
# Grade 10 Compulsory Mathematics: Statistics Study Guide
## Unit: Statistics | Chapter: Measures of Dispersion | Complete Solutions

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## 🎯 Chapter Introduction: Understanding Dispersion

This guide covers the fundamental concepts of **Measures of Dispersion** in statistics. Dispersion helps us understand how spread out or scattered a set of data is. Think of it as measuring the "variety" or "variability" in your data. A low dispersion means the data points are clustered together, while high dispersion indicates they are spread far apart. 📊

### Key Learning Areas
* **Defining Dispersion:** What it means for data to be scattered.
* **Absolute vs. Relative Measures:** Understanding the two main types of dispersion measures.
* **Quartile Deviation (QD):** A measure of dispersion based on the middle 50% of the data.
* **Coefficient of Quartile Deviation:** A relative measure for comparing dispersion between different datasets.
* **Calculations for Continuous Data:** Step-by-step methods to calculate QD and its coefficient for frequency distributions.

---

## 🔑 Key Concepts & Definitions

Before jumping into calculations, let's clarify the core ideas.

### 1. What is Dispersion?
**Dispersion** simply means the extent to which numerical data is likely to vary or scatter around an average value. In other words, it's the measure of the **variability** or **fluctuation** within a dataset.

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### 2. Types of Dispersion Measures
There are two main categories for measuring dispersion:

* **Absolute Measure of Dispersion:** These measures express the variability in the **same units** as the original data (e.g., kg, marks, inches). They are used for describing the amount of spread in a single dataset.
    * Examples: Range, **Quartile Deviation**, Mean Deviation, Standard Deviation.

* **Relative Measure of Dispersion:** These are unit-less ratios used for **comparing the variability** of two or more datasets, especially when their units or averages are different.
    * Examples: Coefficient of Range, **Coefficient of Quartile Deviation**, Coefficient of Variation.

---

### 3. Quartile Deviation (QD)
**Quartile Deviation (QD)**, also known as the **Semi-Interquartile Range**, measures the average spread of the middle 50% of the data around the median. It is calculated as half the difference between the upper quartile ($Q_3$) and the lower quartile ($Q_1$).

* **Formula:**
    $$
    \text{Quartile Deviation (QD)} = \frac{Q_3 - Q_1}{2}
    $$

---

### 4. Coefficient of Quartile Deviation
The **Coefficient of Quartile Deviation** is a relative measure. It expresses the quartile deviation as a percentage of the average of the upper and lower quartiles, making it perfect for comparing the consistency of different datasets.

* **Formula:**
    $$
    \text{Coefficient of QD} = \frac{Q_3 - Q_1}{Q_3 + Q_1}
    $$

---

## 📚 Exercise Solutions

### Short Questions

***

#### Question 1: What is dispersion?
**Answer:** Dispersion refers to the scattering, variability, deviation, or fluctuation of data points around a central value.

***

#### Question 2: Write the various measures of dispersion.
**Answer:**
The various measures of dispersion are:
1.  **Absolute Measures:** Range, Quartile Deviation, Mean Deviation, and Standard Deviation.
2.  **Relative Measures:** Coefficient of Range, Coefficient of Quartile Deviation, Coefficient of Mean Deviation, Coefficient of Variation.

***

#### Question 3: Define quartile deviation and write its formula.
**Answer:**
Half of the difference between the upper quartile ($Q_3$) and the lower quartile ($Q_1$) is called **quartile deviation (QD)** or the semi-interquartile range.
Its formula is:
$$
\text{Quartile Deviation (QD)} = \frac{Q_3 - Q_1}{2}
$$

***

#### Question 4: What do you mean by the coefficient of quartile deviation?
**Answer:**
The coefficient of quartile deviation is the relative measure of dispersion based on the lower and upper quartiles. It is used for comparison purposes.
Its formula is:
$$
\text{Coefficient of Quartile Deviation} = \frac{Q_3 - Q_1}{Q_3 + Q_1}
$$

***

#### Question 5: Differentiate between quartile deviation and its coefficient.
**Answer:**
Quartile deviation is an **absolute measure** of dispersion, calculated as $\frac{Q_3 - Q_1}{2}$, and is expressed in the same units as the data. In contrast, the coefficient of quartile deviation is a **relative measure**, calculated as $\frac{Q_3 - Q_1}{Q_3 + Q_1}$, which is a unit-less ratio used to compare the variability of different datasets.

