
# Grade 10 Compulsory Mathematics: Statistics Study Guide
## Unit: Statistics | Chapter: Standard Deviation & Coefficient of Variation | Complete Solutions

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## 🎯 Chapter Introduction: Standard Deviation

Welcome to the complete guide on **Standard Deviation**, the most widely used and robust measure of dispersion in statistics. While other measures like Quartile and Mean Deviation provide valuable insights, Standard Deviation is considered the most reliable because it gives more weight to the data points that are farther from the mean. It's the cornerstone for many advanced statistical analyses. 🔬

### Key Learning Areas
* **Defining Standard Deviation ($\sigma$):** Understanding the "root mean square deviation."
* **Coefficient of Standard Deviation:** A relative measure for comparing spread.
* **Coefficient of Variation (C.V.):** A standardized measure of dispersion, expressed as a percentage, used to compare consistency.
* **Calculation Methods:** Step-by-step processes for individual, discrete, and continuous series.
* **Consistency and Stability:** Using C.V. to determine which dataset is more consistent.

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## 🔑 Key Concepts & Definitions

Let's dive into the foundational ideas.

### 1. What is Standard Deviation?
**Standard Deviation ($\sigma$)** is the **positive square root of the mean of the squared deviations from the arithmetic mean**. This might sound complex, but it's a very logical process:

1.  **Find the Mean ($\bar{X}$):** Calculate the average of the data.
2.  **Find Deviations:** Find the difference between each data point and the mean.
3.  **Square the Deviations:** Square each difference. This makes all values positive and gives more weight to larger deviations.
4.  **Find the Average of Squares:** Calculate the mean of these squared deviations. This value is called the **Variance ($\sigma^2$)**.
5.  **Take the Square Root:** The square root of the variance is the **Standard Deviation**.

It is also known as the **root mean square deviation** and is denoted by the Greek letter **$\sigma$** (sigma).

### 2. Coefficient of Standard Deviation
This is a **relative measure** of dispersion. It's the ratio of the Standard Deviation to the arithmetic mean, which allows for a standardized comparison of variability between different datasets.

* **Formula:**
    $$
    \text{Coefficient of SD} = \frac{\sigma}{\bar{X}}
    $$

### 3. Coefficient of Variation (C.V.)
The **Coefficient of Variation (C.V.)** is the most common relative measure of dispersion. It's simply the Coefficient of Standard Deviation expressed as a **percentage**.

* **Formula:**
    $$
    C.V. = \frac{\sigma}{\bar{X}} \times 100\%
    $$

**Why is C.V. important?** It's used to compare the consistency or stability of different datasets.
* A **low C.V.** means the data points are close to the mean, indicating **high consistency** or stability.
* A **high C.V.** means the data points are spread out, indicating **low consistency**.

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## 📚 Exercise Solutions

### Short Questions

***

#### Question 1: Define standard deviation.
**Answer:**
Standard deviation is the positive square root of the mean of the squared deviations of data points from their arithmetic mean. It's often called the root mean square deviation and is denoted by $\sigma$.

***

#### Question 2: What is the coefficient of standard deviation and its formula?
**Answer:**
The coefficient of standard deviation is a relative measure of dispersion, calculated as the ratio of the standard deviation ($\sigma$) to the mean ($\bar{X}$). Its formula is:
$$
\text{Coefficient of SD} = \frac{\sigma}{\bar{X}}
$$

***

#### Question 3: Define the coefficient of variation and write its formula.
**Answer:**
The coefficient of variation (C.V.) is the coefficient of standard deviation expressed as a percentage. It is a unit-less measure used for comparing the consistency of data.
$$
C.V. = \frac{\sigma}{\bar{X}} \times 100\%
$$

***

#### Question 4: What does a low coefficient of variation mean?
**Answer:**
A low coefficient of variation indicates that the data points are clustered closely around the mean. This signifies that the data is more **consistent**, uniform, stable, or homogenous.

***

#### Question 5: What is the difference between standard deviation and mean deviation?
**Answer:**
**Standard deviation** is the square root of the average of *squared* deviations from the mean. **Mean deviation** is the average of the *absolute* deviations from the mean or median. Standard deviation gives greater weight to extreme values because it squares the deviations.

***

#### Question 6: What is the difference between the coefficient of standard deviation and the coefficient of variation?
**Answer:**
The coefficient of standard deviation is the ratio $\frac{\sigma}{\bar{X}}$. The coefficient of variation is simply this same ratio multiplied by 100 to express it as a percentage ($C.V. = \frac{\sigma}{\bar{X}} \times 100\%$).

