Measures of Spread (Range, Variance, Standard Deviation)

Introduction

Key Concepts

Range

The range is the simplest measure of spread, representing the difference between the highest and lowest values in a data set. It provides a quick assessment of the data's dispersion but lacks sensitivity to the distribution of values within the range.

Formula:

$$\text{Range} = \text{Maximum value} - \text{Minimum value}$$

Example:

Consider the data set: 3, 7, 8, 5, 12, 14, 21, 13, 18.

The range is calculated as:

$$\text{Range} = 21 - 3 = 18$$

While the range provides a quick snapshot, it does not account for the distribution of the remaining data points.

Variance

Variance measures the average squared deviation of each data point from the mean. It quantifies the degree of spread in the data set, considering how each value varies from the average.

Population Variance Formula:

$$\sigma^2 = \frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}$$

Sample Variance Formula:

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

Where:

• $\sigma^2$ = population variance

Example:

Using the same data set: 3, 7, 8, 5, 12, 14, 21, 13, 18.

First, calculate the mean ($\bar{x}$):

$$\bar{x} = \frac{3 + 7 + 8 + 5 + 12 + 14 + 21 + 13 + 18}{9} = \frac{101}{9} \approx 11.22$$

Next, compute each squared deviation:

Sum of squared deviations:

$$\sum (x_i - \bar{x})^2 \approx 287.32$$

Sample variance:

$$s^2 = \frac{287.32}{9 - 1} = \frac{287.32}{8} \approx 35.91$$

The variance indicates how much the data points deviate from the mean on average.

Standard Deviation

Standard deviation is the square root of variance and provides a measure of spread in the same units as the data, making it more interpretable. It indicates the average distance of each data point from the mean.

Population Standard Deviation Formula:

$$\sigma = \sqrt{\sigma^2} = \sqrt{\frac{\sum_{i=1}^{N} (x_i - \mu)^2}{N}}$$

Sample Standard Deviation Formula:

$$s = \sqrt{s^2} = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}$$

Example:

Using the previously calculated sample variance ($s^2 \approx 35.91$), the sample standard deviation is:

$$s = \sqrt{35.91} \approx 5.99$$

This value signifies that, on average, each data point deviates from the mean by approximately 5.99 units.

Interpreting Measures of Spread

Measures of spread complement measures of central tendency (mean, median, mode) by providing insights into data variability. A higher standard deviation indicates greater dispersion, while a lower standard deviation signifies data points are closer to the mean.

Applications:

• Education: Assessing student performance variability.
• Finance: Evaluating investment risk.
• Healthcare: Analyzing patient recovery times.

Advantages:

• Range is simple to compute and understand.
• Variance and standard deviation account for all data points.
• Standard deviation is in the same units as the data, enhancing interpretability.

Limitations:

• Range is sensitive to outliers and does not reflect data distribution.
• Variance involves squared units, which can be less intuitive.
• Standard deviation assumes a symmetric distribution of data.

Comparison Table

Summary and Key Takeaways