***

#### Question 6: In continuous data, if $Q_1 = 40$ and $Q_3 = 60$, find the quartile deviation.
**Solution:**

| Calculation Step | Formula & Result |
|:---|:---|
| **Find Quartile Deviation (QD)** | $QD = \frac{Q_3 - Q_1}{2} = \frac{60 - 40}{2} = \mathbf{10}$ |

***

#### Question 7: In a continuous series, the lower quartile is 25 and its quartile deviation is 10. Find the upper quartile.
**Solution:**

| Calculation Step | Formula & Result |
|:---|:---|
| **Find Upper Quartile ($Q_3$)** | $10 = \frac{Q_3 - 25}{2} \implies 20 = Q_3 - 25 \implies Q_3 = \mathbf{45}$ |

***

#### Question 8: In a continuous series, the coefficient of quartile deviation is $\frac{1}{2}$ and its upper quartile is 60. Find the first quartile.
**Solution:**

| Calculation Step | Formula & Result |
|:---|:---|
| **Find First Quartile ($Q_1$)** | $\frac{1}{2} = \frac{60 - Q_1}{60 + Q_1} \implies 60 + Q_1 = 120 - 2Q_1 \implies 3Q_1 = 60 \implies Q_1 = \mathbf{20}$ |

***

#### Question 9: If the coefficient of quartile deviation is $\frac{3}{7}$ and $Q_1 = 20$, find $Q_3$.
**Solution:**

| Calculation Step | Formula & Result |
|:---|:---|
| **Find Upper Quartile ($Q_3$)** | $\frac{3}{7} = \frac{Q_3 - 20}{Q_3 + 20} \implies 3Q_3 + 60 = 7Q_3 - 140 \implies 200 = 4Q_3 \implies Q_3 = \mathbf{50}$ |

---

### Long Questions

***

#### Question 10: Find the quartile deviation and its coefficient from the following data.

| Marks obtained | No. of students |
|:---:|:---:|
| 60-65 | 7 |
| 65-70 | 5 |
| 70-75 | 8 |
| 75-80 | 4 |
| 80-85 | 3 |
| 85-90 | 3 |

**Solution:**
1.  **Cumulative Frequency Table:**

| Marks obtained | Frequency (f) | Cumulative Frequency (c.f.) |
|:---:|:---:|:---:|
| 60-65 | 7 | 7 |
| 65-70 | 5 | 12 |
| 70-75 | 8 | 20 |
| 75-80 | 4 | 24 |
| 80-85 | 3 | 27 |
| 85-90 | 3 | 30 |
| **Total** | **N = 30** | |

2.  **Find Quartiles:**
    * **For $Q_1$**: Position is $(\frac{30}{4})^{th} = 7.5^{th}$ item. $Q_1$ class is **65-70**.
        $Q_1 = 65 + \frac{7.5 - 7}{5} \times 5 = \mathbf{65.5}$
    * **For $Q_3$**: Position is $(\frac{3 \times 30}{4})^{th} = 22.5^{th}$ item. $Q_3$ class is **75-80**.
        $Q_3 = 75 + \frac{22.5 - 20}{4} \times 5 = \mathbf{78.125}$

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{78.125 - 65.5}{2} = \mathbf{6.3125}$ |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{78.125 - 65.5}{78.125 + 65.5} = \frac{12.625}{143.625} \approx \mathbf{0.088}$ |

***

#### Question 11: Find the quartile deviation and its coefficient from the following data.

| Weight (kg) | No. of person |
|:---:|:---:|
| 20-30 | 5 |
| 30-40 | 15 |
| 40-50 | 10 |
| 50-60 | 8 |
| 60-70 | 6 |
| 70-80 | 2 |

**Solution:**
1.  **Cumulative Frequency Table:**

| Weight (kg) | Frequency (f) | c.f. |
|:---:|:---:|:---:|
| 20-30 | 5 | 5 |
| 30-40 | 15 | 20 |
| 40-50 | 10 | 30 |
| 50-60 | 8 | 38 |
| 60-70 | 6 | 44 |
| 70-80 | 2 | 46 |
| **Total** | **N = 46** | |

2.  **Find Quartiles:**
    * **For $Q_1$**: Position is $(\frac{46}{4})^{th} = 11.5^{th}$ item. $Q_1$ class is **30-40**.
        $Q_1 = 30 + \frac{11.5 - 5}{15} \times 10 \approx \mathbf{34.33}$
    * **For $Q_3$**: Position is $(\frac{3 \times 46}{4})^{th} = 34.5^{th}$ item. $Q_3$ class is **50-60**.
        $Q_3 = 50 + \frac{34.5 - 30}{8} \times 10 = \mathbf{55.625}$

