Measures of Dispersion

Measures of dispersion help analyze the spread of data within a dataset. It indicates the extent to which data varies from the average. The five measures of dispersion are range, variance, standard deviation, mean deviation, and quartile deviation.

We can use these metrics to identify whether a given dataset is uniform or diverse. The dispersion value is zero when all data points are the same. However, the measures of dispersion increase, as the differences between data points increase.

The measures of dispersion are classified into two main categories. Dispersion methods can be divided into two main categories. Absolute and relative measures. These are further divided into many. Absolute measures have the same units, and the relative measures are always dimensionless. Also, the coefficients of dispersion are the relative measures of dispersion. Now, let us examine the two main types of dispersion measures. These types can be easily understood with the help of the image below.

The absolute measure of dispersion is quantified and expressed in the same units as the data. Meters, kilograms, and dollars are some examples of the absolute measures of dispersion that are represented in the same units as the data. For instance, if the standard deviation of BVT company’s salary distribution is $700, it indicates that the salaries vary by around $700 from the mean. Here are some of the absolute measures of dispersion. 

Range: It is an absolute measure of dispersion that can be defined as the difference between the distribution’s maximum and minimum values. To calculate the range, the formula we can employ is:

\(\text{Range = H − S}\)

Here, H is the highest value and S is the smallest value in the dataset.

Mean deviation: It is the difference between each data point and the mean, calculated as arithmetic mean. The formula for finding the mean deviation is:

\(\text{Mean deviation} = \frac{\sum _{i=1}^n (x_i-\bar x)} {n}\)

Here, x̄ denotes the mean, median, or mode of the dataset and it is the central value.

n is the number of values and x_{​​​​i} is the individual data point. 

Standard deviation: It is the square root of the mean of the squared deviations from the average value of the dataset. The formula for standard deviation is different for population standard deviation and sample standard deviation. Therefore, the formula of standard deviation is given as,

For population standard deviation:
 

\(\sigma = \sqrt{\sigma^{2}} \)

For sample standard deviation: 
 

\(s = \sqrt{s^{2}}\)

Variance: It is the mean or average of the squared deviations from the mean of the provided dataset. The formula for population variance is: 

 \(σ ^2 = {\sum _i ^n (x_i - \mu)^2 \over N}\)

The formula for calculating sample variance is:

\(S^2 = {{\sum_ i ^n (x_i - \bar x)^2 }\over n-1}\)

Here, x_i refers to each value of a dataset.

x̄ is the mean. 

n is the number of values in the dataset. 

Interquartile range: It is defined as the difference between the upper quartile (Q₃) and the lower quartile (Q₁). The formula used is \(Q_3 − Q_1\). Here, Q₃ is the upper or the third quartile and Q₁ is the lower or first quartile.

To compare separate datasets with different units, relative measures of dispersion are used. They are represented as percentages and ratios. Also, these measures are unitless. Some of the few relative measures of dispersion are listed below.

Coefficient of Range: In a dataset, the coefficient of range is calculated as the ratio of the difference between the maximum and the minimum values to the sum of the two values. The formula for calculating the coefficient of range is:
 

\(\frac{(H - S)}{(H + S)}\)

Coefficient of Variation: It is expressed as a percentage. In a dataset, the coefficient of variation (CV) is the ratio of the standard deviation to the mean. The formula for the coefficient of variation (CV) is: 

\(\left( \frac{\text{S.D.}}{\text{Mean}} \right) \times 100 \)

Coefficient of Mean Deviation: It is the ratio of the mean deviation to the central value, from which it is used to calculate. The formula is given below: 

\(\text{Coefficient of mean deviation} = \frac{\text{Mean Deviation}}{\bar{X}}\)

Here, x̄ is the central value, from which the mean deviation is used to calculate. 

Coefficient of Quartile Deviation: The coefficient of quartile deviation is calculated as the ratio of the difference between the first and third quartiles to their sum. The formula for finding the coefficient of quartile deviation is:

 \(\text {Coefficient of Quartile Deviation } = \frac{Q_3-Q_1}{Q_3+Q_1}\) 

Measures of dispersion describe how data values spread around a central value. The mean and the standard deviation formulas for these measures are given in the tables below.​

Absolute measures of dispersion

Relative Measures of Dispersion

The standard deviation is the square root of the variance. It converts the measure back to the original units for more straightforward interpretation, which is why scientists routinely report both the mean and the standard deviation in studies. It is produced as the primary measure of central tendency and spread. The population standard deviation is denoted by the Greek letter sigma (σ). It is the square root of the population variance σ², which follows the same formula as variance but under a square root.

