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![Coefficient of Quartile Deviation
A relative measure of dispersion based on the
quartile deviation is called the coefficient of
quartile deviation.
Formula:
Coefficient of Quartile Deviation =
𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛
𝑀𝑒𝑑𝑖𝑎𝑛
× 100
=
Q3 – Q1
Q3 + Q1
× 100 [By Simplification]](https://image.slidesharecdn.com/group-4-corrected-170506161930/75/Measure-of-Dispersion-in-statistics-46-2048.jpg)
Uploaded byMd. Mehadi Hassan Bappy
Measure of Dispersion in statistics
The document discusses the concept of dispersion in statistics, defining it as the extent to which a distribution is stretched or squeezed, with examples of variance, standard deviation, and interquartile range. It classifies measures of dispersion into absolute and relative categories, explaining their characteristics and applications, along with their merits and demerits. Various specific measures, calculations, and practical applications of these measures are also presented.
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Measure of Dispersion in statistics
- 1.
- 2. We are Group4 Tasnim Ansari Hridi (ID-09) Md. Mehedi Hassan Bappy (ID-21) Debanik Chakraborty (ID-25) Syed Ishtiak Uddin Ahmed (ID-31) Devasish Kaiser (ID-49)
- 3. Definition of Measureof Dispersion In statistics, dispersion (also called variability, scatter, or spread) is the extent to which a distribution is stretched or squeezed. Common examples of measures of statistical dispersion are the variance, standard deviation, and interquartile range.
- 4. Example Centre: Same Variation: Different Year2000: Close Dispersion Year 2015: Wide Dispersion Better Quality Data:Data ofYear 2000
- 5. Why Measure ofDispersion Serve as a basis for the control of the variability To compare the variability of two or more series
- 6. Facilitate the useof other statistical measures. Reliable Determine the reliability of an average Why Measure of Dispersion
- 7. Characteristics of anIdeal Measure of Dispersion Must be based on all observations of the data. It should be rigidly defined It should be easy to understand and calculate.
- 8. Must beleast affected by the sampling fluctuation. Must be easily subjected to further mathematical operations Characteristics of an Ideal Measure of Dispersion It should not be unduly affected by the extreme values.
- 9. Types of Measuresof Dispersion
- 10. Classification of Measures ofdispersion in Statistics
- 11.
- 12. Algebraic Measure ofDispersion × Mathematical way to calculate the measure of dispersion. Example: Calculation of Standard Deviation or Co-efficient of Variance by using numbers and formulas.
- 13. Characteristics of Algebraic Measureof Dispersion • Mathematical Way • Algebraic Variables are used • Numerical Figures are used here • Formulas & Equations are used
- 14. Graphical Measure ofDispersion × The way to calculate the measure of dispersion by figures and graphs. Example: Calculation of Dispersion among the heights of the students of a class from the average height using a graph.
- 15. Characteristics of Graphical Measureof Dispersion • It is a visual way of measuring dispersion • Graphs, figures are used • Sometimes, it cannot give the actual result • It helps the reader to have an idea about the dispersion practically at a glance
- 16. Absolute Measure ofDispersion Absolute Measure of Dispersion gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measures the dispersion in the same units as the units of original data. Absolute measures cannot be used to compare the variation of two or more series/ data set.
- 17. Classification of Algebraic Measureof Dispersion
- 18. Absolute Measure of Dispersion AbsoluteMeasure of Dispersion gives an idea about the amount of dispersion/ spread in a set of observations. These quantities measures the dispersion in the same units as the units of original data. Absolute measures cannot be used to compare the variation of two or more series/ data set.
- 19. Relative Measure ofDispersion These measures are a sort of ratio and are called coefficients. Each absolute measure of dispersion can be converted into its relative measure. It can be used to compare two or more data sets
- 20. Difference Between Absoluteand Relative Measure of Dispersion 3 This is calculated from original data These measure are calculated absolute measures 2 It is not expressed in terms of percentage It is expressed in terms of percentage 1 It has the variable unit It has no unit Absolute Measure Relative Measure
- 21. 6 There is nochange in variables and with the absolute measures. There is changes in variables with relative measures. 5 These measure cannot be used to compare the variation of two or more series These measure can be used to compare the variation of two or more series. 4 No use of ratio Use of ratio Absolute Measure Relative Measures
- 22.
- 23. Classification of Absolutemeasure Mean Deviation Quartile Deviation Standard Deviation Range
- 24.
