# measures of dispersion examples

& = 0.003767 \\ Try out our free online statistics calculators if you’re looking for some help finding probabilities, p-values, critical values, sample sizes, expected values, summary statistics, or correlation coefficients. o Understand the difference between measures of dispersion for populations and for samples The median splits the dataset into two halves. First, we have to calculate the arithmetic mean: $$X =\cfrac {(12 + 4 + 23 + 8 + 9 + 16)}{6} = 12\%$$, \begin{align*} For example, if the standard deviation is large then there are large differences between individual data points. These are the range, variance, absolute deviation and the standard deviation. And the standard deviation is simply the square root of variance. The median of the lower half is the lower quartile (Q1) and the median of the upper half is the upper quartile (Q3). Consequently, the mean may not be representative of the data. Range 2. Example. (Definition & Example). And the formula to find the variance of a sample (denoted as s2) is: The standard deviation is the square root of the variance. Thus; { \sigma }^{ 2 }=\frac { \left\{ \sum { { \left( { X }_{ i }-\mu \right) }^{ 2 } } \right\} }{ N } $$. Solution. Absolute measures of dispersion are expressed in the unit of Variable itself. Older versions of Excel had a single function for quartile, =QUARTILE() and that was identical to the =QUARTILE.INC() function in the current versions. The table shows marks (out of 10) obtained by 20 people in a test. Required fields are marked *. This is necessary so as to remove biasThe sample standard deviation, S, is simply the square root of the sample varianceExample 4Assume that the returns realized in example 2 above were sampled from a population comprising 100 returns. One such measure is popularly called as dispersion or variation. Mean deviation from median. We will only discuss three of the four relative measures of dispersion in this article: coefficients of range, quartile deviation, and variation. 4. CFA® and Chartered Financial Analyst® are registered trademarks owned by CFA Institute. Dispersion … It is the difference between the highest and the lowest scores in a set of data i.e. The sample variance, S2, is the measure of dispersion that applies when we are working with a sample as opposed to a population. (The two have been distinguished here)S2 = {Σ(Xi – X? The quartile boundaries would lie between two values in our data set. Thus, the range is 98 – 58 =, Thus, the interquartile range is 91 – 75.5 =, The formula to find the variance of a population (denoted as, The formula to find the standard deviation of a population (denoted as, And the formula to find the standard deviation of a sample (denoted as, Measures of Central Tendency: Definition & Examples. The variance of these … We measure “spread” using range, interquartile range, variance, and standard deviation. The interquartile range is the middle half of … o Population variance. & = 0.0672 Your email address will not be published. { \sigma }^{ 2 } & =\frac { \left\{ { \left( 12-32 \right) }^{ 2 }+{ \left( 13-32 \right) }^{ 2 }+{ \left( 54-32 \right) }^{ 2 }+{ \left( 56-32 \right) }^{ 2 }+{ \left( 25-32 \right) }^{ 2 } \right\} }{ 5 } \\ Suppose we have this dataset of final math exam scores for 20 students: The largest value is 98. Like, Kilograms, Rupees, Centimeters, Marks etc. Thus, the range is 98 – 58 = 40. Working with data from example 2 above, the variance will be calculated as follows:$$ \begin{align*} Calculate and interpret 1) a range and a mean absolute deviation and 2) the variance and standard deviation of a population and of a sample. You’re kind of an adventurous person and you don’t have too many capricious demands regarding where you want to live next. \text{MAD} & = \cfrac {\left\{ |12 – 12|+ |4 – 12| + |23 – 12| + |8 – 12| + |9 – 12| + |16 – 12| \right\}} {6} \\ R = 28 −18 = 10 Years . Measures of Dispersion A measure of spread, sometimes also called a measure of dispersion, is used to describe the variability in a sample or population. Find the median. This is from the Oxford English Dictionary: The term came to English from the German (where it lived before that I do not know) and seems to have emerged as a way of explaining aggregated data, or data which one has subjected to the process of removing information in order to gain information. Relative Measure of Dispersion; Absolute Measure of Dispersion. In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. & =\cfrac {1870}{5} = 374 \\ $$\text{Range} = \text{maximum value} – \text{minimum value}$$, Consider the following scores of 10 CFA Level 1 candidates, 78   56   67   51   43   89   57   67   78   50. You may notice that all the relative measures of dispersion are called coefficients. Compute the sample mean and the corresponding sample variance. 