measures of dispersions: range, quartile deviation, … measure of central tendency of any series or...

Directorate ofof

DistanceEducation(DDE)

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TOPIC

Program:P.G.

Subject:

Measures of Dispersions: Range, Quartile Deviation, Mean Deviation, Standard Deviation and Lorenz Curve-Their Merits and Limitations j

Geography

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and Specific Uses

Previous

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(QM&CR)(QM&CR)

Lesson:II

Prepared By: Dr. Kanhaiya Lal, Assistant Professor, DDE, K.U.K.

Directorate of

1.1 Introductionof


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The measure of central tendency of any series or datadistribution summarises it into single representative formhi h f l i t b t it f il t t th Program:

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which are useful in many respect but it fails to account thegeneral distribution pattern of data.

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Thus any conclusion only based on central tendencymay be misleading.

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Dispersion tries to fill this gap by stressing on thepattern of data distribution in any series.

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Lesson:II

Dispersion can prove very effective in association withcentral tendency in making any statistical decision.

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2.1 Objectiveof


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Any data series in statistics is usually represented bysome measure of central tendency.

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As the uniformity of a data series increases the value ofcentral measure become more representative. j

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So, it is always meaningful to interpret central measureof a data series keeping in mind the dispersion present in Previous

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of a data series keeping in mind the dispersion present inthat series.Here, an insight has been given on whatdispersion is, what are its importance, what are differentt f di i d h i t t di i (QM&CR)

Lesson:II

types of dispersion and how some important dispersionscan be calculated, their merits‐demerits, etc.

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3.1 What is Measure of Dispersion?of


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Measure of dispersion has two terms, measure anddispersion.

Program:P.G.

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‘Measure’ here means specific method of estimationwhile ‘dispersion’ term means deviation or difference or

d f i l f h i l lj

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spread of certain values from their central value.

Various statisticians have defined it variously. Previous

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According to Simpson and Kafka, “The measurement ofthe scatterness of the mass of figures in a series about an (QM&CR)

Lesson:II

the scatterness of the mass of figures in a series about anaverage is called a measure of variation or dispersion.”

Directorate of

According to W. I. King, “the term dispersion is used toindicate the facts that within a given group the items differ of


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indicate the facts that within a given group, the items differfrom one another in size or in other words, there is lack ofuniformity in their sizes.”

Program:P.G.

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Spiegel defines it as, “The degree to which numericaldata tend to spread about an average value is called the j

Geography

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p gvariation or dispersion of the data.”

Similarly to A L Bowley “Dispersion is a measure of Previous

Paper:V(a)

(QM&CR)

Similarly, to A. L. Bowley, Dispersion is a measure ofvariation of the items.”

All th d fi iti h di i d i d t (QM&CR)

Lesson:II

All these definitions shows dispersion as a spread in dataseries with respect to some representative reference valueof the data series.

Directorate of

Dispersion is a directionless average estimate ofvariations in individual observations from a central of


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variations in individual observations from a centralmeasure to shows the degree of nonuniformity in our datadistribution.

Program:P.G.

Subject:

Dispersion is regarded as directionless measure since itonly shows the magnitude of deviation from central j

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measure, whether deviation is positive or negative it doesdescribe.

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It is also commonly known as scatter or spread or widthor variation or average of second order.

(QM&CR)

Lesson:II

In real problems it is very rare that all observations havethe same values as that of its central tendency.

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For example, two cricketer with same 65 average scoreper year considered for a span of 8 years one with scores of


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per year considered for a span of 8 years, one with scoresclose to average value in all years while the other with veryhigh score in some year and very poor in most of year,

t b t d t l l F i li bl Program:P.G.

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cannot be rated at same level. Former is more reliablebatsman. Similarly, only average annual income of Indiameasured as Gross Domestic Product (GDP) does not j

Geography

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mean each citizen has this income or income near to it ormerely by increasing GDP all citizens can be equallydeveloped Previous

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developed.

The central value gives nearly good results as long as ourd t di t ib ti l i t f b ti (QM&CR)

Lesson:II

data distributions are normal i.e. our most of observationsare close to it.