- 25. Range The difference betweenthe maximum and minimum observations in the data set. R= H-L
- 26. 5, 10 ,15 , 20, 7, 9, 12 , 17 , 13 , 6 , 10 , 11 , 17 , 16 Range = H- L = 20- 5 = 15
- 27. Merits and Demeritsof Range Gives a quick answer Cannot be calculated in open ended distributions Affected by sampling fluctuations Changes from one sample to the next in population Gives a rough answer and is not based on all observationSimple and easy to understand
- 28.
- 29. Mean deviation The averageof the absolute values of deviation from the mean(median or mode) is called mean deviation. = 𝒇 | 𝒙 − 𝒙 | 𝑵
- 30. Merits of Meandeviation Simplifies calculations Can be calculated by mean, median and mode Is not affected by extreme measures Used to make healthy comparisons
- 31. Demerits of MeanDeviation Not reliable Mathematically illogical to assume all negatives as positives Not suitable for comparing series
- 32.
- 33. Quartile Deviation The halfdistance between 75th percentile i.e. 3rd quartile (Q1) and 25th percentile i.e. 1st quartile (Q3) is Quartile deviation or Interquartile range. Q.D = Q3 – Q1 𝟐
- 34. Has better resultthan range mode. Is not affected by extreme items Merits of Quartile Deviation
- 35. Demerits of QuartileDeviation It is completelydependent on thecentral items. All the items of the frequencydistribution are not given equal importance in finding the values of Q1 and Q3 Because it does not take into accountall the items of the series, considered to be inaccurate.
- 36.
- 37. Standard Deviation Standard deviationis calculated as the square root of average of squared deviations taken from actual mean. It is also called root mean square deviation. = √ 𝒙− 𝒙 𝟐 𝒏
- 38.
- 39. Merits of standarddeviation It takes intoaccount all the items and is capableof future algebraic treatment andstatistical analysis. It is possible to calculatestandard deviationfor two or more series This measure is most suitable for makingcomparisonsamong two or more series about variability.
- 40. Demerits of StandardDeviation It is difficult to compute. It assigns more weights to extreme itemsand less weights to items that are nearer to mean.
- 41. Classifications of Relative Measuresof Dispersion
- 42. Chart of classification Relative Measure Coefficientof Range Coefficient of Quartile Deviation Coefficient of Mean Deviation Coefficient of Variation
- 43.
- 44. Coefficient of Range Themeasure of the distribution based on range is the coefficient of range also known as range coefficient of dispersion. Formula: Coefficient of Range= 𝑅𝑎𝑛𝑔𝑒 𝐻𝑖𝑔ℎ𝑒𝑠𝑡 𝑉𝑎𝑙𝑢𝑒+𝐿𝑜𝑤𝑒𝑠𝑡 𝑣𝑎𝑙𝑢𝑒 × 100
- 45.
- 46. Coefficient of QuartileDeviation A relative measure of dispersion based on the quartile deviation is called the coefficient of quartile deviation. Formula: Coefficient of Quartile Deviation = 𝑄𝑢𝑎𝑟𝑡𝑖𝑙𝑒 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑑𝑖𝑎𝑛 × 100 = Q3 – Q1 Q3 + Q1 × 100 [By Simplification]
- 47. Merits & Demeritsof Coefficient of Quartile Deviation Merits 1. Easily understood 2. Not much Mathematical Difficulties 3. Better Result than Coefficient of Range Sampling fluctuation Ignorance of last 25% of data sets. Values being irregular Demerits
- 48.
- 49. Coefficient of MeanDeviation The relative measure of dispersion we get by dividing Mean Deviation by Mean or Median, is called Coefficient of Mean Deviation. Formula: Coefficient of MD= 𝑀𝑒𝑎𝑛 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑑𝑖𝑎𝑛 𝑜𝑟 𝑀𝑒𝑎𝑛 × 100
- 50. Merits & Demeritsof Coefficient of Mean Deviation Merits 1. Better Result than Range & Quartile Coefficient. 2. Least sampling fluctuation. 3. Rigidly defined. Fractional Average. Cannot be used for sociological studies Less reliable than Coefficient of Variation Demerits
- 51.
- 52. Coefficient of Variation Coefficientof Variation is a measure of spread that describes the amount of variability relative to the mean. Formula: Coefficient of Variation= 𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝐷𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑀𝑒𝑎𝑛 × 100
- 53. Merits & Demeritsof Coefficient of Variation Merits 1. Best one 2. Most appropriate one 3. Based on Mean and Standard Deviation 4. COV is dimensionless or non- unitized It is impossible to calculate if Mean is 0 It is difficult to calculate if the values are both positive and negative numbers & if the mean is close to 0. Demerits
- 54. Practical Uses ofCoefficient of Variance INVESTMENT ANALYSIS STOCK MARKET RISK EVALUATION COMBINED STANDARD DEVIATION OF SEVERAL GROUPS PERFORMANCES OF TWO PLAYERS INDUSTRIES & FACTORIES
- 55.
