Descriptive Statistics and Data Normalization Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Descriptive Statistics and Data Normalization short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Revision Checklist: Statistical Constraints

Core Logic

Mean Total Sum (

S = n \times \overset{x}{ˉ}

)

Median Middle Value (

x_{(n + 1) /2}

)

Mode Highest Frequency Value

Step-by-Step Solver

Sort variables: $x_{1} \leq x_{2} \leq \dots \leq x_{n}$ .
Fill Median.
Fill Mode (maximize frequency in valid slots).
Calculate remaining sum.
Enumerate integer solutions for remaining slots.
Verify: Order preserved? Range respected? Mode is unique?

Red Flag

Never assume $x_{1} = x_{2}$ unless necessary; it risks creating a bimodal distribution.

Revision Checklist: Z-Scores

Z = \frac{x - μ}{σ}

Unitless
Compare different variables.

Centered
Mean is always 0.

Scaled
Std Dev is always 1.

Robust
Handles outliers well.

Red Flag: Do not confuse Z-score with Min-Max scaling. Z-score preserves variance; Min-Max destroys it.

Descriptive Statistics and Data Normalization: Solved Questions with Step-by-Step Explanations (2 Problems)

Question 1 · Quantitative Aptitude MCQ

For a sorted data set of 7 numbers represented as $x_{1} \leq x_{2} \leq x_{3} \leq x_{4} \leq x_{5} \leq x_{6} \leq x_{7}$ , which position corresponds to the median?

A.
3rd
B.
4th
C.
5th
D.
Average of 3rd and 5th

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition recall question, recognisable because it asks for the position of the median in a sorted list of a specific size.

Step 1: Recall the definition of the median. The median is the middle value of a sorted data set.

Step 2: Determine the middle position for an odd number of elements $n$ . The formula for the middle position is $\frac{n + 1}{2}$ .

Step 3: Substitute $n = 7$ into the formula: $\frac{7 + 1}{2} = \frac{8}{2} = 4$ .

Step 4: The 4th position corresponds to the variable $x_{4}$ .

Answer: B

Question 2 · Quantitative Aptitude MCQ

For a sorted data set of 9 variables $x_{1} \leq x_{2} \leq x_{3} \leq x_{4} \leq x_{5} \leq x_{6} \leq x_{7} \leq x_{8} \leq x_{9}$ , which variable represents the median?

A.
x_3
B.
x_4
C.
x_5
D.
x_6

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition recall question, recognisable because it asks for the position of the median in a sorted list of a specific size.

Step 1: Recall the definition of the median. The median is the middle value of a sorted data set.

Step 2: Determine the middle position for an odd number of elements $n$ . The formula for the middle position is $\frac{n + 1}{2}$ .

Step 3: Substitute $n = 9$ into the formula: $\frac{9 + 1}{2} = \frac{10}{2} = 5$ .

Step 4: The 5th position corresponds to the variable $x_{5}$ .

Answer: C

More short notes in this unit

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Descriptive Statistics and Data Normalization Short Notes for GATE DA: Concepts, Formulas, Worked Examples & Practice

Descriptive Statistics and Data Normalization short notes for GATE DA: 2 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practi

A question from this chapter

Question 1

For a sorted data set of 7 numbers represented as x_{1} \leq x_{2} \leq x_{3} \leq x_{4} \leq x_{5} \leq x_{6} \leq x_{7}, which position corresponds to the median?

Question 2

For a sorted data set of 9 variables x_{1} \leq x_{2} \leq x_{3} \leq x_{4} \leq x_{5} \leq x_{6} \leq x_{7} \leq x_{8} \leq x_{9}, which variable represents the median?

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