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4.1 Objectives of Measure of Dispersionof


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It is the value of dispersion which says how muchreliable a central tendency is?

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Usually, a small value of dispersion indicates thatmeasure of central tendency is more reliable j

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representative of data series and vice‐versa.

Many powerful analytical tools in statistics such as Previous

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Many powerful analytical tools in statistics such ascorrelation analysis, the testing of hypothesis, analysis ofvariance, the statistical quality control, regression analysis

b d f i ti f ki d th (QM&CR)

Lesson:II

are based on measure of variation of one kind or another(Gupta, 2004).

Directorate of

The degree of data spread also helps in analysingimportance of different components of a system For of


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importance of different components of a system. Forexample, if we take agricultural productivity to dependupon input of fertilizer, hybrid seeds, irrigation,i ti id ti id d hi Th f Program:

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insecticides, pesticides and machinery. The cause of anyabruptness in productivity can be analysed by comparingcentral measure of different inputs with its variation and j

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thus it helps in taking corrective measures.

Measure of dispersion is also used to compare Previous

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Measure of dispersion is also used to compareuniformity of different data like income, temperature,rainfall, weight, height, etc.

(QM&CR)

Lesson:II

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5.1 Properties of a Good Measure of Dispersionof


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Like a good measure of central tendency the goodmeasure of dispersion should also have similarh t i ti Program:

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characteristics.

A good measure of dispersion should be clearly defined jGeography

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so that there should not be any scope of subjectivity incomputation as well as its interpretation.

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It should be easy to compute, understand and interpretand further, all individual observations should be used init ti ti d l it h ld b f f bi (QM&CR)

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its estimation and also it should be free from any biasnessor biasness due to any extreme value.

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Since dispersion is also used to estimate many statisticalcomplex properties of data so a dispersion should be easily of


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complex properties of data so a dispersion should be easilyapplicable in any algebraic operations.

Fi ll h f di i h ld b l t Program:P.G.

Subject:

Finally, such measure of dispersion should be leastaffected by sampling or have high degree of samplingstability. j

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6.1 Measuring DispersionPrevious

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Measure of dispersion is always a real number. If allvalues of individual observations are identical with centraltendenc then dispersion is al a s ero and as de iation (QM&CR)

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tendency then dispersion is always zero and as deviationin observation from central tendency increases, dispersionalso increases but it never become negative.

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Further a measure of dispersion is absolute or relative.of


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In the case of absolute measure of dispersion the unit ofindividual observations and the unit of dispersion remainsth Thi di i i f l i i diff t Program:

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the same. This dispersion is useful in comparing differentdata set in same unit and with same average size.

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In relative dispersion unit of original observationbecomes irrelevant since it is a ratio of absolute dispersionto some central tendency It is useful in comparison of Previous

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to some central tendency. It is useful in comparison ofdata series in different units or with different size of data.

S f di i R (QM&CR)

Lesson:II

Some common measures of dispersion are: Range,Quartile Deviation, Mean Deviation, Standard Deviationand Lorenz Curve.

Directorate of

6.1.1.1 Rangeof


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Range is the quickest and simplest measure ofdispersion.

Program:P.G.

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Being a positional measure it accounts only thedifference between the highest and the lowest observationi d i d d k i ll

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in any data series and does not take into account allindividual observations and so it is quickest but at thesame time a rough or crude measure of dispersion. Previous

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Range always has the same unit as original observationshave (QM&CR)

Lesson:II

have.

Directorate of

Symbolically for ungrouped data it is represented by:of


PR = H – L

Wh R R Program:P.G.

Subject:

Where, R = RangeH = Highest value in the observationL = Lowest value in the observation j

Geography

Class:Previous

In the case of a grouped data range is estimated bytaking the difference of upper limit of highest class Previous

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taking the difference of upper limit of highest classinterval and lower limit of lowest class interval.

Also in such case the difference of mid values of highest (QM&CR)

Lesson:II

Also in such case the difference of mid‐values of highestand lowest class interval are used as range.

