TOPIC. Directorate of Distance Education (DDE) Program: P.G. Subject: Geography. Class: Previous. Paper: V(a)

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1 TOPIC Measures Dispersions: Range, Quartile Deviation, Mean Deviation, Standard Deviation and Lorenz Curve-Their Merits and Limitations and Specific Uses Prepared By: Dr. Kanhaiya Lal, Assistant Pressor, DDE, K.U.K.

2 1.1 Introduction P The measure central tendency any series or data distribution summarises it into single representative form which h are useful inmany respect but it fails toaccountt the general distribution pattern data. Thus any conclusion only based on central tendency may be misleading. Dispersion tries to fill this gap by stressing on the pattern data distribution in any series. Dispersion can prove very effective in association with central tendency in making any statistical decision.

3 2.1 Objective Any data series in statistics is usually represented by some measure central tendency. As the uniformity a data series increases the value central measure become more representative. So, it is always meaningful to interpret central measure a data series keeping in mind the dispersion present in that series.here, an insight has been given on what dispersion is, what are its importance, what are different types dispersioni and how some important t dispersionsi can be calculated, their merits demerits, etc.

4 3.1 What is Measure Dispersion? Measure dispersion has two terms, measure and dispersion. Measure here means specific method estimation while dispersion term means deviation or difference or spread certain values from their hi central value. Various statisticians have defined it variously. According to Simpson and Kafka, The measurement the scatterness the mass figures in a series about an average is called a measure variation or dispersion.

5 According to W. I. King, the term dispersion is used to indicate the facts that within a given group, the items differ from one another in size or in other words, there is lack uniformity in their sizes. Spiegel defines it as, The degree to which numerical data tend to spread about an average value is called the variation or dispersion the data. Similarly, to A. L. Bowley, Dispersion is a measure variation the items. All these dfiiti definitions shows dispersioni as a spread in dt data series with respect to some representative reference value the data series.

6 Dispersion is a directionless average estimate variations in individual observations from a central measure to shows the degree nonuniformity in our data distribution. Dispersion is regarded as directionless measure since it only shows the magnitude deviation from central measure, whether deviation is positive or negative it does describe. It is also commonly known as scatter or spread or width or variation or average second order. In real problems it is very rare that all observations have the same values as that its central tendency.

7 For example, two cricketer with same 65 average score per year considered for a span 8 years, one with scores close to average value in all years while the other with very high score in some year and very poor in most year, cannot be rated at same level. l Former is morereliable batsman. Similarly, only average annual income India measured as Gross Domestic Product (GDP) does not mean each citizen has this income or income near to it or merely by increasing GDP all citizens can be equally developed. The central value gives nearly good results as long as our dt data distributions ib ti are normal i.e. our most observations are close to it.

8 4.1 Objectives Measure Dispersion It is the value dispersion which says how much reliable a central tendency is? Usually, a small value dispersion indicates that measure central tendency is more reliable j representative data series and vice versa. Many powerful analytical tools in statistics such as correlation analysis, the testing hypothesis, analysis variance, the statistical quality control, regression analysis are based on measure variation iti one kind or another (Gupta, 2004).

9 The degree data spread also helps in analysing importance different components a system. For example, if we take agricultural productivity to depend upon input fertilizer, hybrid seeds, irrigation, insecticides, id pesticides id and machinery. The cause any abruptness in productivity can be analysed by comparing central measure different inputs with its variation and thus it helps in taking corrective measures. Measure dispersion is also used to compare uniformity different data like income, temperature, rainfall, weight, height, etc.

10 5.1 Properties a Good Measure Dispersion P Like a good measure central tendency the good measure dispersion should also have similar characteristics. ti A good measure dispersion should be clearly defined so that there should not be any scope subjectivity in computation as well as its interpretation. It should be easy to compute, understand and interpret and further, all individual observations should be used in its estimation and also it should be free from any biasness or biasness due to any extreme value.

11 Since dispersion is also used to estimate many statistical complex properties data soa dispersion should be easily applicable in any algebraic operations. P Finally, such measure dispersioni should be least affected by sampling or have high degree sampling stability. 6.1 Measuring Dispersion Measure dispersion is always a real number. If all values individual observations are identical with central tendency then dispersion is alwaysa zero and as deviation in observation from central tendency increases, dispersion also increases but it never become negative.

12 Further a measure dispersion is absolute or relative. P In the case absolute measure dispersion the unit individual observations and the unit dispersion remains thesame.this dispersioni isuseful in comparing different data set in same unit and with same average size. In relative dispersion unit original observation becomes irrelevant since it is a ratio absolute dispersion to some central tendency. It is useful in comparison data series in different units or with different size data. Some common measures dispersioni are: Range, Quartile Deviation, Mean Deviation, Standard Deviation and Lorenz Curve.

13 Range Range is the quickest and simplest measure dispersion. Being a positional measure it accounts only the difference between the highest and the lowest observation in any data series and does not take into account all individual observations and so it is quickest but at the same time a rough or crude measure dispersion. Rangealwayshasthesameunitasoriginalobservations have.

