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**Variance and Standard Deviation for Grouped Data**

- Explanation of variance for ungrouped and grouped data.
- Calculation methods and example.

**Coefficient of Variation \((CV)\)**

- Definition and calculation.
- Example demonstrating its application.

**Simplified Formula for Variance**

- A more convenient method for calculating variance.

**Mean Deviation**

- Definition for both ungrouped and grouped data.
- Example calculation.

**Measures of Skewness**

- Types of skewness (symmetric, positively skewed, negatively skewed).
- Coefficient of skewness calculation and example.
- Pearsonâ€™s skewness coefficient.

### Variance and Standard Deviation for Grouped Data

**Variance for Ungrouped Data:**

For ungrouped data, the variance (\(\sigma^2\)) is calculated as:

\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^N (x_i – \mu)^2 \]

where \(N\) is the total number of data points, \(x_i\) is each individual data point, and \(\mu\) is the mean of the data.

**Variance for Grouped Data:**

Assuming we have \(k\) groups with frequencies \(f_i\) and the central value of each group is \(w_i\), the variance of the population can be calculated as:

\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^k f_i (w_i – \mu)^2 \]

where \(N\) is the sum of all frequencies.

The standard deviation (\(\sigma\)) is the square root of the variance:

**For a Sample:**

When dealing with a sample, replace \(\mu\) with the sample mean \((\bar{x})\), and the formula for variance becomes:

\[ s^2 = \frac{1}{n-1} \sum_{i=1}^k f_i (w_i – \bar{x})^2 \]

where \(n\) is the sample size.

### Example Calculation

Consider an example where we have weights of 30 individuals classified into 5 groups:

Class | Frequency \((f_i)\) | Central Value \((w_i)\) | Deviation from Mean \((\bar{x} – w_i)\) |
---|---|---|---|

65-69 | 4 | 67.5 | -9.5 |

70-74 | 6 | 72.5 | -4.5 |

75-79 | 12 | 77.5 | 0.5 |

80-84 | 5 | 82.5 | 5.5 |

85-89 | 3 | 87.5 | 10.5 |

For variance \((s^2)\):

\[ s^2 = \frac{1}{29} \left[ 4(-9.5)^2 + 6(-4.5)^2 + 12(0.5)^2 + 5(5.5)^2 + 3(10.5)^2 \right] = 33.362 \]

The standard deviation \((s)\) is:

\[ s = \sqrt{33.362} \approx 5.776 \]

### Coefficient of Variation (CV)

The coefficient of variation for the sample is given by:

\[ CV = \frac{s}{\bar{x}} \times 100 = \frac{5.776}{77} \times 100 \approx 7.50% \]

### Simplified Formula for Variance

A more convenient formula for variance that does not require calculating deviations is:

\[ \sigma^2 = \frac{1}{N} \sum_{i=1}^k f_i w_i^2 – \mu^2 \]

For samples:

\[ s^2 = \frac{1}{n-1} \left( \sum_{i=1}^k f_i w_i^2 – n \bar{x}^2 \right) \]

### Mean Deviation

For ungrouped data, mean deviation \((A)\) is calculated as:

\[ A = \frac{1}{N} \sum_{i=1}^N |x_i – \mu| \]

For grouped data:

\[ A = \frac{1}{N} \sum_{i=1}^k f_i |w_i – \mu| \]

**Example for Mean Deviation:**

Using the same weights example:

Class | Frequency \((f_i)\) | Central Value \((w_i)\) | Deviation \((|\bar{x} – w_i|)\) |
---|---|---|---|

65-69 | 4 | 67.5 | 9.5 |

70-74 | 6 | 72.5 | 4.5 |

75-79 | 12 | 77.5 | 0.5 |

80-84 | 5 | 82.5 | 5.5 |

85-89 | 3 | 87.5 | 10.5 |

Mean deviation \((A)\):

\[ A = \frac{1}{30} \left[ 4 \times 9.5 + 6 \times 4.5 + 12 \times 0.5 + 5 \times 5.5 + 3 \times 10.5 \right] = 4.333 \]

### Measures of Skewness

Distributions can be symmetric, positively skewed, or negatively skewed.

**Coefficient of Skewness \((\beta_1)\):**

For a dataset \(x_1, x_2, \ldots, x_n\), the skewness coefficient is:

\[

\beta_1 = \frac{n \sum_{i=1}^n (x_i – \bar{x})^3}{(n-1)(n-2)s^3}

\]

**Example Calculation:**

For the data (1, 0, 1, 3, 5, 14):

\[ \bar{x} = 4 \]

\[ \beta_1 = \frac{6 \times 882}{5 \times (5.22)^3} = 1.036 \]

For grouped data, the skewness coefficient can be written as:

\[ \beta_1 = \frac{1}{n} \sum_{i=1}^k f_i \left( \frac{w_i – \bar{x}}{s} \right)^3 \]

**Pearsonâ€™s Skewness Coefficient:**

\[ \text{Pearsonâ€™s Skewness} = \frac{3(\bar{x} – \text{Median})}{s} \]

### Key Takeaways

- Variance and standard deviation provide measures of dispersion for both grouped and ungrouped data.
- The coefficient of variation helps compare variability across different datasets.
- Skewness measures the asymmetry of data distributions, helping to understand the dataâ€™s shape relative to the mean.

This summary covers the main concepts and calculations related to variance, standard deviation, mean deviation, and skewness for grouped and ungrouped data.