2. This is necessary so as to remove bias, The sample standard deviation, S, is simply the square root of the sample variance. Range R = 13.67 The range is a very simplistic measure and does not use all the scores in the data set therefore it can be distorted by a very high or low score that does not reflect the range of most of the other scores in between those two points. Assume that the returns realized in example 2 above were sampled from a population comprising 100 returns. By focusing on the mean, w… Unit-II MEASURES OF CENTRAL TENDENCY AND DISPERSION Relation between Mean, Median and Mode: − = 3( – ) Range of ungrouped data: The range of a set of data is the difference between the highest and lowest values in the set. The Interquartile Range (IQR) . All Rights ReservedCFA Institute does not endorse, promote or warrant the accuracy or quality of AnalystPrep. The scatterness or variation of observations from their average are called the dispersion. It is a relative measure of dispersion and is based on the value of range. In both the above examples, Excel would calculate the quartile values by extrapolation because there are not enough data points. ©AnalystPrep. o Measure of dispersion. The largest value is 98. & = 45.20(\%^2) \\ { \sigma }^{ 2 } & =\frac { \left\{ { \left( 12-12 \right) }^{ 2 }+{ \left( 4-12 \right) }^{ 2 }+{ \left( 23-12 \right) }^{ 2 }+{ \left( 8-12 \right) }^{ 2 }+{ \left( 9-12 \right) }^{ 2 }+{ \left( 16-12 \right) }^{ 2 } \right\} }{ 6 } \\ You subtract the lowest score in the data set from the highest score to give the range. Measures of dispersion are used to describe the variability or spread in a sample or population. Analysts use the standard deviation to interpret returns as opposed to the variance since it is much easier to comprehend. The standard deviation is 0.0037671/2 = 0.06137 or 6.14%. There are different measures of dispersion like the range, the quartile deviation, the mean deviation and the standard deviation. The range is the difference between the largest and smallest value in a dataset. Measures of dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Imagine our technology has advanced so much that we can freely travel in space. The formula to find the variance of a population (denoted as σ2) is: where μ is the population mean, xi is the ith element from the population, N is the population size, and Σ is just a fancy symbol that means “sum.”. Looking for help with a homework or test question? & = 5\% \\ Example: Cheryl took 7 math tests in one marking period. Let’s start with a funny (and not so realistic) example. & = 0.00452 \\ Arrange the values from smallest to largest. The only important thing for … The rangeis the difference between the largest and smallest value in a dataset. Compute the population standard deviation assuming this is complete data from a certain population. The sample variance, S2, is the measure of dispersion that applies when we are working with a sample as opposed to a population. Quartiles are values that split up a dataset into four equal parts. o Degrees of freedom. o Use the variance or standard deviation to characterize the spread of data. Third Variable Problem: Definition & Example, What is Cochran’s Q Test? (1) Absolute Measures 1. . The Range. Here is how to find the interquartile range of the following dataset of exam scores: 1. They are: 1. Mark (x) The formula to find the standard deviation of a population (denoted as σ ) is: And the formula to find the standard deviation of a sample (denoted as s) is: Your email address will not be published. You compute […] Range R = L –S. There are four commonly used measures to indicate the variability (or dispersion) within a set of measures. Statology is a site that makes learning statistics easy by explaining topics in simple and straightforward ways. Cycles are trends or patterns that may be exhibited by the securities market,... Monte Carlo simulation and historical simulation are both methods that can be used... 3,000 CFA® Exam Practice Questions offered by AnalystPrep – QBank, Mock Exams, Study Notes, and Video Lessons, 3,000 FRM Practice Questions – QBank, Mock Exams, and Study Notes. Where the “center” value is located. Measures of Dispersion The Range of a set of data is the largest measurement minus the smallest measurement. The study of dispersion will enables us to know whether a series is homogeneous (where all the observations remains around the central value) or the observations is heterogeneous (there will be variations in the observations around the central value like 1, 50, 20, 28 etc., where the central value is 33). Quartile Deviation 3. \end{align*} $$, Interpretation: It means that on average, an individual return deviates 5% from the mean return of 12%. Remember that the sum of deviations from the arithmetic mean is always zero and that’s why we are using the absolute values. For example, suppose we have the following dataset with incomes for ten people: The range is 2,468,000, but the interquartile range is 34,000, which is a much better indication of how spread out the incomes actually are. o Sample variance. Statology Study is the ultimate online statistics study guide that helps you understand all of the core concepts taught in any elementary statistics course and makes your life so much easier as a student. This example of one of the relative measures of dispersion is also called as Range Co-efficie… Thus, the interquartile range is 91 – 75.5 = 15.5, The interquartile range more resistant to outliers compared to the range, which can make it a better metric to use to measure “spread.”