Concept of Dispersion

The central tendency measures are insufficient to describe data. The mean of two data sets can be the same, but they can be completely different. To describe data, one must first understand the degree of variability. The three most common measures of dispersion are range, interquartile range, and standard deviation. The term “dispersion” refers to the degree to which a set of data is dispersed. When all the data is the same, the measure of dispersion is zero, and it rises as the data becomes more varied. Dispersion measures determine whether the scattered data is homogeneous or heterogeneous. It also refers to the differences between data sets.

Definition of dispersion

“In statistics, dispersion refers to the degree to which a distribution is stretched or squeezed. Dispersion differs from location or central tendency, and the two together are one of the most commonly used properties of distributions.”

Measures of dispersion

Dispersion measures aid in describing data variability. Data dispersion is a statistical term that describes the degree to which data is dispersed. Measures of dispersion are thus specific types of measures used to quantify data dispersion.

                   Figure 1

The following are the most common statistical dispersion measures:

• Mean deviation 
• Variation 
• Calculate the standard deviation 
• Interquartile range and range

Dispersion Measures Types

The Measure of Dispersion in Statistics is divided into two categories, each with its own set of methods for measuring the various types of data. Its primary application is in biological statistics. We can quickly classify them by determining whether or not they contain units.

We can categorizes the information into two groups:

1. Absolute Measurements of Dispersion
2. Relative Measures of Dispersion

Absolute Measures of Dispersion – Absolute Dispersion Measures is a dataset with units; it uses the same unit as the original dataset. The average of dispersion quantities such as Standard or Mean deviation is used to calculate the absolute measure of dispersion. The Absolute Measure of Dispersion can be expressed in a variety of units, including rupees, centimeters, marks, kilograms, and other weights and measures.

In statistics, there are several different types of absolute measures of dispersion:

• Range: The difference between the largest and smallest value of data variability is measured by range. The simplest form of Measures of Dispersion is the range. 

Example: 1,2,3,4,5,6,7 Range = Highest value – Lowest value = (7 – 1) = 6 

• Mean (μ): The average of the numbers is calculated as the mean. To find the Mean, add all of the results together and divide by the total number of terms.

Example: 1,2,3,4,5,6,7,8 = Mean = (sum of all terms divided by total number of terms) = (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) / 8 = 36 / 8 = 4.5 

• Variance (σ2): In simple terms, the variance is calculated by adding the squared distances of each term in the distribution from the Mean, then dividing by the total number of terms in the distribution.

(σ2) = ∑ ( X – μ)2 / N 

• Standard Deviation: The square root of Variance can be used to calculate Standard Deviation. To calculate the standard deviation of any data, you must first calculate the variance. The standard deviation is the best measure of dispersion is the standard deviation.

Standard Deviation = √σ 

• Mean deviation: Mean deviation is also known as average deviation, and it can be calculated using the data’s Mean or Median. The arithmetic deviation of a different item that follows the central tendency is represented as mean deviation.

1. Mean Deviation using Mean: ∑ | X – M | / N 
2. Mean Deviation using Median: ∑ | X – X1 | / N 

Relative measures of dispersion:  In statistics, relative measures of dispersion are values without units. When comparing the distribution of two or more datasets, a relative measure of dispersion is used.

• Standard Deviation Coefficient: The standard deviation coefficient is the ratio of standard deviation to the mean of a distribution of terms.

                                                σ = ( √( X – X1)) / (N – 1) 

                          Deviation = ( X – X1)   , σ = standard deviation  , N= total number  

• Coefficient of Quartile Deviation:  The ratio of the difference between the upper quartile and the lower quartile to the sum of the upper quartile and lower quartile is known as the coefficient of quartile deviation.

(Q3 – Q3) / (Q3 + Q1), Upper Quartile (Q3) ,Lower Quartile (Q1)

• Coefficient of Mean Deviation: The mean or median of the data can be used to calculate the mean Deviation Coefficient.

Mean Deviation using Mean: ∑ | X – M | / N 

Mean Deviation using Mean: ∑ | X – X1 | / N 

Dispersion Measures: Characteristics

• It must be simple to calculate and comprehend.
• The fluctuations in observations haven’t had much of an impact.
• Based on everything you’ve seen
• It should be able to be subjected to further mathematical analysis.
• It should be influenced by sampling fluctuations.

Conclusion

In statistics Dispersion refers to the measure of the variability of data or terms. Like variability may give random measurement errors where some of the instrumental measurements are found to be imprecise. Understanding dispersion is important for understanding statistics. It aids comprehension of concepts such as data diversification, data spread, data maintenance, and data maintenance over the central value or central tendency. Furthermore, statistical dispersion recognize us to gain a better understanding of data distribution.