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GNDU Question Paper-2024

B.A 3

Semester

COMPUTER SCIENCE

(Computer Oriented Numerical & Statistical Methods)

Time Allowed: Three Hours Max. Marks: 75

Note: Attempt Five questions in all, selecting at least One question from each section. The

Fifth question may be attempted from any section. All questions carry equal marks

SECTION-A

1. What is difference between numerical methods and numerical analysis? Which are

different types of errors and their measures ?

2. Find a root of equation x

- x

+ x - 7 = 0 near x = 2 using bisection method.

SECTION-B

3. Solve the following system of equations using Guass Jordon method:

2x - 3y + z = - 1

x + 4y + 5z = 25

3x - 4y + z = 2

4. Which are different methods to solve non-homogenous system of linear equations?

Write the algorithm of Guass Siedel method.

SECTION-C

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5. Using following table that gives the normal weights of a baby during the first six months

of age estimate the weight of a baby at age of 4 months:

Age in

Months

Weight (Kg.)

6. (a) Write the algorithm of Simpson's 1/3 rule to find area under curve.

(b) Using trapezoidal rule find the area between x-axis and the lines x = 1 and x = 4 and the

curve through the points given by:

1.5

2.5

3.5

2.3

2.9

3.6

4.5

5.2

5.8

SECTION-D

7. Calculate mean and mode of the distribution given below:

X less

Frequency

115

135

145

8. What is Dispersion? Which are various measures of dispersion ? Compare

Dispersion and Skewness.

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GNDU Answer Paper-2024

B.A 3

Semester

COMPUTER SCIENCE

(Computer Oriented Numerical & Statistical Methods)

Time Allowed: Three Hours Max. Marks: 75

Note: Attempt Five questions in all, selecting at least One question from each section. The

Fifth question may be attempted from any section. All questions carry equal marks

SECTION-A

1. What is difference between numerical methods and numerical analysis? Which are

different types of errors and their measures ?

Ans: 󷈷󷈸󷈹󷈺󷈻󷈼 A New Beginning: The Curious Student

Once upon a time in a quiet classroom, a student raised his hand and asked,

Sir, why do we learn numerical methods when calculators and computers can already give

us answers?

The teacher smiled and replied,

Ah, good question! But let me tell you somethingcomputers dont magically know the

answers. They use methods to calculate them, and it is analysis that checks if those methods

are correct, efficient, and reliable. Lets take a journey into this world.

And with that, the teacher began the story of numerical methods, numerical analysis, and

errors.

󹼧 Numerical Methods: The Recipe Book

Imagine you are hungry and want to eat a cake. What do you need? A recipe!

Numerical methods are like recipesstep-by-step procedures that tell you how to

approximate solutions of mathematical problems.

For example:

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Want to solve an equation like x2−5x+6=0? Use the Bisection Method or Newton

Raphson Method.

Want to compute an integral where formulas fail? Use Trapezoidal Rule or

Simpsons Rule.

Need to solve big systems of equations in engineering? Use Gauss Elimination

Method or Jacobi Iteration.

So in short:

󷷑󷷒󷷓󷷔 Numerical methods = Tools or step-by-step techniques to get approximate answers.

󹼧 Numerical Analysis: The Food Critic

Now, imagine youve baked the cake using a recipe. But waithow do you know if it tastes

good, if it was made efficiently, or if something went wrong? Thats where a food critic

comes in.

Numerical analysis is like that critic. It doesnt just accept the recipe; it asks:

Will this recipe always work?

How fast will it give results?

Will the result be reliable if the ingredients (data) change a little?

Numerical analysis studies the accuracy, efficiency, stability, and reliability of numerical

methods.

So in short:

󷷑󷷒󷷓󷷔 Numerical analysis = The study of how good, stable, and accurate those methods are.

󷘹󷘴󷘵󷘶󷘷󷘸 Difference in One Glance

Aspect

Numerical Methods

Numerical Analysis

What it

Practical techniques to approximate

solutions

Study of the quality and behavior of

those techniques

Purpose

To get the solution

To check how good the solution is

Analogy

Recipe book

Food critic

Focus

Implementation

Accuracy, error, stability, efficiency

So, if numerical methods are the how, numerical analysis is the how good.

󹼧 Now, Enter the Villain: Errors

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Every good story has a villain, right? Here, the villain is error.

