Measures of Dispersion - SS2 Economics Lesson Note

The measure of dispersion is a statistical concept that helps to quantify how spread out or varied a set of data is. It is used to describe the variability of a dataset and to understand how much the individual data points deviate from the mean or central value of the dataset. The most common measures of dispersion include range, variance, and standard deviation. These measures help to indicate how much the data deviates from the central value or mean.

A large measure of dispersion indicates that the data points are widely spread out, while a small measure of dispersion indicates that the data points are more tightly clustered around the central value. Understanding the measure of dispersion is important in many fields, including finance, economics, and social sciences, where it is used to analyze data and make predictions

Range:

The range is simply the difference between the largest and smallest values in a dataset.

Merits:

Easy to calculate and understand.

Provides a simple measure of the spread of the data.

Demerits:

Only considers the difference between the maximum and minimum values, and ignores the distribution of values within that range.

Sensitive to outliers, which can distort the results.

Variance:

The variance is the average of the squared differences between each data point and the mean of the dataset.

Merits:

Takes into account all the values in the dataset, not just the maximum and minimum.

Provides a measure of how much the data points are spread out from the mean.

Demerits:

Sensitive to outliers, which can have a significant impact on the result.

The units of variance are squared, making it difficult to compare with other variables that may have different units.

Standard Deviation:

The standard deviation is the square root of the variance.

Merits:

Provides a measure of the spread of the data that is easy to interpret.

Can be used to identify outliers in the data.

The units of standard deviation are the same as the units of the original data, making it easy to compare with other variables.

Demerits:

Sensitive to outliers, which can have a significant impact on the result.

Can be affected by the shape of the distribution of the data..

Word Problem on Range:

Suppose the test scores for a group of students are: 85, 75, 90, 80, and 95. What is the range of the scores?

Solution:

The smallest score is 75, and the largest score is 95. Therefore, the range is 95 - 75 = 20.

Word Problem on Variance:

Suppose the test scores for a group of students are: 85, 75, 90, 80, and 95. What is the variance of the scores?

Solution:

The mean score is (85+75+90+80+95)/5 = 85

The squared differences between each score and the mean are: (85-85)², (75-85)², (90-85)^ (80-85)², and (95-85)²

The average of these squared differences is (0² + 100² + 25² + 25²+ 100²)/5 = 250.

Therefore, the variance is 250.

Word Problem on Standard Deviation:

Suppose the test scores for a group of students are: 85, 75, 90, 80, and 95. What is the standard deviation of the scores?

Solution:

From the previous example, we know that the variance is 250.

Therefore, the standard deviation is the square root of 250, which is approximately 15.81.