Directorate of

In case of open ended grouped data, the width ofadjacent class is used i e it is assumed that highest/lowest of


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adjacent class is used, i.e. it is assumed that highest/lowestclass has same width as that of adjacent class, to estimateupper/lower limit of highest/ lowest class.

Program:P.G.

Subject:

Above method is equally applicable with grouped datahaving equal or unequal class intervals. j

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g q q

To make it free from unit, coefficient of range issometimes calculated Previous

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sometimes calculated.

Coefficient of Range = (H – L)/ (H+L)(QM&CR)

Lesson:II

Where, H = Highest value in the observationL = Lowest value in the observation

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6.1.1.2 Merits and Demerits of Rangeof


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It is simplest and easiest to compute, understand andinterpret.

Program:P.G.

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It is a crude measure since it does not take into accountall individual observation. Addition/removal of a single j

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extreme value at upper/lower end of data series can alterthe range to great extent.

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In the case of open ended grouped class true estimationof range becomes impossible.

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Sampling may affect it adversely and its value may varymarkedly from sample to sample.

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Since it does not take into account of any observationbetween the highest and the lowest value and so it tells of


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between the highest and the lowest value and so it tellsnothing about actual distribution of data between thesetwo extremes.

Program:P.G.

Subject:

It gives nearly good result only if sample size issufficiently large and data are fairly continuous or regular. j

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In grouped data it rarely happens that data distributionactually touches upper/lower class limit of highest/lowest Previous

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actually touches upper/lower class limit of highest/lowestclass interval. So range calculated using these values isusually not accurate. Range if calculated using mid‐valuesf hi h t d l t l t t l thi bl t (QM&CR)

Lesson:II

of highest and lowest class try to resolve this problem tosome extent but only if data are fairly and sufficientlydistributed in these two classes.

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The value of range changes with the transformation ofscale For example between Rs 1 and Rs 10 the difference is of


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scale. For example between Rs.1 and Rs.10 the difference is9 only but if we take paisa as unit this difference willbecome 900.

Program:P.G.

Subject:

6.1.2.1 Quartile Deviationj

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Quartile deviation is another positional and absolutemeasure of data dispersion in any series which try tominimise the error of range as a measure of dispersion Previous

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minimise the error of range as a measure of dispersion.

Unlike range it avoids the use of extreme values and init l th diff f fi t d thi d til (QM&CR)

Lesson:II

its place uses the difference of first and third quartile as ameasure of dispersion.

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It is also called semi‐interquartile range or semi‐quartilerange or interquartile range of


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range or interquartile range.

Thus, this measure of dispersion ignores fifty per cent(fi t t d l t t) f b ti Program:

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(first 25 per cent and last 25 per cent) of observations.

Symbolically it is estimated using following formula, jGeography

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Quartile Deviation (QD) = (Q3 – Q1)/2Previous

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Where, Q3 = third quartileQ1 = first quartile

(QM&CR)

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Thus quartile deviation gives the average amount bywhich two quartiles differ from the median (Gupta, 2004).

Directorate of

Further in any symmetrical or non‐skewed or normaldata distribution median (Q ) plus/minus QD exactly of


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data distribution median (Q2) plus/minus QD exactlycovers 50 per cent of the data distribution on either side ofthe median since in such case Q3 – Q2 = Q2 – Q1 or

l Q Q Q d Q Q Q Program:P.G.

Subject:

conversely Q2 + QD = Q3 and Q2 – QD = Q1.

In reality, rarely a business, economic or social data are jGeography

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perfectly symmetrical. So quartile deviation as a measureof dispersion should be preferably used only where datadistribution are moderately skewed Previous

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distribution are moderately skewed.

A lower/higher value of quartile deviation in less skewedd t fl t th t /l di t ib ti d th (QM&CR)

Lesson:II

data reflects that more/less distributions are around themedian value.

Directorate of

A relative counterpart of quartile deviation is calledcoefficient of quartile deviation and it is represented by of


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coefficient of quartile deviation and it is represented byfollowing formula,

C ffi i t f Q (Q Q )/ (Q Q ) Program:P.G.