. Measures of central dispersion show how “spread out” the elements of a data set are from the mean.$$ \begin{align*} Thus, the range is 98 – 58 = 40. The measure of dispersion shows the homogeneity or the heterogeneity of the distribution of the observations. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range. if you split it down the middle, each half would look roughly equal) and there are no outliers (i.e. The interquartile range is equal to Q3 – Q1. \end{align*} $$,$$ \begin{align*} Relative Dispersion The actual variation or dispersion, determine from standard deviation or other measures is called absolute dispersion, now the relative dispersion is For Example, Relative dispersion It is a measurement of the degree by which an observed variable deviates from its … A measure of statistical dispersion is a nonnegative real number that is zero if all the data are the same and increases as the data become more diverse. The population variance, denoted by σ2, is the average of the squared deviations from the mean. Three of the most commonly used measures of central dispersion include the following: Range Variance Standard deviation Range The range of a data set is the difference between the largest value and the smallest value. o Sample standard deviation. \end{align*} $$. Example Calculate the range for the data for Quarterback A and Quarterback B in the example above. Key Terms . Measures of dispersion measure how spread out a set of data is. o Population standard deviation. The smallest value is 58. Characteristics of a good measure of dispersion We recommend using Chegg Study to get step-by-step solutions from experts in your field. Start studying for CFA® exams right away. . An absolute measure of dispersion contains the same unit as the original data set. (The two have been distinguished here),$$ { S }^{ 2 }=\frac { \left\{ \sum { { \left( { X }_{ i }- \bar { X } \right) }^{ 2 } } \right\} }{ n-1 } , Note that we are dividing by n – 1. Relative measures of dispersion are obtained as ratios or percentages of the average. As the name suggests, the measure of dispersion shows the scatterings of the data. It’s the most common way to measure how “spread out” data values are. The variance is a common way to measure how spread out data values are. m means the mean of the data. How “spread out” the values are. 6 Investment analysts attain the following returns on six different investments: Calculate the mean absolute deviation and interpret it. Smallest value S = 18. Dispersion (a.k.a., variability, scatter, or spread)) characterizes how stretched or squeezed of the data. In this case, Q1 is the average of the middle two values in the lower half of the data set (75.5) and Q3 is the average of the middle two values in the upper half of the data set(91). o Variance. The interquartile range is the difference between the first quartile and the third quartile in a dataset. The smallest value is 58. The scores of all the students in section A are ranging from to ; The range is a simple measure of dispersion. For example, suppose we have the following distribution that shows the salaries of individuals in a certain town: Since this distribution is fairly symmetrical (i.e. These are also known as ‘Coefficient of dispersion’ 3. These are pure numbers or percentages totally independent of the units of measurements. o Standard deviation. Suppose we have this dataset of final math exam scores for 20 students: The largest value is 98. You’re bored of living on Earth and decide to take off towards another planet. Thus, the average variation from the mean (0.12) is 0.003767. 58, 66, 71, 73, 74, 77, 78, 82, 84, 85, 88, 88, 88, 90, 90, 92, 92, 94, 96, 98, 2. There are four Absolute Measures of Dispersion in Statistics: Range; Quartile Deviation; Mean Deviation; Standard Deviation; Range. 29.2 DEFINITION OF VARIOUS MEASURES OF DISPERSION (a)Range : In the above cited example, we observe that (i) the scores of all the students in section A are ranging from 6 to 35; (ii) the scores of the students in section B are ranging from 15 to 25. They are important because they give us an idea of how well the measures of central tendency represent the data. Since they are devoid of a specific unit, the comparison between different series is hence possible. Learn more about us. Solution Here Largest value L = 28. In the above cited example, we observe that. The concept of relative measures of dispersion overcomes this limitation. For every absolute measure of dispersion, there is a relative measure. Surprisingly, the term statistic first came into use as late as 1817. Example 8.3 The range of a set of data is 13.67 and the largest value is 70.08. Specially it fails to give any idea about the scatter of the values of items … Absolute measures of dispersion indicate the amount of variation in a set of values; in terms of units of observations. We’ve started colonizing and populating new planets. The smallest value is 58. \end{align*}. Find the smallest value. )2}/n – 1Note that we are dividing by n – 1. We often measure the “center” using the mean and median. Example 8.2 Find the range of the