Measurements of Dispersion (Variation)

In descriptive statistics, several common variation measurements are used to describe the spread or dispersion of a group of observations. Some of the most common measurements of variation include:

1. Range: The range is the difference between the largest and smallest values in a group of observations. It is a simple and easy-to-understand measure of variation, but it is sensitive to outliers and can be affected by extreme values.
2. Interquartile range (IQR): The interquartile range is the difference between the third and the first quartile in a group of observations. It is calculated by subtracting the value of the first quartile (Q1) from the value of the third quartile (Q3). The IQR is a more robust measure of variation than the range because it is not affected by extreme values.
3. Standard deviation: The standard deviation measures the observations' average distance from the data's mean. It is calculated by taking the square root of the average squared differences between each observation and the mean. The standard deviation is a commonly used measure of variation because it is not affected by extreme values and considers the variability of all the observations in the group. The formula for the standard deviation of a sample is:
   $$s = \sqrt{\frac{\sum(x_i - \overline{x})^2}{n-1}}$$. The formula for the standard deviation of the population is calculated using this formula: $$\sigma = \sqrt{\frac{\sum(x_i - \mu)^2}{N}}$$
4. Variance: The variance measures the average squared difference of the observations from the mean of the data. It is calculated by taking the average squared differences between each observation and the mean. The variance is the square of the standard deviation.The formula for the variance of a sample is: $$s^2 = \frac{\sum(x_i - \overline{x})^2}{n-1}$$ The formula for the standard deviation of the population is calculated using this formula: $$\sigma^2 = \frac{\sum(x_i - \mu)^2}{N}$$

Conclusion

Dispersion, also known as variation, refers to the spread or distribution of values in a data set. It is an essential concept in statistics, as it allows us to understand how much the values in a data set vary. Several measurements of dispersion can be used to quantify the amount of variation in a data set. Some of the most common measurements include the range, interquartile range, variance, and standard deviation.  Understanding and using these measurements can help understand the characteristics of a data set and make informed decisions based on the data.