But unlike in fairy tales, this villain cant be destroyed completelyit can only be minimized

or controlled.

Whenever we use numerical methods, the answers are approximate, not exact. This

difference between the true value and the approximate value is called error.

The teacher now leaned in and whispered,

If you understand errors, you understand the heart of numerical analysis.

󹼧 Types of Errors:

Lets meet the different villains in our story.

1. Absolute Error

This is the simplest one. Imagine the true answer is ₹100 and you calculate ₹98.

Error = |100 98| = 2.

󷷑󷷒󷷓󷷔 Absolute Error = |True Value Approximate Value|

It tells you how far your answer is from the truth.

2. Relative Error

But waitwhat if the true value is huge, like ₹1,00,000, and your error is just ₹2? Thats

hardly a problem.

So instead of just looking at the difference, we compare the error with the true value:

Relative Error = Absolute Error ÷ True Value

This tells us the size of error relative to the true value.

3. Percentage Error

To make it more human-friendly, we often multiply relative error by 100.

So:

Percentage Error = (Relative Error × 100)%

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This is like telling someone,

Your cake is 99% correct, only 1% off.

4. Truncation Error

Imagine you want to calculate π, but instead of writing 3.1415926, you stop at 3.14. That

small loss of accuracy is truncation error.

󷷑󷷒󷷓󷷔 It happens when we cut off a series or process without completing it.

For example, when we approximate functions using formulas or ignore higher-order terms

in expansions.

5. Rounding Error

This happens due to limitations of machines. Computers cant store infinite digits, so they

round numbers.

Example: Writing 0.333333 instead of the exact fraction 1/3.

󷷑󷷒󷷓󷷔 It is small but accumulates when thousands of operations are performed.

6. Propagation of Errors

Like a rumor in a classroom, an error can spread. If your initial calculation is slightly wrong,

and you use it in the next step, the error grows. This is propagation of errors, and numerical

analysis studies how stable a method is against it.

󹼧 Measures of Errors

Just identifying errors is not enoughwe need ways to measure and manage them.

1. Absolute Error (AE):

o AE = |True Value Approximate Value|

o Tells exact difference.

2. Relative Error (RE):

o RE = AE ÷ True Value

o Tells error size compared to true value.

3. Percentage Error (PE):

o PE = RE × 100%

o Tells error in percentage, easy to understand.

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󷷑󷷒󷷓󷷔 These measures help us say, How good is our approximation?

󹼧 Why Errors Are Not Always Bad

Here comes the twist in the story: Errors are not always villains. Without errors, we would

never get quick, usable results. Exact answers are often impossible or take forever to

compute. Approximations (with small errors) save time and effort.

Think of it this way:

Would you prefer waiting 5 hours for the exact square root of 2?

Or use NewtonRaphson to get 1.414 with 0.0001 error in a few seconds?

Most engineers, scientists, and even examiners prefer the second option.

󷈷󷈸󷈹󷈺󷈻󷈼 The Teachers Closing Words

The teacher finally said:

Numerical methods are our recipes to cook up solutions. Numerical analysis is our food

critic who makes sure those recipes actually work well. And errorsthough villainsare

necessary characters in the story. They remind us that perfection is rare, but good

approximations are powerful.

The student smiled, finally understanding.

2. Find a root of equation x

- x

+ x - 7 = 0 near x = 2 using bisection method.

Ans: A different beginning

Imagine standing at the edge of a river, trying to find the exact stepping stone that lets you

cross safely. You cant see the whole path, but you can test spots: too far left, you slip; too

far right, you sink. So you start in the middle, and keep halvingleft, right, closer each

timeuntil your foot lands on solid stone. Thats the spirit of the bisection method: careful,

steady, and guaranteed to find the crossing if one exists.

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SECTION-B

3. Solve the following system of equations using Guass Jordon method:

2x - 3y + z = - 1

x + 4y + 5z = 25

3x - 4y + z = 2

Ans: Imagine youre an algebra detective on the trail of three mysterious numbers x, y, and z

hiding behind three clues (the equations). Your job is to use the GaussJordan method: a neat,

mechanical detective technique that turns those clues into straight-up answers by transforming the

system into a simple, one-variable-per-line form. Ill tell the whole procedure like a little story so

its easy to follow and enjoyable to read.

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4. Which are different methods to solve non-homogenous system of linear equations?

Write the algorithm of Guass Siedel method.

