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The range is the simplest measure of the spread or variability in a dataset. It is calculated by finding the difference between the highest and lowest values. For the given measurements, which are 4, 1, 3, 1, 3, 1, 2, and 2, we identify the highest number as 4 and the lowest as 1.
Subsequently, the range is calculated by subtracting the lowest value from the highest value:

• Range = Highest value - Lowest value = 4 - 1 = 3

The range provides a quick snapshot of the total spread of the data, yet it does not reflect how data points are distributed between these extreme values.

The mean, often referred to as the average, is a key concept in descriptive statistics that indicates the central tendency of a dataset. It offers a way to summarize all data points with a single value. To find the mean (\(\bar{x}\)), sum all the data points and divide by the number of values, \(n\).
For our data consisting of the values 4, 1, 3, 1, 3, 1, 2, and 2, the mean is calculated as follows:

• Sum of values = 4 + 1 + 3 + 1 + 3 + 1 + 2 + 2 = 17
• Number of values, \(n = 8\)
• Mean, \(\bar{x} = \frac{17}{8} = 2.125\)

The mean provides an overall summary, but it can sometimes be affected by extreme values, also known as outliers.

Variance is a statistical metric that quantifies the degree of variation or dispersion in a dataset. Unlike the range, it takes into account all data points and how they relate to the mean. The formula for variance (\(s^{2}\)) is:

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

To determine variance:

• Calculate the square of each data value, sum these squares: \(\sum_{i=1}^n x_{i}^2 = 4^2 + 1^2 + 3^2 + 1^2 + 3^2 + 1^2 + 2^2 + 2^2 = 45\)
• Divide the total by \(n\): \(\frac{45}{8} = 5.625\)
• Subtract the square of the mean: \(5.625 - 2.125^2 = 5.625 - 4.515625 = 1.109375\)

The calculated variance is 1.109375, capturing how tightly or loosely data values are clustered around the mean.

Standard deviation (\(s\)) is derived from variance and provides a more intuitive measure of data dispersion. While variance is expressed in square units, standard deviation is in the same units as the data, making it easier to interpret. It shows the average distance data points are from the mean. The formula for standard deviation is the square root of the variance:

• \(s = \sqrt{1.109375} \approx 1.053\)

This means data points typically deviate from the mean by approximately 1.053 in the dataset.
The standard deviation is a key indicator of variability and is more informative than the range, especially when comparing different datasets. It's widely used across fields, making it crucial for understanding data behavior.