- 56. Coefficient of range Let1,2,4,6,7 is a set of values of a distribution. Here, Highest Value, XH=7 & Lowest Value, XL=1 So, Range, R= 7-1 = 6 Now, Coefficient of Range = 𝐑 XH + XL × 100 = 𝟔 𝟕+𝟏 × 100 =75%
- 57. Coefficient of Quartile deviationLetthe number of students in 5 classes are 110, 150, 180, 190, 240 is a set of values. Here, Q1= size of 𝐍+𝟏 𝟒 th item = 130 And, Q3 = size of 𝟑(𝐍+𝟏) 𝟒 th item = 215 So, Coefficient of Quartile Deviation =Q3 – Q1 Q3 + Q1 × 100 = 215−130 215+130 × 100= 24.64 %
- 58. Coefficient of MeanDeviation Let the ages of 5 boys in a class is 12, 14, 14, 15, 18. So their Mean, 𝐱 = 𝟏𝟐+𝟏𝟒+𝟏𝟒+𝟏𝟓+𝟏𝟖 𝟓 = 14.6 Mean Deviation, MD = | 𝒙 − 𝒙 | 𝑵 =|12−14.6| + |14−14.6| + |14− 14.6|+ |15−14.6| + |18−14.6| 𝟓 = 1.52 Now, the Coefficient of MD= 𝐌𝐃 𝐱 × 𝟏𝟎𝟎 = 𝟏.𝟓𝟐 𝟏𝟒.𝟔 × 𝟏𝟎𝟎 = 10.41%
- 59. Coefficient of VariationSuppose thereturns on an investment for 4 years is Tk.1000, Tk.3000, Tk.4500 & Tk.5000. So, Mean, 𝐱 = 3375 Standard Deviation, SD = 1796.99 So, Coefficient of Variation, CV= 𝐒𝐃 𝐱 × 100 = 𝟏𝟕𝟗𝟔.𝟗𝟗 𝟑𝟑𝟕𝟓 × 100 = 53.24%
- 60. The daily saleof sugar in a certain grocery shop is given below : Monday Tuesday Wednesday Thursday Friday Saturday 75 kg 120 kg 12 kg 50 kg 70.5 kg 140.5 kg respectively.
- 61. “ No of Dayssale of sugar Monday 60 Tuesday 120 Wednesday 10 Thursday 50 Friday 70 Saturday 140 𝜮 𝒐𝒇 𝑫𝒂𝒚𝒔 = 𝟔 𝜮𝒙 = 𝟒𝟓𝟎 Mean, 𝑥 = 𝑥 𝑛 = 4𝟓𝟎 6 = 7𝟓
- 62. “ x 𝒙 𝟐 603600 120 14400 10 100 50 2500 70 4900 140 19600 𝜮𝒙 = 𝟒𝟓𝟎 𝜮𝒙 𝟐 = 45100 Standard deviation: 𝝈 = 𝜮𝒙 𝟐 𝒏 − 𝜮𝒙 𝒏 𝟐 = 𝟒𝟓𝟏𝟎𝟎 𝟔 − 𝟒𝟓𝟎 𝟔 𝟐 = 𝟕𝟓𝟏𝟔. 𝟔𝟔 − 𝟓𝟔𝟐𝟓 = 𝟒𝟑. 𝟒𝟗
- 63. Quartile Deviation The marksof 7 students in Mathematics result are given below : 70, 85, 92,68, 75, 96, 84 Find out- • First Quartile Deviation • Third Quartile Deviation
- 64. Quartile deviation × Firstquartile 𝐐 𝟏 = 𝐬𝐢𝐳𝐞 𝐨𝐟 𝐧 + 𝟏 𝟒 𝐭𝐡 𝐢𝐭𝐞𝐦 = size of 𝟕+𝟏 𝟒 𝐭𝐡 𝐢𝐭𝐞𝐦 = size of 2nd item. = 70 ×Third Quartile 𝑸 𝟑 = 𝒔𝒊𝒛𝒆 𝒐𝒇 𝟑 𝒏 + 𝟏 𝒕𝒉 𝟒 𝒊𝒕𝒆𝒎 = size of 𝟑 𝟕+𝟏 𝒕𝒉 𝟒 𝒊𝒕𝒆𝒎 = size of 6th item =92 Arranging the data in ascending order we get, 68,70,75,84,85,92,96
- 65.