***

#### Question 7: If $\sum f(m - \bar{X})^2 = 484$, $N = 24$, and $\bar{X} = 25$, find the standard deviation and its coefficient.
**Solution:**
1.  **Standard Deviation ($\sigma$):**
    $$
    \sigma = \sqrt{\frac{\sum f(m - \bar{X})^2}{N}} = \sqrt{\frac{484}{24}} \approx \sqrt{20.167} \approx 4.49
    $$
2.  **Coefficient of SD:**
    $$
    \text{Coefficient} = \frac{\sigma}{\bar{X}} = \frac{4.49}{25} \approx 0.18
    $$

***

#### Question 8: If $\sum fd^2 = 848$, $\sum fd = 0$, $N=100$, and Assumed Mean A=12, find the standard deviation.
**Solution:**
Since $\sum fd = 0$, the assumed mean (A) is equal to the actual mean ($\bar{X}$).
$$
\sigma = \sqrt{\frac{\sum fd^2}{N} - \left(\frac{\sum fd}{N}\right)^2} = \sqrt{\frac{848}{100} - \left(\frac{0}{100}\right)^2} = \sqrt{8.48} \approx 2.91
$$

***

#### Question 9: If $\sum fd'^2 = 125$, $\sum fd' = -4$, $N=20$, and class size $C=10$, find the standard deviation.
**Solution:**
Using the step-deviation formula:
$$
\sigma = \sqrt{\frac{\sum fd'^2}{N} - \left(\frac{\sum fd'}{N}\right)^2} \times C = \sqrt{\frac{125}{20} - \left(\frac{-4}{20}\right)^2} \times 10
$$
$$
\sigma = \sqrt{6.25 - (-0.2)^2} \times 10 = \sqrt{6.25 - 0.04} \times 10 = \sqrt{6.21} \times 10 \approx 2.492 \times 10 \approx 24.92
$$

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### Long Questions (Worked Examples)

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#### Question 10: Find the standard deviation and its coefficient for the data: 5, 10, 15, 20, 25.
**Solution:**
1.  **Find Mean:** $N=5$. $\sum X = 5+10+15+20+25 = 75$.
    $\bar{X} = \frac{75}{5} = 15$.
2.  **Calculations:**
| X | $D = X - \bar{X}$ | $D^2$ |
|:---:|:---:|:---:|
| 5 | -10 | 100 |
| 10 | -5 | 25 |
| 15 | 0 | 0 |
| 20 | 5 | 25 |
| 25 | 10 | 100 |
| **Total** | | **$\sum D^2 = 250$** |

* **Standard Deviation ($\sigma$)** = $\sqrt{\frac{\sum D^2}{N}} = \sqrt{\frac{250}{5}} = \sqrt{50} \approx \mathbf{7.07}$
* **Coefficient of SD** = $\frac{\sigma}{\bar{X}} = \frac{7.07}{15} \approx \mathbf{0.47}$

***

#### Question 11: Find the standard deviation and C.V. for the data: 30, 33, 37, 40, 44, 50.
**Solution:**
1.  **Find Mean:** $N=6$. $\sum X = 30+33+37+40+44+50 = 234$.
    $\bar{X} = \frac{234}{6} = 39$.
2.  **Calculations:**

| X | $D = X - 39$ | $D^2$ |
|:---:|:---:|:---:|
| 30 | -9 | 81 |
| 33 | -6 | 36 |
| 37 | -2 | 4 |
| 40 | 1 | 1 |
| 44 | 5 | 25 |
| 50 | 11 | 121 |
| **Total**| | **$\sum D^2 = 268$** |

* **Standard Deviation ($\sigma$)** = $\sqrt{\frac{268}{6}} = \sqrt{44.67} \approx \mathbf{6.68}$
* **Coefficient of Variation (C.V.)** = $\frac{6.68}{39} \times 100\% \approx \mathbf{17.13\%}$

***

#### Question 12: Find the standard deviation from the given data.

| x | f |
|:---:|:---:|
| 6 | 1 |
| 8 | 14 |
| 10 | 25 |
| 12 | 27 |
| 14 | 18 |
| 16 | 9 |
| 18 | 4 |
| 20 | 2 |

**Solution:**
1.  **Find Mean ($\bar{X}$):** N=100. $\sum fx = (6 \times 1) + (8 \times 14) + ... + (20 \times 2) = 1200$.
    $\bar{X} = \frac{1200}{100} = 12$.
2.  **Calculations for SD:**

| x | f | $D = x-12$ | $D^2$ | $fD^2$ |
|:---:|:---:|:---:|:---:|:---:|
| 6 | 1 | -6 | 36 | 36 |
| 8 | 14 | -4 | 16 | 224 |
| 10 | 25 | -2 | 4 | 100 |
| 12 | 27 | 0 | 0 | 0 |
| 14 | 18 | 2 | 4 | 72 |
| 16 | 9 | 4 | 16 | 144 |
| 18 | 4 | 6 | 36 | 144 |
| 20 | 2 | 8 | 64 | 128 |
| **Total** | **N=100** | | | **$\sum fD^2=848$** |