Subject:

Coefficient of QD = (Q3 – Q1)/ (Q3 + Q1)

6.1.2.2 Merits and Demerits of Quartile Deviation jGeography

Class:Previous

Quartile deviation considers only middle 50 per cent ofobservations and so it is not affected by extreme values as Previous

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observations and so it is not affected by extreme values asin the case of range.

It l lik t id ll b ti i (QM&CR)

Lesson:II

It also, like range, not considers all observations inestimating dispersion and so its result may be misleading.

Directorate of

Since it not considers extreme values and so it is usefulin estimating dispersion in grouped data with open ended of


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in estimating dispersion in grouped data with open endedclass.

S li d l ff t it ti ti lik i th Program:P.G.

Subject:

Sampling may adversely affect its estimation like in thecase of range.

jGeography

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Quartile deviation as a measure of dispersion is mostreliable only with symmetrical data series. Unfortunately,in social sciences most of data distributions are generally Previous

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in social sciences most of data distributions are generallyasymmetrical in nature. So, its use in social sciences isusually limited to data which are moderately skewed.

(QM&CR)

Lesson:II

Directorate of

6.1.3.1 Mean Deviationof


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A proper approach to the measurement of dispersion orvariability would require that all the values in a series aret k i t id ti O f th th d f d i it Program:

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Subject:

taken into consideration. One of the methods of doing itis through average deviation or mean deviation.

jGeography

Class:Previous

As the very name indicates, this measure is an average orthe mean of the deviations of the values from a fixedpoint which is usually the arithmetic mean and Previous

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point, which is usually the arithmetic mean andsometimes the median (Bhat and Mahmood, 1993).

Th ti ll th i d t i t ki th (QM&CR)

Lesson:II

Theoretically there is an advantage in taking thedeviations from median because the sum of deviations ofitems from median is minimumwhen sign are ignored.

Directorate of

However, in practice, the arithmetic mean is morefrequently used in calculating the value of average of


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frequently used in calculating the value of averagedeviation and this is the reason why it is more commonlycalled mean deviation (Gupta, 2004).

Program:P.G.

Subject:

Mean deviations are computed first by summing theabsolute differences of each observation from mean and j

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then dividing it by number of observations. The sign ofdeviations are ignored i.e. only absolute values are used,since sum of deviations from mean is always zero (Hooda Previous

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since sum of deviations from mean is always zero (Hooda,2002; Levin and Fox, 2006).

B t t th f b l t d i ti t d t (QM&CR)

Lesson:II

By contrast the sum of absolute deviations tends tobecome larger as the variability of a distribution increases(Levin and Fox, 2006).

Directorate of

Following mathematical formula is used to estimatemean deviation of a data series of


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mean deviation of a data series,

Case I: Ungrouped data seriesProgram:

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Subject:

Mean Deviation (MD) =j

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Where, = sum of absolute deviations from mean= total number of observations

Case II: Grouped data series Previous

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Case II: Grouped data seriesMean Deviation (MD) =

Wh f f b i (QM&CR)

Lesson:II

Where, = frequency of observations= mid‐value of each class

Other symbols have same meaning as in the case I.

Directorate of

6.1.3.2 Merits and Demerits of Mean Deviationof


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Mean deviation is an absolute measure of dispersion butunlike range or quartile deviation it is a calculative

f di i hi h i it d t Program:P.G.

Subject:

measure of dispersion which gives it some advantages overthem.

jGeography

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It took into account all values of observations inestimating dispersion and truly tries to give scatter in data.

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Paper:V(a)

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In most of cases mean deviation measures dispersionfrom mean. Mean is not only easy to compute but also easyt d t d d th h t i li d i (QM&CR)

Lesson:II

to understand and even those who are not specialised instatistics can understand it and a dispersion based on it isalso appealing to them.

Directorate of

Mean deviation is an absolute measure of dispersion so acomparison against data series represented in different of


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comparison against data series represented in differentunit is difficult.