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{55.625 - 34.33}{2} \approx \mathbf{10.65}$ kg |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{55.625 - 34.33}{55.625 + 34.33} \approx \mathbf{0.237}$ |

***

#### Question 12: Find the quartile deviation and its coefficient from the following data.

| Class interval | Frequency |
|:---:|:---:|
| 20-30 | 8 |
| 30-40 | 16 |
| 40-50 | 4 |
| 50-60 | 4 |
| 60-70 | 3 |

**Solution:**
1.  **Cumulative Frequency Table:**

| Class interval | Frequency (f) | c.f. |
|:---:|:---:|:---:|
| 20-30 | 8 | 8 |
| 30-40 | 16 | 24 |
| 40-50 | 4 | 28 |
| 50-60 | 4 | 32 |
| 60-70 | 3 | 35 |
| **Total** | **N = 35** | |

2.  **Find Quartiles:**
    * **For $Q_1$**: Position is $(\frac{35}{4})^{th} = 8.75^{th}$ item. $Q_1$ class is **30-40**.
        $Q_1 = 30 + \frac{8.75 - 8}{16} \times 10 \approx \mathbf{30.47}$
    * **For $Q_3$**: Position is $(\frac{3 \times 35}{4})^{th} = 26.25^{th}$ item. $Q_3$ class is **40-50**.
        $Q_3 = 40 + \frac{26.25 - 24}{4} \times 10 = \mathbf{45.625}$

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{45.625 - 30.47}{2} \approx \mathbf{7.58}$ |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{45.625 - 30.47}{45.625 + 30.47} \approx \mathbf{0.20}$ |

***

#### Question 13: Find the quartile deviation and its coefficient from the following data.

| Size | Frequency |
|:---:|:---:|
| 4-8 | 6 |
| 8-12 | 10 |
| 12-16 | 18 |
| 16-20 | 30 |
| 20-24 | 15 |
| 24-28 | 12 |
| 28-32 | 10 |
| 32-36 | 6 |
| 36-40 | 5 |

**Solution:**
1.  **Find N and Quartile Positions:** $N = 112$.
    * $Q_1$ position is $28^{th}$ item.
    * $Q_3$ position is $84^{th}$ item.

2.  **Find Quartiles from c.f. table:**
    * **For $Q_1$**: c.f. is (6, 16, **34**,...). $Q_1$ class is **12-16**.
        $Q_1 = 12 + \frac{28 - 16}{18} \times 4 \approx \mathbf{14.67}$
    * **For $Q_3$**: c.f. is (..., 64, 79, **91**,...). $Q_3$ class is **24-28**.
        $Q_3 = 24 + \frac{84 - 79}{12} \times 4 \approx \mathbf{25.67}$

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{25.67 - 14.67}{2} = \mathbf{5.5}$ |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{25.67 - 14.67}{25.67 + 14.67} = \frac{11}{40.34} \approx \mathbf{0.273}$ |

***

#### Question 14: Find the quartile deviation and its coefficient from the following data.

| Height (inches) | No. of students |
|:---:|:---:|
| 60-62 | 4 |
| 62-64 | 6 |
| 64-66 | 8 |
| 66-68 | 12 |
| 68-70 | 7 |
| 70-72 | 2 |

**Solution:**
1.  **Find N and Quartile Positions:** $N = 39$.
    * $Q_1$ position is $9.75^{th}$ item.
    * $Q_3$ position is $29.25^{th}$ item.

2.  **Find Quartiles from c.f. table:**
    * **For $Q_1$**: c.f. is (4, **10**,...). $Q_1$ class is **62-64**.
        $Q_1 = 62 + \frac{9.75 - 4}{6} \times 2 \approx \mathbf{63.92}$
    * **For $Q_3$**: c.f. is (..., 18, **30**,...). $Q_3$ class is **66-68**.
        $Q_3 = 66 + \frac{29.25 - 18}{12} \times 2 \approx \mathbf{67.88}$

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{67.88 - 63.92}{2} = \mathbf{1.98}$ inches |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{67.88 - 63.92}{67.88 + 63.92} = \frac{3.96}{131.8} \approx \mathbf{0.03}$ |

***

#### Question 15: Find the quartile deviation and its coefficient from the following data.

| Expenditure | No. of workers |
|:---:|:---:|
| 0-10 | 5 |
| 10-20 | 15 |
| 20-30 | 10 |
| 30-40 | 8 |
| 40-50 | 6 |

**Solution:**
1.  **Find N and Quartile Positions:** $N = 44$.
    * $Q_1$ position is $11^{th}$ item.
    * $Q_3$ position is $33^{rd}$ item.

