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In statistics, dispersion is the degree to which a distribution is stretched or squeezed. Measures of Dispersion differs with location or central tendency, and together they are one of the most used properties of distributions. Most common measures of statistical dispersion are: 1. Mean deviation2. Variance3. Standard deviation 4. Range & Inter-quartile range.  Let us discuss the measures of dispersion and there examples Mean DeviationMean Deviation is a statistical measure to find the average deviation of values from the mean in a sample. How to calculate Mean Deviation?1. Calculate the average of the observations2. Calculate the difference of each observation from the mean (deviations of all the observations). 3. Average all the deviations. Mean Deviation Example If we take nos. 1 to 7 and find their mean deviation then, The mean of these terms will be 4 And their deviation from the mean will be
And mean deviation comes out to be 1.714 VarianceThe Variance is a degree of variability. It is determined by taking the average of squared deviations from the mean. Variance represents the degree of spread in your data set. The further spread the data, the larger the variance to the mean. Formula \(\sigma^2 = \frac{1}{n} \sum^n_{i=1}(x_i – \bar{x})^2\)Variance Example Sample A: -2, 2, 2, -2, -2, 2, -2, 2 Sample B: -1, 1, 2, -2, 2, -1, 1, -2 The mean for both these samples is a ‘0’. But variance for sample A is ‘4’ And that for sample B is ‘5’. We can see that Sample B had a more widespread set of data, with the mean same as Sample A. Standard DeviationThe Standard Deviation is a measure of the amount of variation of a set of values. The standard deviation of a random variable is the square root of its variance. A low standard deviation shows that the values tend to be approaching the mean of the set, while a high standard deviation indicates that the values are spread out across a wider range. Formula \(\sigma = \sqrt{\frac{1}{n}\sum^n_{i=1}(x_i – \bar{x})^2}\), Here,\(n\) = no. of terms\(x_i\) = mean \(\bar{x}\) = elements of data Standard Deviation Example Sample A: -2, 2, 2, -2, -2, 2, -2, 2 Sample B: -1, 1, 2, -2, 2, -1, 1, -2 The mean for both these samples is a ‘0’. Here Standard deviation will be Sample A: \(2\) Sample B: \(\sqrt{5}\) RangeThe range is the difference between the highest and lowest values of a dataset. Range Example For the dataset {4, 6, 9, 3, 7}The lowest value -> 3The highest value -> 9 So, range -> 9-3=6. Interquartile RangeThe Interquartile Range(IQR), also called the mid-spread, is a measure of statistical dispersion, being equal to the difference between 75th and 25th percentiles, or between upper and lower quartiles, IQR = Q3 − Q1. In other words, the IQR is the first quartile subtracted from the third quartile. These quartiles can be seen on a box plot on the data. It is a trimmed estimator, defined as the 25% trimmed range, and is a commonly used robust measure of scale. Applications of Measures of Dispersion
FAQsWhat do you mean by measures of dispersion? The measure of dispersion shows the spread of data. It explains the data differs from one another, delivering a precise picture of the distribution of data. Dispersion is the degree to which values in a distribution differ from the average of the distribution. What are the four measures of dispersion? Measures of dispersion describe the spread of the data. They include the mean deviation, range,interquartile range, standard deviation, and variance. What are the uses of measures of dispersion? Measures of central tendency are used to calculate “normal” values of a dataset, measures of dispersion are crucial in explaining the range of the data, or its variation around a mean value. Why is standard deviation called the best measure of dispersion? Standard deviation is studied as the most reliable measure of dispersion. It is based on all values of given data and thus presents data about the entire series, a change in one value affects the value of the standard deviation and hence is the most widely trusted and used. Dispersion in statistics is a way of describing how to spread out a set of data is. Dispersion is the state of data getting dispersed, stretched, or spread out in different categories. It involves finding the size of distribution values that are expected from the set of data for the specific variable. The meaning of dispersion in statistics is “numeric data that is likely to vary at any instance of average value assumption”. Dispersion of data in Statistics helps one to easily understand the dataset by classifying them into their own specific dispersion criteria like variance, standard deviation and ranging. Dispersion is a set of measures that helps one to determine the quality of data in an objectively quantifiable manner. Most often data science courses start with the basics of statistics and dispersion is one such concept that you cannot afford to skip. Measures of DispersionThe measures of dispersion contain almost the same unit as the quantity being measured. There are many Measures of Dispersion found that help us to get more insights into the data:
Image Source Types of Measures of DispersionThe Measure of Dispersion in Statistics is divided into two main categories and offer ways of measuring the diverse nature of data. It is mainly used in biological statistics. We can easily classify them by checking whether they contain units or not. So as per the above, we can divide the data into two categories which are:
Absolute Measures of DispersionAbsolute Measures of Dispersion is one with units; it has the same unit as the initial dataset. Absolute Measure of Dispersion is expressed in terms of the average of the dispersion quantities like Standard or Mean deviation. The Absolute Measure of Dispersion can be expressed in units such as Rupees, Centimetre, Marks, kilograms, and other quantities that are measured depending on the situation. Types of Absolute Measure of Dispersion in Statistics:
Example: 1,2,3,4,5,6,7,8
= (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8) / 8 = 36 / 8 = 4.5
It basically shows how far a number, for example, a student’s mark in an exam, is from the Mean of the entire class. Formula: (σ2) = ∑ ( X − μ)2 / N
Formula: Standard Deviation = √σ
Formula: Interquartile Range: Q3 – Q1.