* **Standard Deviation ($\sigma$)** = $\sqrt{\frac{\sum fD^2}{N}} = \sqrt{\frac{848}{100}} = \sqrt{8.48} \approx \mathbf{2.91}$

***

#### Question 13: Calculate the coefficient of variation from the data given below.

| Class Interval | Frequency |
|:---:|:---:|
| 0-10 | 5 |
| 10-20 | 8 |
| 20-30 | 15 |
| 30-40 | 16 |
| 40-50 | 6 |

**Solution:**
1.  **Find Mean ($\bar{X}$):** N=50. $\sum fm = 1350$.
    $\bar{X} = \frac{1350}{50} = 27$.
2.  **Find Standard Deviation ($\sigma$):**

| m | f | $D = m - 27$ | $D^2$ | $fD^2$ |
|:---:|:---:|:---:|:---:|:---:|
| 5 | 5 | -22 | 484 | 2420 |
| 15 | 8 | -12 | 144 | 1152 |
| 25 | 15 | -2 | 4 | 60 |
| 35 | 16 | 8 | 64 | 1024 |
| 45 | 6 | 18 | 324 | 1944 |
| **Total** | **N=50** | | | **$\sum fD^2=6600$** |
    $\sigma = \sqrt{\frac{\sum fD^2}{N}} = \sqrt{\frac{6600}{50}} = \sqrt{132} \approx 11.49$.
4.  **Calculate C.V.:**
    **C.V.** = $\frac{\sigma}{\bar{X}} \times 100\% = \frac{11.49}{27} \times 100\% \approx \mathbf{42.56\%}$

***

#### Question 14: Calculate the coefficient of variation from the data given below.

| Class Interval | Frequency |
|:---:|:---:|
| 0-20 | 2 |
| 20-40 | 3 |
| 40-60 | 4 |
| 60-80 | 5 |
| 80-100 | 6 |

**Solution:**
1.  **Find Mean ($\bar{X}$):** N=20.
2. 
| C.I. | m | f | fm |
|:---:|:---:|:---:|:---:|
| 0-20 | 10 | 2 | 20 |
| 20-40 | 30 | 3 | 90 |
| 40-60 | 50 | 4 | 200 |
| 60-80 | 70 | 5 | 350 |
| 80-100 | 90 | 6 | 540 |
| **Total** | | **N=20** | **$\sum fm = 1200$** |

    $\bar{X} = \frac{1200}{20} = 60$.
    
3.  **Find Standard Deviation ($\sigma$):**
| m | f | $D=m-60$ | $D^2$ | $fD^2$ |
|:---:|:---:|:---:|:---:|:---:|
| 10 | 2 | -50 | 2500 | 5000 |
| 30 | 3 | -30 | 900 | 2700 |
| 50 | 4 | -10 | 100 | 400 |
| 70 | 5 | 10 | 100 | 500 |
| 90 | 6 | 30 | 900 | 5400 |
| **Total** | **N=20** | | | **$\sum fD^2 = 14000$** |
    $\sigma = \sqrt{\frac{14000}{20}} = \sqrt{700} \approx 26.46$.
4.  **Calculate C.V.:**
    **C.V.** = $\frac{26.46}{60} \times 100\% \approx \mathbf{44.1\%}$

***

#### Question 15: Find the standard deviation of the data given below.

| Class Interval | Frequency |
|:---:|:---:|
| 5-15 | 7 |
| 15-25 | 3 |
| 25-35 | 6 |
| 35-45 | 4 |
| 45-55 | 5 |

**Solution:**

1.  **Find Mean ($\bar{X}$):** N=25. $\sum fm = (10\times7) + (20\times3) + (30\times6) + (40\times4) + (50\times5) = 720$.
    $\bar{X} = \frac{720}{25} = 28.8$.
2.  **Calculations for SD:**

| m | f | $D = m-28.8$ | $D^2$ | $fD^2$ |
|:---:|:---:|:---:|:---:|:---:|
| 10 | 7 | -18.8 | 353.44 | 2474.08 |
| 20 | 3 | -8.8 | 77.44 | 232.32 |
| 30 | 6 | 1.2 | 1.44 | 8.64 |
| 40 | 4 | 11.2 | 125.44 | 501.76 |
| 50 | 5 | 21.2 | 449.44 | 2247.2 |
| **Total** | **N=25** | | | **$\sum fD^2 = 5464$** |
* **Standard Deviation ($\sigma$)** = $\sqrt{\frac{5464}{25}} = \sqrt{218.56} \approx \mathbf{14.78}$

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This guide now contains a comprehensive set of examples. Practice these thoroughly to master the concepts of Standard Deviation and Coefficient of Variation. Good luck! 🌟