14 Symbolically for ungrouped data it is represented by: P R=H L Where, R = Range H = Highest value in the observation L = Lowest value in the observation In the case a grouped data range is estimated by taking the difference upper limit highest class interval and lower limit lowest class interval. Also in such case the difference mid values highest and lowest class interval are used as range.

15 In case open ended grouped data, the width adjacentclass is used, i.e. it is assumed that highest/lowest class has same width as that adjacent class, to estimate upper/lower limit highest/ lowest class. Above method is equally applicable with grouped data having equal or unequal class intervals. To make it free from unit, coefficient range is sometimes calculated. Coefficient Range = (H L)/ (H+L) Where, H = Highest value in the observation L =Lowest value in the observation

16 Merits and Demerits Range It is simplest and easiest to compute, understand and interpret. It is a crude measure since it does not take into account all individual observation. Addition/removal a single extreme value at upper/lower end data series can alter the range to great extent. In the case open ended grouped class true estimation range becomes impossible. Sampling may affect it adversely and its value may vary markedly from sample to sample.

17 Since it does not take into account any observation between the highest and the lowest value and so it tells nothing about actual distribution data between these two extremes. It gives nearly good result only if sample size is sufficiently large and data are fairly continuous or regular. In grouped data it rarely happens that data distribution actually touches upper/lower class limit highest/lowest class interval. So range calculated using these values is usually not accurate. Range if calculated using mid values highesth and lowest class try to resolve this problem to some extent but only if data are fairly and sufficiently distributed in these two classes.

18 The value range changes with the transformation scale. For example between Rs.1 and Rs.10 the difference is 9 only but if we take paisa as unit this difference will become Quartile Deviation Quartile deviation is another positional and absolute measure data dispersion in any series which try to minimise the error range as a measure dispersion. Unlikerangeitavoidstheuseextremevaluesandin its place uses the difference first and third quartile asa measure dispersion.

19 It is also called semi interquartile range or semi quartile range or interquartile range. P Thus, this measure dispersion ignores fifty per cent (first 25 per cent and last 25 per cent) observations. Symbolically it is estimated using following formula, Quartile Deviation (Q D )=(Q 3 Q 1 )/2 Where, Q 3 =thirdquartile Q 1 = first quartile Thus quartile deviation gives the average amount by which two quartiles differ from the median (Gupta, 2004).

20 Furtherinanysymmetricalornon skewed or normal data distribution median (Q 2 ) plus/minus Q D exactly covers 50 per cent the data distribution on either side the median since in such case Q 3 Q 2 = Q 2 Q 1 or conversely Q 2 + Q D = Q 3 and Q 2 Q D = Q 1. P In reality, rarely a business, economic or social data are perfectly symmetrical. So quartile deviation as a measure dispersion should be preferably used only where data distribution are moderately skewed. A lower/higher value quartile deviation in less skewed dt data reflects thatt more/less distributions ib ti are around the median value.

21 A relative counterpart quartile deviation is called coefficient quartile deviation and it is represented by following formula, P Coefficient i Q D = (Q 3 Q 1 )/ (Q 3 + Q 1 ) Merits and Demerits Quartile Deviation j Quartile deviation considers only middle 50 per cent observations and so it is not affected by extreme values as in the case range. It also, like range, not considers all observations in estimating dispersion and so its result may be misleading.

22 Since it not considers extreme values and so it is useful in estimating dispersion in grouped data with open ended class. P Sampling may adversely affect its estimation like in the case range. Quartile deviation as a measure dispersion is most reliable only with symmetrical data series. Unfortunately, in social sciences most data distributions are generally asymmetrical in nature. So, its use in social sciences is usually limited to data which are moderately skewed.

23 Mean Deviation P A proper approach to the measurement dispersion or variability would require that all the values in a series are taken into consideration. One themethods doing it is through average deviation or mean deviation. Asthevery name indicates, this measure is an average or the mean the deviations the values from a fixed point, which is usually the arithmetic mean and sometimes the median (Bhat and Mahmood, 1993). Theoretically there is an advantage in tki taking the deviations from median because the sum deviations items from median is minimum when sign are ignored.

24 However, in practice, the arithmetic mean is more frequently used in calculating the value average deviation and this is the reason why it is more commonly called mean deviation (Gupta, 2004). Mean deviations are computed first by summing the absolute differences each observation from mean and then dividing it by number observations. The sign deviations are ignored i.e. only absolute values are used, since sum deviations from mean is always zero (Hooda, 2002; Levin and Fox, 2006). By contrast t the sum absolute deviations tends to become larger as the variability a distribution increases (Levin and Fox, 2006).

25 Following mathematical formula is used to estimate mean deviation a data series, Case I: Ungrouped data series Mean Deviation (MD) = Where, = sum absolute deviations from mean = total number observations Case : Grouped data series Mean Deviation (MD) = Where, = frequency observations =mid value each class Other symbols have same meaning as in the case I.