Formula: As mentioned, the Mean Deviation can be calculated using Mean and Median.
Relative Measures of DispersionRelative Measure of Dispersion in Statistics are the values without units. A relative measure of dispersion is used to compare the distribution of two or more datasets. The definition of the Relative Measure of Dispersion is the same as the Absolute Measure of Dispersion; the only difference is the measuring quantity. Types of Relative Measure of Dispersion: Relative Measure of Dispersion is the calculation of the co-efficient of Dispersion, where 2 series are compared, which differ widely in their average. The main use of the co-efficient of Dispersion is when 2 series with different measurement units are compared. 1. Co-efficient of Range: it is calculated as the ratio of the difference between the largest and smallest terms of the distribution, to the sum of the largest and smallest terms of the distribution. Formula:
2. Co-efficient of Variation: The coefficient of variation is used to compare the 2 data with respect to homogeneity or consistency. Formula:
3. Co-efficient of Standard Deviation: The co-efficient of Standard Deviation is the ratio of standard deviation with the mean of the distribution of terms. Formula:
4. Co-efficient of Quartile Deviation: The co-efficient of Quartile Deviation is the ratio of the difference between the upper quartile and the lower quartile to the sum of the upper quartile and lower quartile. Formula:
5. Co-efficient of Mean Deviation: The co-efficient of Mean Deviation can be computed using the mean or median of the data. Mean Deviation using Mean: ∑ | X – M | / N Mean Deviation using Mean: ∑ | X – X1 | / N These formulas come in handy a lot while calculating different aspects of data and when you use python with data science, achieving this gets easier as the programming language offers various statistical packages for these. Why dispersion is important in a statisticThe knowledge of dispersion is vital in the understanding of statistics. It helps to understand concepts like the diversification of the data, how the data is spread, how it is maintained, and maintaining the data over the central value or central tendency. Moreover, dispersion in statistics provides us with a way to get better insights into data distribution. For example, 3 distinct samples can have the same Mean, Median, or Range but completely different levels of variability. How to Calculate DispersionDispersion can be easily calculated using various dispersion measures, which are already mentioned in the types of Measures of Dispersion described above. Before measuring the data, it is important to understand the diversion of the terms and variations. One can use the following method to calculate the dispersion:
For example, let us consider two datasets:
On calculating the mean and median of the two datasets, both have the same value, which is 100. However, the rest of the dispersion measures are totally different as measured by the above methods. The range of B is 10 times higher, for instance. How to represent Dispersion in StatisticsDispersion in Statistics can be represented in the form of graphs and pie-charts. Some of the different ways used include:
Example: What is the variance of the values 3,8,6,10,12,9,11,10,12,7?Variation of the values can be calculated using the following formula:
What is an example of dispersion?One of the examples of dispersion outside the world of statistics is the rainbow- where white light is split into 7 different colours separated via wavelengths. Some statistical ways of measuring it are-
Conclusion:Dispersion in statistics refers to the measure of the variability of data or terms. Such variability may give random measurement errors where some of the instrumental measurements are found to be imprecise. It is a statistical way of describing how the terms are spread out in different data sets. The more sets of values, the more scattered data is found, and it is always directly proportional. This range of values can vary from 5 - 10 values to 1000 - 10,000 values. This spread of data is described by the range of descriptive range of statistics. Measures of Dispersion in statistics can be represented using a Dot Plot, Box Plot, and other different ways. Learn dispersion and other concepts in statistics as the introductory course of knowledgehut python with data science program. Abhresh is specialized as a corporate trainer, He has a decade of experience in technical training blended with virtual webinars and instructor-led session created courses, tutorials, and articles for organizations. He is also the founder of Nikasio.com, which offers multiple services in technical training, project consulting, content development, etc. |
What are the 5 Measures of Dispersion?
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