 9 + 11 + 12 + 7 + 16 + 23 + 13 + 18 + 14 + 5 + 8}{12}
\]

\[
\bar{x} = \frac{144}{12} = 12
\]

**Mean** = 12

---

### 2. **Median**  
To find the median, we first need to arrange the numbers in ascending order:
\[
5, 7, 8, 8, 9, 11, 12, 13, 14, 16, 18, 23
\]
Since there are 12 values (even number), the median is the average of the 6th and 7th values:
\[
\text{Median} = \frac{11 + 12}{2} = \frac{23}{2} = 11.5
\]

**Median** = 11.5

---

### 3. **Mode**  
The mode is the value that appears most frequently in the data. In this case, the number 8 appears twice.

**Mode** = 8

---

### 4. **Range**  
The range is the difference between the maximum and minimum values:
\[
\text{Range} = 23 - 5 = 18
\]

**Range** = 18

---

### 5. **Coefficient of Variation (C.V.)**  
The coefficient of variation is calculated as:
\[
\text{C.V.} = \left( \frac{\text{Standard Deviation}}{\text{Mean}} \right) \times 100
\]
Given that the standard deviation \( S = 5 \) and the mean \( \bar{x} = 12 \):
\[
\text{C.V.} = \left( \frac{5}{12} \right) \times 100 = 41.67\%
\]

**Coefficient of Variation (C.V.)** = 41.67%

---

### 6. **Z18 (Z-score for the value 18)**  
The Z-score for a given value \( x \) is calculated as:
\[
Z_x = \frac{x - \bar{x}}{S}
\]
For \( x = 18 \), \( \bar{x} = 12 \), and \( S = 5 \):
\[
Z_{18} = \frac{18 - 12}{5} = \frac{6}{5} = 1.2
\]

**Z18** = 1.2

---

Would you like further details on any of these calculations?

### Related Questions:
1. How do you calculate variance from the given data?
2. What is the significance of the Z-score in statistics?
3. How do you determine if data is normally distributed using the mean and standard deviation?
4. What is the difference between population standard deviation and sample standard deviation?
5. How does the coefficient of variation help in comparing data sets?

**Tip**: Always verify if the given standard deviation is sample or population-based before performing further calculations.

The following numbers represent the number of students registered in the English course: 8, 9, 11, 12, 7, 16, 23, 13, 18, 14, 5, 8. Calculate the following (given the standard deviation S = 5): 1- Mean, 2- Median, 3- Mode, 4- Range, 5- Coefficient of Variation (C.V.), 6- Z18

Learn how to calculate the mean, median, mode, range, coefficient of variation (C.V.), and Z-score using a real-world dataset of students registered for an English course. Understand key concepts in descriptive statistics, including the Z-score and coefficient of variation. Suitable for Grades 10-12.

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