26 Merits and Demerits Mean Deviation P Meandeviationisanabsolutemeasuredispersionbut unlike range or quartile deviation it is a calculative measure dispersioni which h gives it some advantages over them. It took into account all values observations in estimating dispersion and truly tries to give scatter in data. In most cases mean deviation measures dispersion from mean. Mean is not only easy to compute but also easy to understand d and even those who arenot specialised ili di in statistics can understand it and a dispersion based on it is also appealing to them.

27 Mean deviation is an absolute measure dispersion so a comparison against data series represented in different unit is difficult. P Also a change in unit or scale measurement changes the value dispersion. As meanisleastaffected by sampling so use mean in measuring dispersion also retains this property. But if median is used then it holds good or nearly good as long as data series are symmetrical or moderately skewed. Since identification mode isnot possible with all dt data series so it cannot be used frequently in measuring mean deviation.

28 Mean always has a tendency upward biasness and also accurate mean identification in open ended grouped data is not possible. This problem also percolates in mean deviation where deviation is measured using mean. In estimating mean deviation we take into account only absolute value (only magnitude and not sign) deviation from central tendency. It is mathematically a unsound practice and so it limits it s further algebraic use Standard Deviation This method measuring dispersioni is most widely acclaimed by statisticians since it nearly have all properties a good measure dispersion.

29 This method is not based on absolute value deviation individual data from the mean so it is algebraically tenable. P This problem in standard d deviation has been overcome by squaring the individual deviation from mean. These squared individual deviations are summed up, then averaged and finally its square root has been identified as a measure standard deviation. This is why it is also known as root mean square deviation.

30 Mathematically following formula represents the concept standard deviation, Case I: Ungrouped data series Where, S 2 = variance; S 2 = = = standard deviation =sum square deviations from mean = total number observations

31 Case : Grouped data series S 2 = = Where, S 2 = variance; = standard deviation = frequency observations =mid value each class

32 Merits and Demerits Standard Deviation Its most important beauty is that it is free from the compulsion taking only absolute value in estimating mean deviation. So it is frequently applicable in different algebraicoperations. P It took into account allindividualobservationsandso any slight variation in any observations automatically got representation in standard deviation. Through variance it easily reflects the aberration in data series. It is the basis relative measure dispersion coefficient variation (CV).

33 It is also an absolute measure dispersion and so comparisons data series in different units measurement are not tenable. P Its value changes if unit measurement changed. In a normal distribution, data are symmetrically j distributed around mean(mean, median or mode all become identical) and mean σ covers per cent observations; mean 2σ covers per cent observations and mean 3σ covers per cent observations. This property is useful in dividing a data series into suitable groupsor class.

34 Lorenz Curve Lorenz curve is a graphical way showing dispersion in any data distribution. It was developed by Max O. Lorenz in 1905 as a representative distribution in wealth. It is a useful measure to show the distribution any phenomena and it is frequently used to show distribution wealth, assets, biodiversity, land holdings, etc. It assumes thatt in the case equal distribution ib ti n per cent should have n per cent share in the total.

35 It is graphically constructed using cumulative frequencies parameterand its distribution, one along y axis and other along x axis, respectively, which together add up individually to 100 per cent. The line making an angle 45 from horizontal shows the case equal distribution. As any curved line moves away from this line equal distribution the inequality in distribution that phenomena tends to increase. Graphically this concept has been explained li din the next slide.

36 Source: A shows the area deviation from equal distribution and B is the actual distribution any Phenomena. Thus A and B always add up to give the line equal distribution.

37 Merits and Demerits Lorenz Curve Lorenz Curve is a graphic way representing distribution and so it makes the complex data in visual format which is easy to grasp by any one. Lorenz Curve can be used to calculate Gini Coefficient an another measure inequality. It is graphical way representation and so have no algebraic utility. Note: Out all discussed measures, first two are positional measure dispersion while next two are calculative measure dispersion and the last one is a graphic way estimating dispersion.

38 7.1 Suitability a Dispersion Measure P Which measure in a particular case should be used depends upon the nature data series, our purpose and course quality good measure dispersion. i For normal data distribution almost all gives good result but it is standard deviation which is useful in many estimations higher order. Where gap in data exist due care is required in the use positional measure or better it should be avoided.

39 Similarly in the case open ended data mean deviation can be avoided. P If our purpose is to inform public at large the selected should ldbesimple likerange, meandeviation, etc. Thus there is no concrete hard and fast rule by which it can be said that this measure dispersion should be used in this case; it all depends upon the purpose and nature data.

40 8.1 Summary P Central tendency gives a representative value to complex data series by which a series not only become easily understandable d but also become comparable with other series. But a central tendency is how much reliable representative it depends upon the variation in data distribution. Different measures dispersion explains the variability in data series and thus only in association with it a central tendency become true representative a data series. But a suitable method measuring dispersion depends itself upon the data distribution and the purpose. So it should be used carefully to avoid any misleading il interpretation data series.