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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 something—computers don’t magically know the

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

are correct, efficient, and reliable. Let’s 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 recipes—step-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

Simpson’s 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 you’ve baked the cake using a recipe. But wait—how do you know if it tastes

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

comes in.

Numerical analysis is like that critic. It doesn’t 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 can’t be destroyed completely—it 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:

Let’s 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 wait—what if the true value is huge, like ₹1,00,000, and your error is just ₹2? That’s

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 can’t 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 enough—we 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 Newton–Raphson to get 1.414 with 0.0001 error in a few seconds?

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

󷈷󷈸󷈹󷈺󷈻󷈼 The Teacher’s 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 errors—though villains—are

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 can’t 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 halving—left, right, closer each

time—until your foot lands on solid stone. That’s 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 you’re 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 Gauss–Jordan 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. I’ll tell the whole procedure like a little story so

it’s 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

iterating—guessing, checking, improving—until the puzzle settles into place. Solving a

system of linear equations is like that. When the system is non-homogeneous, meaning it’s

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 problem’s 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, it’s crucial to know

whether a solution exists and whether it’s unique. The Rouché–Capelli 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

Small–medium

dense systems

Deterministic,

exact (up to

round-off)

Can be costly; round-

off accumulation

Cramer’s 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

Gauss–Seidel

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, Gauss–Seidel, 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.

Gauss–Seidel method: intuition and algorithm

Convergence considerations

Gauss–Seidel 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 Gauss–Seidel and the reason it usually outpaces

Jacobi under similar conditions

Where Gauss–Seidel shines (and where it doesn’t)

• 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 available—convergence 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 Gauss–Seidel.

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 matrices—half the work of LU.

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

comparison.

• SOR: A “boosted” Gauss–Seidel 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, Gauss–Seidel 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 finesse—approaching the solution step

by steady step, guided by structure and good guesses. Gauss–Seidel, with its “use the

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

moves the whole group forward—one 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 baby’s weight at different ages.

Here’s 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 baby’s weight. But the diary had no entry for 4 months! What to do

now?

This is where mathematics becomes the “detective.” We can’t 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 don’t know an exact value but have some nearby known values, we try

to interpolate. That’s 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. That’s interpolation!

We are going to do the same with the baby’s weight.

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󹵍󹵉󹵎󹵏󹵐 Finding the Right Neighbours

We want the baby’s weight at 4 months. Let’s 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 baby’s 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, we’ll assume the baby’s

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 baby’s estimated weight = 9 kilograms. 󷔬󷔭󷔮󷔯󷔰󷔱󷔴󷔵󷔶󷔷󷔲󷔳󷔸

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󼐧󼐨󼐩󼐪󼐫󷹩󼐬󼐭󷹫󼐮󷹬󼐯󼐰󼐱󼐲󼐳 Making It Feel Real

Imagine the baby’s mom picks him up and says, “Wow, you’ve become a healthy 9 kg at 4

months!” Even though the diary didn’t 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 baby’s 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. That’s how we landed on 9 kg at 4 months.

󽆪󽆫󽆬 Turning It into a Lesson for Life

This story is not just about a baby’s weight. It’s about life in general. Many times, we don’t

have exact answers in real life—whether it’s predicting tomorrow’s temperature, estimating

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

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

mathematics—it 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 baby’s 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 baby’s 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: Simpson’s 1/3 rule and trapezoidal area: an intuitive walk through numerical

integration

A different beginning

Imagine you’re watching rain fall on a field. You can’t 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 mathematician’s toolkit are Simpson’s

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.

We’ll do two things:

• Explain the algorithm of Simpson’s 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.

Simpson’s 1/3 rule: the idea, conditions, and algorithm

Simpson’s 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, Simpson’s 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 Simpson’s

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

with uniform spacing.

• Write formulas cleanly: Examiners appreciate correct structure—endpoints,

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 function’s 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 you’re a teacher in a classroom. You’ve 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, here’s 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 between—yet 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 difference—the way data points are spread around the average—is what we call

dispersion.

And if you look even closer, sometimes the scores aren’t just spread—they 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).

Let’s 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 student’s 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 it’s

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 we’ve understood dispersion (the spread), let’s 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

Let’s 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

Pearson’s 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 dataset’s distribution.

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

have suggestions, feel free to share your feedback.”