5.3 Measures of Dispersion

Measures of dispersion describe how spread out the data is. The key measures include range, quartiles, variance, mean deviation, and standard deviation.

Key Concepts:

• Range:

  The difference between the maximum and minimum values in a dataset. \[ \text{Range} = \max(x) - \min(x) \]

• Quantiles:

  Values that divide a data set into equal parts. Common quantiles include:

  • Quartiles:

    Divide the data into four equal parts.

  • Median (Q2):

    The middle value of an ordered data set.

  • Lower Quartile (Q1):

    The median of the lower half of data.

  • Upper Quartile (Q3):

    The median of the upper half of data.

  • Interquartile Range (IQR):

    Measures the spread of the middle 50% of data. \[ \text{IQR} = Q3 - Q1 \]

  • Semi-Interquartile Range:

    Half of the interquartile range. \[ \text{SIQR} = \frac{Q3 - Q1}{2} \]

• Mean Deviation:

  The average absolute deviation of each data point from the mean. \[ \text{Mean Deviation} = \frac{\sum |x - \bar{x}|}{n} \]

• Variance:

  Measures the average squared deviation from the mean. \[ \sigma^2 = \frac{\sum (x - \bar{x})^2}{n} \]

• Standard Deviation:

  The square root of variance, showing the spread of data. \[ \sigma = \sqrt{\sigma^2} \]

Examples:

Example 1: Find the range, quartiles, and interquartile range for the following data set.

Given Data:

\\ \(4, 8, 15, 16, 23, 42, 50, 55, 60\)

Solution:

Step 1: Find the Range

\[ \text{Range} = 60 - 4 = 56. \]

Step 2: Find the Quartiles

• Median (Q2): The middle value is \(23\).
• Lower Quartile (Q1): The median of the lower half (\(4, 8, 15, 16\)) is: \[ Q1 = \frac{8+15}{2} = 11.5. \]
• Upper Quartile (Q3): The median of the upper half (\(42, 50, 55, 60\)) is: \[ Q3 = \frac{50+55}{2} = 52.5. \]

Step 3: Calculate Interquartile Range (IQR)

\[ \text{IQR} = Q3 - Q1 = 52.5 - 11.5 = 41. \]

Step 4: Compute Semi-Interquartile Range

\[ \text{SIQR} = \frac{IQR}{2} = \frac{41}{2} = 20.5. \] Thus, the range is **56**, IQR is **41**, and SIQR is **20.5**.

Example 2: Find the Mean Deviation, Variance, and Standard Deviation for the following data set.

Given Data:

\\ \(5, 10, 15, 20, 25\)

Solution:

Step 1: Compute the Mean

\[ \bar{x} = \frac{5+10+15+20+25}{5} = \frac{75}{5} = 15. \]

Step 2: Compute Mean Deviation

\[ \sum |x - \bar{x}| = |5-15| + |10-15| + |15-15| + |20-15| + |25-15| \] \[ = 10 + 5 + 0 + 5 + 10 = 30. \] \[ \text{Mean Deviation} = \frac{30}{5} = 6. \]

Step 3: Compute Variance

\[ \sigma^2 = \frac{\sum (x - \bar{x})^2}{n} \] \[ = \frac{(5-15)^2 + (10-15)^2 + (15-15)^2 + (20-15)^2 + (25-15)^2}{5} \] \[ = \frac{100 + 25 + 0 + 25 + 100}{5} = \frac{250}{5} = 50. \]

Step 4: Compute Standard Deviation

\[ \sigma = \sqrt{50} \approx 7.07. \] Thus, the mean deviation is **6**, variance is **50**, and standard deviation is **7.07**.