Al h i it l f t h Program:P.G.

Subject:

Also a change in unit or scale of measurement changesthe value of dispersion.

jGeography

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As mean is least affected by sampling so use of mean inmeasuring dispersion also retains this property. But ifmedian is used then it holds good or nearly good as long Previous

Paper:V(a)

(QM&CR)

median is used then it holds good or nearly good as longas data series are symmetrical or moderately skewed.

Si id tifi ti f d i t ibl ith ll d t (QM&CR)

Lesson:II

Since identification of mode is not possible with all dataseries so it cannot be used frequently in measuring meandeviation.

Directorate of

Mean always has a tendency of upward biasness and alsoaccurate mean identification in open ended grouped data of


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accurate mean identification in open ended grouped datais not possible. This problem also percolates in meandeviation where deviation is measured using mean.

Program:P.G.

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In estimating mean deviation we take into account onlyabsolute value (only magnitude and not sign) of deviation j

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Class:Previous

from central tendency. It is mathematically a unsoundpractice and so it limits it’s further algebraic use.

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6.1.4.1 Standard Deviation

Thi th d f i di i i t id l (QM&CR)

Lesson:II

This method of measuring dispersion is most widelyacclaimed by statisticians since it nearly have allproperties of a good measure of dispersion.

Directorate of

This method is not based on absolute value of deviationof individual data from the mean so it is algebraically of


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of individual data from the mean so it is algebraicallytenable.

Thi bl i t d d d i ti h b Program:P.G.

Subject:

This problem in standard deviation has been overcomeby squaring the individual deviation from mean.

jGeography

Class:Previous

These squared individual deviations are summed up,then averaged and finally its square root has beenidentified as a measure of standard deviation Previous

Paper:V(a)

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identified as a measure of standard deviation.

This is why it is also known as root mean squared i ti (QM&CR)

Lesson:II

deviation.

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Mathematically following formula represents theconcept of standard deviation of


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concept of standard deviation,

Case I: Ungrouped data seriesProgram:

P.G.

Subject:

S2 =j

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=Previous

Paper:V(a)

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Where, S2 = variance; = standard deviation

f f d i ti f (QM&CR)

Lesson:II

= sum of square of deviations from mean= total number of observations

Directorate of

Case II: Grouped data seriesof


PS2 =

Program:P.G.

Subject:= jGeography

Class:PreviousWhere S2 = variance; = standard deviation Previous

Paper:V(a)

(QM&CR)

Where, S = variance; = standard deviation

= frequency of observations(QM&CR)

Lesson:II

= mid‐value of each class

Directorate of

6.1.4.2 Merits and Demerits of Standard DeviationIts most important beauty is that it is free from the of


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Its most important beauty is that it is free from thecompulsion of taking only absolute value in estimatingmean deviation. So it is frequently applicable in differentl b i ti Program:

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Subject:

algebraic operations.

It took into account all individual observations and so jGeography

Class:Previous

any slight variation in any observations automatically gotrepresentation in standard deviation.

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Through variance it easily reflects the aberration in dataseries.

(QM&CR)

Lesson:II

It is the basis of relative measure of dispersioncoefficient of variation (CV).

Directorate of

It is also an absolute measure of dispersion and socomparisons of data series in different units of of


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comparisons of data series in different units ofmeasurement are not tenable.

It l h if it f t h d Program:P.G.

Subject:

Its value changes if unit of measurement changed.

In a normal distribution, data are symmetrically jGeography

Class:Previous

distributed around mean(mean, median or mode allbecome identical) and mean σ covers 68.27 per cent ofobservations; mean 2σ covers 95 45 per cent of Previous

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observations; mean 2σ covers 95.45 per cent ofobservations and mean 3σ covers 99.73 per cent ofobservations. This property is useful in dividing a data

i i t it bl l (QM&CR)

Lesson:II

series into suitable groups or class.

Directorate of

6.1.5.1 Lorenz Curveof


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Lorenz curve is a graphical way of showing dispersion inany data distribution.

Program:P.G.