Ans: A different beginning

Imagine a team trying to solve a puzzle together. Some members work by carefully

dismantling the puzzle piece by piece until the picture appears clearly; others prefer

iteratingguessing, checking, improvinguntil the puzzle settles into place. Solving a

system of linear equations is like that. When the system is non-homogeneous, meaning its

of the form Ax=b with a nonzero right-hand side, we can either dismantle directly or

iterate our way to the truth, depending on the problems size, structure, and accuracy

needs.

What non-homogeneous means and when a solution exists

A non-homogeneous system is written as Ax=b, where AA is the coefficient matrix, xx the

unknown vector, and bb a nonzero vector. Before choosing a method, its crucial to know

whether a solution exists and whether its unique. The RouchéCapelli theorem states that a

system is consistent if and only if the rank of the coefficient matrix equals the rank of the

Major methods to solve non-homogeneous systems

Comparison table

Method

Type

When to use

Pros

Cons

Gaussian

elimination /

Direct

Smallmedium

dense systems

Deterministic,

exact (up to

round-off)

Can be costly; round-

off accumulation

Cramers rule

Direct

Very small

systems

Conceptually

simple

Numerically

inefficient, unstable

for large n

Jacobi

Iterative

Large, sparse;

easy

parallelization

Simple, parallel-

friendly

Slower convergence;

needs conditions

GaussSeidel

Iterative

Large, sparse;

diagonally

dominant

Faster than

Jacobi; uses fresh

updates

Convergence needs

structure; ordering

matters

SOR

(Successive

over-

relaxation)

Iterative

Like GS, but

tuned with

ω\omega

Accelerates

convergence

Choosing ω\omega is

nontrivial

Conjugate

Gradient (SPD)

Iterative

Symmetric

positive definite

Fast for large

sparse SPD

Requires SPD;

preconditioning

often needed

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Direct vs iterative: how to choose

Direct methods (Gaussian elimination, LU/Cholesky): Great for small to medium

dense systems where a one-shot, numerically stable result is needed. Cholesky is

optimal when AA is symmetric positive definite.

Iterative methods (Jacobi, GaussSeidel, SOR, CG): Ideal for large, sparse systems

where memory and speed benefit from matrix sparsity; they offer controllable

accuracy and are often the default in scientific computing, with convergence

accelerated by structure (e.g., diagonal dominance or SPD) and good

preconditioners.

GaussSeidel method: intuition and algorithm

Convergence considerations

GaussSeidel converges for a broad class of matrices; common sufficient conditions include

strictly diagonally dominant matrices and symmetric positive definite matrices. Ordering of

equations (and variables) can also affect convergence speed, and techniques like SOR

introduce a relaxation parameter ω\omega to accelerate convergence beyond Gauss

Seidel.

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This is the standard iterative procedure; it exploits newly updated components within the

same iteration, which is the hallmark of GaussSeidel and the reason it usually outpaces

Jacobi under similar conditions

Where GaussSeidel shines (and where it doesnt)

Shines:

o Large, sparse systems from discretized PDEs, circuit networks, structural

models.

o When diagonal dominance or SPD structure is present, and when memory

constraints favor simple iterations.

o When a decent initial guess is availableconvergence can be very quick.

Watch-outs:

o Poorly scaled or non-dominant matrices may converge slowly or stall.

o Sensitive to ordering; reordering can improve diagonal dominance.

o SOR may be preferred when tuning ω\omega yields faster convergence than

plain GaussSeidel.

The bigger picture: a quick tour of other methods

Gaussian elimination / LU: Stable, direct, and predictable; great when the matrix is

modest in size and dense.

Cholesky: The go-to for symmetric positive definite matriceshalf the work of LU.

Jacobi: Conceptually simple and parallel-friendly; good teaching tool and baseline for

comparison.

SOR: A boosted GaussSeidel with relaxation; practical in engineering simulations.

Conjugate Gradient (CG): The weapon of choice for large SPD problems; typically

paired with preconditioners for speed.

In practice, you choose based on matrix structure, size, sparsity, available memory, and

required precision. For many real-world, structured systems, GaussSeidel offers an elegant

balance between simplicity and performance.