2.  **Find Quartiles from c.f. table:**
    * **For $Q_1$**: c.f. is (5, **20**,...). $Q_1$ class is **10-20**.
        $Q_1 = 10 + \frac{11 - 5}{15} \times 10 = \mathbf{14}$
    * **For $Q_3$**: c.f. is (..., 30, **38**,...). $Q_3$ class is **30-40**.
        $Q_3 = 30 + \frac{33 - 30}{8} \times 10 = \mathbf{33.75}$

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{33.75 - 14}{2} = \mathbf{9.875}$ |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{33.75 - 14}{33.75 + 14} = \frac{19.75}{47.75} \approx \mathbf{0.414}$ |

***

#### Question 16: Prepare a frequency distribution table (10-20, etc.) and find the QD and its coefficient.
**Data:** 21, 45, 60, 57, 15, 41, 48, 50, 34, 29, 56, 40, 14, 62, 28, 70, 22, 30, 38, 74, 13, 47, 20, 53, 64, 34, 75, 66.

**Solution:**
1.  **Frequency Distribution Table:** ($N=28$)

| Marks | Freq (f) | c.f. |
|:---:|:---:|:---:|
| 10-20 | 3 | 3 |
| 20-30 | 5 | 8 |
| 30-40 | 4 | 12 |
| 40-50 | 5 | 17 |
| 50-60 | 4 | 21 |
| 60-70 | 4 | 25 |
| 70-80 | 3 | 28 |

2.  **Find Quartiles:**
    * **For $Q_1$**: Position is $(\frac{28}{4})^{th} = 7^{th}$ item. c.f. is (3, **8**,...). $Q_1$ class is **20-30**.
        $Q_1 = 20 + \frac{7 - 3}{5} \times 10 = \mathbf{28}$
    * **For $Q_3$**: Position is $(\frac{3 \times 28}{4})^{th} = 21^{st}$ item. c.f. is exactly 21 at the end of the 50-60 class, so $Q_3$ is the upper boundary. $Q_3 = \mathbf{60}$.

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{60 - 28}{2} = \mathbf{16}$ |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{60 - 28}{60 + 28} = \frac{32}{88} \approx \mathbf{0.364}$ |

***

#### Question 17: The following are the marks obtained by class 9 students. Prepare a frequency distribution table (10-20, etc.) and find the QD and its coefficient.
**Data:** 22, 25, 46, 34, 57, 69, 44, 36, 12, 27, 50, 36, 35, 62, 46, 52, 54, 61, 66, 55, 29, 39, 40, 33, 14, 41, 25, 20, 16.

**Solution:**
1.  **Frequency Distribution Table:** ($N=29$)

| Marks | Freq (f) | c.f. |
|:---:|:---:|:---:|
| 10-20 | 3 | 3 |
| 20-30 | 6 | 9 |
| 30-40 | 6 | 15 |
| 40-50 | 5 | 20 |
| 50-60 | 5 | 25 |
| 60-70 | 4 | 29 |

2.  **Find Quartiles:**
    * **For $Q_1$**: Position is $(\frac{29}{4})^{th} = 7.25^{th}$ item. c.f. is (3, **9**,...). $Q_1$ class is **20-30**.
        $Q_1 = 20 + \frac{7.25 - 3}{6} \times 10 \approx \mathbf{27.08}$
    * **For $Q_3$**: Position is $(\frac{3 \times 29}{4})^{th} = 21.75^{th}$ item. c.f. is (..., 20, **25**,...). $Q_3$ class is **50-60**.
        $Q_3 = 50 + \frac{21.75 - 20}{5} \times 10 = \mathbf{53.5}$

3.  **Final Calculation:**

| Calculation Step | Formula & Result |
|:---|:---|
| **1. Quartile Deviation (QD)** | $QD = \frac{53.5 - 27.08}{2} = \mathbf{13.21}$ |
| **2. Coefficient of QD** | $\text{Coefficient} = \frac{53.5 - 27.08}{53.5 + 27.08} = \frac{26.42}{80.58} \approx \mathbf{0.328}$ |