Subject:

It was developed by Max O. Lorenz in 1905 as arepresentative of distribution in wealth. j

Geography

Class:Previous

It is a useful measure to show the distribution of anyphenomena and it is frequently used to show distribution Previous

Paper:V(a)

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phenomena and it is frequently used to show distributionof wealth, assets, biodiversity, land holdings, etc.

It th t i th f l di t ib ti ‘ ’ (QM&CR)

Lesson:II

It assumes that in the case of equal distribution ‘n’ percent should have ‘n’ per cent share in the total.

Directorate of

It is graphically constructed using cumulativefrequencies of parameter and its distribution one along of


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frequencies of parameter and its distribution, one alongy‐axis and other along x‐axis, respectively, which togetheradd up individually to 100 per cent.

Program:P.G.

Subject:

The line making an angle of 45° from horizontal showsthe case of equal distribution. j

Geography

Class:Previous

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As any curved line moves away from this line of equaldistribution the inequality in distribution of that Previous

Paper:V(a)

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distribution the inequality in distribution of thatphenomena tends to increase.

G hi ll thi t h b l i d i th t (QM&CR)

Lesson:II

Graphically this concept has been explained in the nextslide.

Directorate ofof


PProgram:P.G.

Subject:jGeography

Class:PreviousPrevious

Paper:V(a)

(QM&CR)Source: http://www.rrh.org.au/publishedarticles/article457_1.gi

(QM&CR)

Lesson:II

‘A’ shows the area of deviation from equal distribution and ‘B’ is theactual distribution of any Phenomena. Thus ‘A’ and ‘B’ always add up to givethe line of equal distribution.

Directorate of

6.1.5.2 Merits and Demerits of Lorenz Curveof


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Lorenz Curve is a graphic way of representingdistribution and so it makes the complex data in visualformat which is easy to grasp by any one Program:

P.G.

Subject:

format which is easy to grasp by any one.

Lorenz Curve can be used to calculate Gini Coefficienth f i li

jGeography

Class:Previous

an another measure of inequality.

It is graphical way of representation and so have no Previous

Paper:V(a)

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g p y palgebraic utility.Note: Out of all discussed measures, first two are

positional measure of dispersion while next two are (QM&CR)

Lesson:II

positional measure of dispersion while next two arecalculative measure of dispersion and the last one is agraphic way of estimating dispersion.

Directorate of

7.1 Suitability of a Dispersion Measureof


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Which measure in a particular case should be useddepends upon the nature of data series, our purpose andf lit f d f di i Program:

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of course quality of good measure of dispersion.

For normal data distribution almost all gives good result jGeography

Class:Previous

but it is standard deviation which is useful in manyestimations of higher order.

Previous

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Where gap in data exist due care is required in the use ofpositional measure or better it should be avoided.

(QM&CR)

Lesson:II

Directorate of

Similarly in the case of open ended data mean deviationcan be avoided of


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can be avoided.

If our purpose is to inform public at large the selectedh ld b i l lik d i ti t Program:

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Subject:

should be simple like range, mean deviation, etc.

Thus there is no concrete hard and fast rule by which it jGeography

Class:Previous

can be said that this measure of dispersion should be used in this case; it all depends upon the purpose and nature of data Previous

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data.

(QM&CR)

Lesson:II

Directorate of

8.1 Summaryof


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Central tendency gives a representative value to complexdata series by which a series not only become easily

d t d bl b t l b bl ith th Program:P.G.

Subject:

understandable but also become comparable with otherseries. But a central tendency is how much reliablerepresentative it depends upon the variation in data j

Geography

Class:Previous

distribution. Different measures of dispersion explains thevariability in data series and thus only in association withit a central tendency become true representative of a data Previous

Paper:V(a)

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it a central tendency become true representative of a dataseries. But a suitable method of measuring dispersiondepends itself upon the data distribution and the purpose.S it h ld b d f ll t id i l di (QM&CR)

Lesson:II

So it should be used carefully to avoid any misleadinginterpretation of data series.

measures of dispersions: range, quartile deviation, … measure of central tendency of any series or...

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