Closing thought

Solving Ax=bA x = b is less about brute force and more about strategy. Direct methods

provide clarity in one sweep; iterative methods offer finesseapproaching the solution step

by steady step, guided by structure and good guesses. GaussSeidel, with its use the

freshest information now mindset, is that reliable teammate who listens, adjusts, and

moves the whole group forwardone variable at a time

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SECTION-C

5. Using following table that gives the normal weights of a baby during the first six months

of age estimate the weight of a baby at age of 4 months:

Age in

Months

Weight (Kg.)

Ans: 󷊆󷊇 A New Beginning: Meeting the Baby

Once upon a time, in a cozy town, a baby was born. At birth, the baby weighed 5 kilograms.

Everyone in the family was excited to watch how the baby would grow each month. As the

days passed, the parents kept a careful record of the babys weight at different ages.

Heres the little diary they made:

At 0 months (birth) → 5 kg

At 2 months → 7 kg

At 3 months → 8 kg

At 5 months → 10 kg

At 6 months → 12 kg

Now, a curious cousin came to visit the family when the baby was exactly 4 months old. He

wanted to know the babys weight. But the diary had no entry for 4 months! What to do

now?

This is where mathematics becomes the detective. We cant weigh the baby directly (at

least in our problem), so we use the information we already have to estimate the weight.

󹺔󹺒󹺓 How Do We Estimate?

In real life, when we dont know an exact value but have some nearby known values, we try

to interpolate. Thats just a fancy word meaning to guess intelligently between two known

points.

Think of it like this: if you know your friend walked 2 km in 20 minutes and 3 km in 30

minutes, then you can easily guess that after 25 minutes, your friend must have walked

about 2.5 km. Thats interpolation!

We are going to do the same with the babys weight.

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󹵍󹵉󹵎󹵏󹵐 Finding the Right Neighbours

We want the babys weight at 4 months. Lets see which numbers from the table are

closest:

At 3 months → 8 kg

At 5 months → 10 kg

These are the neighbours of 4 months. The babys weight at 4 months should lie

somewhere between 8 kg and 10 kg.

󽀼󽀽󽁀󽁁󽀾󽁂󽀿󽁃 Step-by-Step Interpolation (Linear)

Since the growth looks fairly steady between 3 and 5 months, well assume the babys

weight grows linearly (like a straight line) between those two points.

The formula for linear interpolation is:

󼪔󼪕󼪖󼪗󼪘󼪙 Putting Numbers into the Formula

So, at 4 months, the babys estimated weight = 9 kilograms. 󷔬󷔭󷔮󷔯󷔰󷔱󷔴󷔵󷔶󷔷󷔲󷔳󷔸

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󼐧󼐨󼐩󼐪󼐫󷹩󼐬󼐭󷹫󼐮󷹬󼐯󼐰󼐱󼐲󼐳 Making It Feel Real

Imagine the babys mom picks him up and says, Wow, youve become a healthy 9 kg at 4

months! Even though the diary didnt have that exact number, math helped us predict it

correctly using the simple idea of steady growth.

󹶜󹶟󹶝󹶞󹶠󹶡󹶢󹶣󹶤󹶥󹶦󹶧 Why Does This Work?

This works because between 3 and 5 months, the babys weight increases smoothly (not in

sudden jumps). The method of interpolation assumes that the baby is putting on weight

evenly during that period. So halfway between 3 and 5 months, the weight will also be

halfway between 8 kg and 10 kg. Thats how we landed on 9 kg at 4 months.

󽆪󽆫󽆬 Turning It into a Lesson for Life

This story is not just about a babys weight. Its about life in general. Many times, we dont

have exact answers in real lifewhether its predicting tomorrows temperature, estimating

how long a journey will take, or guessing how tall youll grow. But by looking at patterns and

using logical steps, we can estimate values that are close to reality. Thats the magic of

mathematicsit gives us the power to predict!

󷈷󷈸󷈹󷈺󷈻󷈼 Wrapping Up with a Human Touch

So, what did we learn?

At 0, 2, 3, 5, and 6 months, the babys weights were recorded.

At 4 months, the data was missing.

Using interpolation, we estimated the missing weight.

The final answer: 9 kg at 4 months.

But more importantly, we learned that numbers are not just dry symbols. They tell stories

stories of growth, change, and life itself. In this case, the story of a little baby growing

healthier day by day. And with just a bit of math, we stepped into the role of a caring

observer, filling in the missing page of the babys growth diary.

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6. (a) Write the algorithm of Simpson's 1/3 rule to find area under curve.