Standard deviation is an essential measure of variability for normal or approximately normal distributions. It enables calculating the proportion of data within specific standard deviations from the mean.

Let us understand the concept of standard deviation with the help of an example. Let us take a problem.

Calculate standard deviation of the numbers 4, 7, 8, 9, 10.

Step 1: Find the mean of the data.

\(x = 4+7+8+9+105=385=7.6\)

Step 2: Next, let's find each deviation from the mean and square it.
 

\(4-7.6 = -3.6, \text{squared} = 12.96\\[1em] 7-7.6=-0.6, \text{squared} = 0.36\\[1em] 8 - 7.6 = 0.4, \text{squared}= 0.16\\[1em] 9 - 7.6 =1.4, \text{squared}= 1.96\\[1em] 10 - 7.6 = 2.4, \text{squared}= 5.76\)

Step 3: Now, let us find the average of the squared deviations. 

\(\text{Sum of squares} = 12.96+0.36+0.16+1.96+5.76 = 21.2\)

\(\text{Variance} = \frac{(21.2)}{5} = 4.24\)

Step 4: Calculating standard deviation.

Take the square root (calculating for standard deviation)

\(=4.242.06\)

In a normal distribution, about 68% of data falls within one standard deviation of the mean, roughly 95% within two standard deviations, and over 99% within three standard deviations. This rule is known as the empirical rule or the 68-95-99 rule. This can be better understood with the help of an image, as shown below:
Similarly, as the standard deviation decreases, the distribution becomes much narrower. This happens regardless of where the mean is present in the graph. We can understand this with the help of an image, as shown below:

In statistics, data can be described by fundamental concepts such as measures of central tendency and measures of dispersion.

Measures of dispersion quantify data variability, while central tendency describes its average behavior. There are five common measures of dispersion, they are range, variance, standard deviation, mean deviation, and quartile deviation. Mean, median, and mode are the three common measures of central tendency. Measures of dispersion are always non-negative and increase as data variability rises.

The value of central tendency can be any real number, which is based on its data distribution. To understand how data values deviate from the average, measures of dispersion are helpful. On the other hand, measures of central tendency find a single representative value to summarize the given data.

Measures of dispersion are a vast topic in mathematics that are helpful in many analytical situations and in statistics. Here are some tips and tricks to help learners master the concept of measures of dispersion.
 

• Teachers can use some everyday situations to explain how “spread” works. Take the examples of students' heights in a class, daily temperatures, and marks scored in different subjects. Show them that the means of the sets might be the same, but the spreads are different.
   
• Graphs play a very important role in understanding measures of dispersion. It is much easier for young learners to understand the topic when we use graphs to explain it, along with definitions. Parents can help them visualize dispersion through line plots, bar graphs, dot plots, etc.
   
• Teachers can start with a simple concept and gradually increase the difficulty level. They can follow the order of teaching range, interquartile range, mean deviation, variance, and standard deviation. This helps build students' understanding capacity.
   
• Parents can encourage their children to use Excel or Google Sheets, Python, and graphing tools to help them focus on understanding rather than mentally overload with numbers.
   
• Learners should start practicing with small data sets and gradually increase the difficulty. Early success builds confidence, and it promotes good performance in more challenging situations.
   
• Learn by connecting dispersion to decision-making. Look at examples, like when athletes are consistent, the standard deviation is low. When the temperature is stable, the range is low.

To know how much the given data deviates from a central value point, the fundamental concept of measures of dispersion is used. The real-world applications of the measures of dispersion are limitless. 

• Business professionals and marketing teams use measures of dispersion to assess the risk of starting a new business, investing in mutual funds, or trading in stock markets. For instance, they can assess the risk of stocks with low standard deviation. 

• Teachers and education analysts use measures of dispersion to evaluate student performance. For example, if the student’s performance is consistent with low dispersion, the teachers can identify that there is no need for any new tutorial methods. 

• The manufacturing companies can use the measures of dispersion, such as standard deviation, to monitor the quality and consistency of their product manufacturing. 

• Weather forecasters can apply the measures of dispersion to analyze the climatic change of a city over time. For instance, if the temperature range is highly inconsistent, meteorologists can forecast a climatic change. 

• In the fields of medicine and sports analysis, the measures of dispersion play a crucial role in evaluating the performance of drugs or athletes.