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

Descriptive Statistics and Data Normalization notes for GATE DA: 15 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice questions.

Chapter Roadmap: Descriptive Statistics and Data Normalization

Chapter Roadmap

Descriptive Statistics and Data Normalization

Mean, Median and Mode Constraints

Reconstructing data sets from statistical properties. Solving for unknown integers using mean, median, and mode definitions.

Standardization and Z-Score Normalization

Transforming data to have zero mean and unit variance. Comparing data from different scales using Z-scores.

What You Will Master

Constraint Satisfaction: Finding specific numbers from average, middle, and most frequent values.
Statistical Rigour: Understanding precise definitions governing these measures.
Data Scaling: Converting raw values into standardized scores.

The Interplay of Central Tendencies

In many exam problems, you are not given the data set. You are given its properties. Treat these properties as constraints in a logic puzzle.

Mean

Fixes the sum of the elements.

Median

Fixes the middle element in sorted order.

Mode

Fixes the frequency of the most common element.

The Strategy

Use the Mean to calculate the total sum required.
Use the Median to place a specific value in the center of your sorted list.
Use the Mode to populate the remaining slots with the most frequent value.
Solve for the unknowns using simple arithmetic.

Solving Integer Constraint Problems

Method: Solving Integer Constraint Problems

Define Variables Let the $n$ integers be $x_{1} \leq x_{2} \leq \dots \leq x_{n}$ .
Apply Median Set the middle element(s). For $n = 5$ , $x_{3} = Median$ .
Apply Mode Place the mode values. If the mode is $M$ and it's unique, ensure it appears more times than any other number. For a large mode, typically $x_{4} = x_{5} = M$ .
Apply Mean Calculate Total Sum $S = n \times Mean$ . $\sum x_{i} = S$
Solve for Unknowns $x_{1} + x_{2} = S - (x_{3} + x_{4} + x_{5})$

Check Constraints

Integers must be within the given range (e.g., 0 to 20).
Order must be maintained ( $x_{1} \leq x_{2} \leq x_{3}$ ).
Mode must remain unique (check frequencies of $x_{1}, x_{2}$ ).

Reconstructing the Five Integers

Example: Reconstructing the Five Integers

Problem

Five integers are picked from 0 to 20 (repetitions allowed). Find the number of ways to pick these integers (ignoring permutations).

Mean = 12
Median = 18
Single Mode = 20

Setup: Sorted integers

x_{1} \leq x_{2} \leq x_{3} \leq x_{4} \leq x_{5}

Median Constraint:

x_{3} = 18

Mode Constraint: Mode is 20. Since

x_{3} = 18

x_{4}

and

x_{5}

must be 20. Frequency of 20 is 2. For a single mode, no other number can appear 2+ times. Thus,

x_{1} \neq = x_{2}

Mean Constraint: Sum =

5 \times 12 = 60

x_{1} + x_{2} + 18 + 20 + 20 = 60 ⟹ x_{1} + x_{2} = 2

Finding Valid Pairs $(x_{1}, x_{2})$ : Constraints:

0 \leq x_{1} \leq x_{2} \leq 18

x_{1} \neq = x_{2}

(0, 2) Valid

(1, 1) Invalid (Bimodal)

Answer: There is exactly 1 way.

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 notes in this unit

chapter

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

Descriptive Statistics and Data Normalization notes for GATE DA: 15 study cards covering concepts, formulas, shortcuts and exam traps, plus solved practice qu

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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