Measures of dispersion are statistical tools used to describe the spread or variability within a set of data. They provide insights into how much the data values differ from the central tendency (mean, median, or mode). Understanding dispersion is crucial for interpreting data accurately and making informed decisions.

1. Range
2. Variance
3. Standard Deviation
4. Interquartile Range (IQR)
5. Mean Absolute Deviation (MAD)

The range is the simplest measure of dispersion. It is the difference between the maximum and minimum values in a data set.

Formula: \[ \text{Range} = \text{Maximum Value} - \text{Minimum Value} \]

Example: Consider the data set: \( {3, 7, 8, 5, 12, 14, 21, 13, 18} \)

• Maximum Value = 21
• Minimum Value = 3

\[ \text{Range} = 21 - 3 = 18 \]

Variance measures the average squared deviation of each data point from the mean. It gives an idea of how spread out the data points are.

Formula: \[ \text{Variance} (\sigma^2) = \frac{\sum (x_i - \mu)^2}{N} \]

Where:

• \( x_i \) = each data point
• \( \mu \) = mean of the data
• \( N \) = number of data points

Example: Consider the data set: \( {4, 8, 6, 5, 3} \)

1. Calculate the mean (\(\mu\)): \[ \mu = \frac{4 + 8 + 6 + 5 + 3}{5} = 5.2 \]

2. Calculate each squared deviation from the mean and sum them: \[ (4 - 5.2)^2 + (8 - 5.2)^2 + (6 - 5.2)^2 + (5 - 5.2)^2 + (3 - 5.2)^2 \] \[ = 1.44 + 7.84 + 0.64 + 0.04 + 4.84 = 14.8 \]

3. Divide by the number of data points (N): \[ \text{Variance} = \frac{14.8}{5} = 2.96 \]

Standard deviation is the square root of the variance. It is expressed in the same units as the data, making it more interpretable.

Formula: \[ \text{Standard Deviation} (\sigma) = \sqrt{\text{Variance}} \]

Example: Using the variance calculated above (2.96): \[ \text{Standard Deviation} = \sqrt{2.96} \approx 1.72 \]

The IQR measures the spread of the middle 50% of the data. It is the difference between the third quartile (Q3) and the first quartile (Q1).

Formula: \[ \text{IQR} = Q3 - Q1 \]

Example: Consider the data set: \( {1, 3, 5, 7, 9, 11, 13} \)

1. Find Q1 (first quartile) and Q3 (third quartile):
   • Q1 = 3
   • Q3 = 11

\[ \text{IQR} = 11 - 3 = 8 \]

MAD measures the average absolute deviation of each data point from the mean.

Formula: \[ \text{MAD} = \frac{\sum |x_i - \mu|}{N} \]

Example: Consider the data set: \( {4, 8, 6, 5, 3} \)

1. Calculate the mean (\(\mu\)): \[ \mu = 5.2 \]

2. Calculate each absolute deviation from the mean and sum them: \[ |4 - 5.2| + |8 - 5.2| + |6 - 5.2| + |5 - 5.2| + |3 - 5.2| \] \[ = 1.2 + 2.8 + 0.8 + 0.2 + 2.2 = 7.2 \]

3. Divide by the number of data points (N): \[ \text{MAD} = \frac{7.2}{5} = 1.44 \]

Given the data set: \( {10, 15, 20, 25, 30} \)

1. Identify the maximum and minimum values.
2. Calculate the range.

Solution:

• Maximum Value = 30
• Minimum Value = 10

\[ \text{Range} = 30 - 10 = 20 \]

Given the data set: \( {2, 4, 6, 8, 10} \)

1. Calculate the mean.
2. Calculate the variance.
3. Calculate the standard deviation.

Solution:

1. Mean (\(\mu\)): \[ \mu = \frac{2 + 4 + 6 + 8 + 10}{5} = 6 \]

2. Variance: \[ (2 - 6)^2 + (4 - 6)^2 + (6 - 6)^2 + (8 - 6)^2 + (10 - 6)^2 \] \[ = 16 + 4 + 0 + 4 + 16 = 40 \] \[ \text{Variance} = \frac{40}{5} = 8 \]

3. Standard Deviation: \[ \text{Standard Deviation} = \sqrt{8} \approx 2.83 \]

Given the data set: \( {5, 7, 8, 12, 14, 18, 21} \)

1. Find Q1 and Q3.
2. Calculate the IQR.

Solution:

1. Q1 = 7, Q3 = 18
2. IQR: \[ \text{IQR} = 18 - 7 = 11 \]

Given the data set: \( {1, 3, 5, 7, 9} \)

1. Calculate the mean.
2. Calculate the MAD.

Solution:

1. Mean (\(\mu\)): \[ \mu = \frac{1 + 3 + 5 + 7 + 9}{5} = 5 \]

2. MAD: \[ |1 - 5| + |3 - 5| + |5 - 5| + |7 - 5| + |9 - 5| \] \[ = 4 + 2 + 0 + 2 + 4 = 12 \] \[ \text{MAD} = \frac{12}{5} = 2.4 \]

• Mistake: Confusing variance and standard deviation.

  • Tip: Remember that standard deviation is the square root of variance.

• Mistake: Forgetting to square the deviations when calculating variance.

  • Tip: Always square the deviations before summing them up for variance.

• Mistake: Not ordering data when calculating quartiles for IQR.

  • Tip: Always sort the data set before finding quartiles.

Understanding measures of dispersion is essential for interpreting the variability in data. By mastering range, variance, standard deviation, IQR, and MAD, you can gain deeper insights into the data's spread and make more informed decisions. Practice these concepts with various data sets to strengthen your grasp and prepare for more advanced statistical analyses.