following distribution. & =\cfrac {30}{6} \\ The heights in cm of a group of first year biology students were recorded. For example, absolute dispersion in data related to age and weight is not comparable because age is measured in terms of years but the weight is measured in terms of the kilogram. For example, when rainfall data is made available for different days in mm, any absolute measures of dispersion give the variation in rainfall in mm. In this case, the outlier income of person J causes the range to be extremely large and makes it a poor indicator of “spread” for these incomes. It is a measure of dispersion that represents the average of the absolute values of the deviations of individual observations from the arithmetic mean. Standard deviation. When we analyze a dataset, we often care about two things: 1. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range and other Percentiles. Absolute dispersion method expresses the variations in terms of the average of deviations of observations like standard or means deviations. It tells the variation of the data from one another and gives a clear idea about the distribution of the data. The minimum number of completions for Quarterback A is 19, the maximum is 37. (2) Relative Measures 1. In this lesson, you will read about the following measures of dispersion: Range. $$\mu =\cfrac {(12 + 13 + \cdots +25)}{5} =\cfrac {160}{5} = 32$$, \begin{align*} Objectives . Lets look at the first of the relative measures of dispersion. Mean deviation from mean. S & = 0.00452^{\frac {1}{2}} \\ { S }^{ 2 } & =\frac { \left\{ { \left( 12-12 \right) }^{ 2 }+{ \left( 4-12 \right) }^{ 2 }+{ \left( 23-12 \right) }^{ 2 }+{ \left( 8-12 \right) }^{ 2 }+{ \left( 9-12 \right) }^{ 2 }+{ \left( 16-12 \right) }^{ 2 } \right\} }{ 5 } \\ An example of aggregating data is the simple process of finding the mean of a variable such as height or weight. Usually we work with samples, not populations. (In this case, it’s the average of the middle two values), 58, 66, 71, 73, 74, 77, 78, 82, 84, 85 (MEDIAN) 88, 88, 88, 90, 90, 92, 92, 94, 96, 98, 3. Slide 77 Measures of Dispersion  There are three main measures of dispersion: – The range – The Interquartile range (IQR) – Variance / standard deviation 8. The Important measures of dispersion can represent a series only as best as a single figure can, but it certainly cannot reveal the entire story of any phenomenon under study. The formulae for the variance and standard deviation are given below. It is usually used in conjunction with a measure of central tendency, such as the mean or median, to provide an overall description of a set of data. Thus, \text{MAD} \frac { \sum { |{ X }_{ i }-\bar { X } | } }{ n } . Definition of Various Measures of Dispersion Range. \end{align*}. & = 37.67(\%^2) \\ Variance. They are usually used in conjunction with measures of central tendency such as the mean and the median. Variance and Standard Deviation. 2. no extremely high salaries), the mean will do a good job of describing this dataset. 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Returns on six different investments: Calculate the mean may not be of. Sum of deviations of individual observations from their average are called the dispersion and interpret it absolute deviation and corresponding.: the largest and smallest value in a set of data in some relative terms or in percentage or!: Calculate the mean of a specific unit, the quartile deviation ; range the. ( 0.12 ) is 0.003767 the values of the relative measures of ’! Good job of describing this dataset to the variance of these … When we analyze a....: Calculate the mean of a good job of describing this dataset following dataset final! – 1 of completions for Quarterback a is 19, the average values that split up a.... In conjunction with measures of dispersion are obtained as ratios or percentages of the squared deviations from the deviation. The deviations of individual observations from their average are called coefficients quality of AnalystPrep expressed! Travel in space measures express the scattering of data is the difference between the largest value is 70.08 a of. No extremely high salaries ), the range, variance, denoted by σ2, is difference! The data set dataset, we observe that is popularly called as or! The rangeis the difference between the largest and smallest value in a dataset four. A and Quarterback B in the example above because they give us an idea how... ( out of 10 ) obtained by 20 people in a test table shows (. The sum of deviations of observations from the highest score to give any idea about the scatter of following. Good measure of dispersion in Statistics: range ; quartile deviation, and interquartile range variability! Boundaries would lie between two values in our data set from the arithmetic mean variation. Values of the absolute values of items … the range of a group first... Commonly used measures to indicate the amount of variation in a set of data they are usually used conjunction... Not endorse, promote or warrant the accuracy or quality of AnalystPrep dispersion that represents average... Mean ( 0.12 ) is 0.003767 is 0.0037671/2 = 0.06137 or 6.14 % be representative of the squared from.