(b) Using trapezoidal rule find the area between x-axis and the lines x = 1 and x = 4 and the

curve through the points given by:

1.5

2.5

3.5

2.3

2.9

3.6

4.5

5.2

5.8

Ans: Simpsons 1/3 rule and trapezoidal area: an intuitive walk through numerical

integration

A different beginning

Imagine youre watching rain fall on a field. You cant measure every single drop, but you

can place buckets along the field, peek into them, and estimate the total water collected.

Numerical integration is just like that: when a curve is too complicated to integrate exactly,

we sample the function at chosen points and estimate the area beneath it with structured

buckets. Two of the most trusted buckets in the mathematicians toolkit are Simpsons

1/3 rule and the trapezoidal rule. One is a bit fancier, drawing smooth parabolic arcs

through points; the other is straightforward, drawing slanted lines to make trapezoids. Both,

used wisely, turn a messy curve into a clean, reliable area estimate.

Well do two things:

Explain the algorithm of Simpsons 1/3 rule (the parabola bucket).

Use the trapezoidal rule (the slanted bucket) to find the area under the curve

between x=1x = 1 and x=4x = 4 using the given data.

Simpsons 1/3 rule: the idea, conditions, and algorithm

Simpsons 1/3 rule is a composite rule that approximates a function by a sequence of

quadratic polynomials (parabolas) over consecutive pairs of subintervals. If the function is

smooth and the data points are equally spaced, Simpsons rule can be impressively accurate

for a modest number of points.

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Step-by-step algorithm

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Tips for presenting these methods in exams

Be clear on conditions: Mention equal spacing and even subintervals for Simpsons

rule; note that trapezoidal works with any spacing, but composite form is simplest

with uniform spacing.

Write formulas cleanly: Examiners appreciate correct structureendpoints,

odd/even weights for Simpson; endpoint average plus interior sum for trapezoidal.

Show working: Brief, accurate arithmetic for the trapezoidal computation

demonstrates mastery without clutter.

Interpret the result: A single sentence explaining whether your number seems

reasonable (based on the functions behavior) adds maturity to your answer.

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SECTION-D

7. Calculate mean and mode of the distribution given below:

X less

Frequency

115

135

145

Ans:

Value (X)

Frequency (f)

115

135

145

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8. What is Dispersion? Which are various measures of dispersion ? Compare

Dispersion and Skewness.

Ans: Dispersion and Skewness: A Story of Spread and Shape

󷈷󷈸󷈹󷈺󷈻󷈼 A Different Beginning

Imagine youre a teacher in a classroom. Youve just given your students a math test out of

100 marks. When you collect the answer sheets and calculate the average, you find the class

average is 60.

Now, heres the twist: in one class, most students scored between 58 and 62. In another

class, some scored 20, some 95, and others scattered everywhere in betweenyet the

average is still 60.

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Both classes have the same mean, but clearly, the story of their performance is very

different. The first class is consistent, tightly packed around the mean. The second class is

scattered, unpredictable, and spread out.

This differencethe way data points are spread around the averageis what we call

dispersion.

And if you look even closer, sometimes the scores arent just spreadthey lean more to one

side. Maybe most students scored low, with a few very high marks pulling the average up.

Or maybe the opposite. That leaning or asymmetry is what we call skewness.

So, dispersion tells us how wide the classroom is, while skewness tells us which side of the

classroom the students are crowding toward.

󹺢 What is Dispersion?

In statistics, dispersion refers to the degree to which data values are spread out around a

central value (like the mean or median). It answers the question: Are the data points tightly

clustered, or are they widely scattered?

If dispersion is low, the data points are close to each other and to the mean.

If dispersion is high, the data points are spread out, showing more variability.

󷷑󷷒󷷓󷷔 In simple words: Dispersion measures consistency.

󹶜󹶟󹶝󹶞󹶠󹶡󹶢󹶣󹶤󹶥󹶦󹶧 Why is Dispersion Important?

1. Beyond averages: Averages alone can be misleading. Two datasets may have the

same mean but very different spreads.

2. Risk and reliability: In finance, dispersion tells us how risky an investment is. In

education, it tells us how consistent students are.

3. Comparison: Dispersion helps compare two groups more meaningfully.

4. Foundation for advanced stats: Variance and standard deviation (measures of

dispersion) are the backbone of probability, regression, and hypothesis testing.

󺬣󺬡󺬢󺬤 Measures of Dispersion

There are several ways to measure dispersion. They can be grouped into absolute measures

(expressed in the same units as the data) and relative measures (ratios or percentages,

useful for comparison).

Lets walk through them like characters in a story.

1. Range The Simplest Measure

Definition: Difference between the maximum and minimum values.

Formula:

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Range=Maximum−Minimum

Example: If marks range from 20 to 95, the range = 95 20 = 75.

Pros: Easy to calculate. Cons: Only depends on two values; ignores the rest.

󷷑󷷒󷷓󷷔 Think of range as the length of the playground. It tells you how wide the field is, but

not how the children are scattered inside it.

2. Quartile Deviation (Semi-Interquartile Range)

Definition: Half the difference between the third quartile (Q3) and the first quartile

(Q1).

Formula:

Interpretation: Focuses on the middle 50% of the data, ignoring extremes.

󷷑󷷒󷷓󷷔 Imagine ignoring the top and bottom performers in the class and only looking at the

middle group. Quartile deviation tells you how spread out that middle group is.

3. Mean Deviation

Definition: The average of the absolute deviations of each value from the mean (or

median).

Formula:

Interpretation: Shows average scatter of data points around the mean.

󷷑󷷒󷷓󷷔 Think of it as asking: On average, how far is each students score from the class

average?

4. Variance and Standard Deviation (The Most Powerful Measures)

Variance: The average of squared deviations from the mean.

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Standard Deviation (SD): Square root of variance.

Why square the deviations?

To avoid negative values cancelling positives.

To give more weight to larger deviations.

󷷑󷷒󷷓󷷔 Standard deviation is the most widely used measure of dispersion. It tells us, in the

same units as the data, how much variation exists.

Example: If the SD of test scores is small, most students scored close to the mean. If its

large, scores are widely spread.

5. Relative Measures of Dispersion

Sometimes we want to compare variability across datasets with different units or scales. For

this, we use coefficients (unit-free ratios).

Coefficient of Range:

Coefficient of Quartile Deviation:

Coefficient of Mean Deviation:

Coefficient of Variation (CV):

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󷷑󷷒󷷓󷷔 CV is especially important in finance and economics. A lower CV means more stability; a

higher CV means more risk.

󷘧󷘨 Enter Skewness: The Shape of the Distribution

Now that weve understood dispersion (the spread), lets meet its cousin: skewness.

Definition: Skewness measures the asymmetry of a distribution around its mean.

If the distribution is perfectly symmetric (like a bell curve), skewness = 0.

If the tail is longer on the right, skewness is positive.

If the tail is longer on the left, skewness is negative.

󷷑󷷒󷷓󷷔 In our classroom story:

If most students scored low but a few scored very high, the average is pulled up →

positive skewness.

If most scored high but a few scored very low, the average is pulled down →

negative skewness.

󹵍󹵉󹵎󹵏󹵐 Comparison: Dispersion vs Skewness

Lets put them side by side for clarity:

Aspect

Dispersion

Skewness

Definition

Measures the spread of data

around a central value

Measures the asymmetry of the

distribution

Focus

Variability/consistency

Shape and direction of lean

Values

Always non-negative

Can be positive, negative, or zero

Measures

Range, Q.D., M.D., Variance,

S.D., CV

Pearsons coefficient, moment

coefficient

Interpretation

High dispersion = more

variability

Positive skew = long right tail; Negative

skew = long left tail

Example

Two classes with same mean

but different spreads

Two classes with same mean and

spread, but one has more low scorers

and one has more high scorers

󷷑󷷒󷷓󷷔 In short:

Dispersion = How wide is the playground?

Skewness = Which side of the playground are the kids crowding toward?

󷇮󷇭 Real-Life Applications

Education: Dispersion shows consistency of student performance; skewness shows

whether most students are weak or strong.

Finance: Dispersion (SD, CV) measures risk; skewness shows whether extreme losses

or extreme gains are more likely.

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Quality Control: Low dispersion means consistent product quality; skewness can

reveal bias in production errors.

󽆪󽆫󽆬 Conclusion

Dispersion and skewness are like two storytellers of data. Dispersion tells us how far the

data wander from the center, while skewness tells us which way they lean.

Dispersion measures variability.

Skewness measures asymmetry.

Together, they give a complete picture of a datasets distribution.

This paper has been carefully prepared for educational purposes. If you notice any mistakes or

have suggestions, feel free to share your feedback.