Descriptive Statistics and Data Normalization Short Notes for GATE DA

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 (10 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

Question 3 · Quantitative Aptitude MCQ

Assertion (A): For any dataset, the sum of the Z-scores of all its data points is exactly zero.
Reason (R): The sum of the deviations of all data points from their mean is always zero.

A.
Both A and R are true, and R is the correct explanation of A.
B.
Both A and R are true, but R is not the correct explanation of A.
C.
A is true, but R is false.
D.
A is false, but R is true.

Correct Answer:

Step-by-Step Solution

Key idea: This is an assertion-reason question testing the fundamental algebraic properties of the mean and Z-scores.

Step 1: Evaluate Assertion (A). The sum of Z-scores is $\sum \frac{x _{i} - μ}{σ} = \frac{1}{σ} \sum (x_{i} - μ)$ . Since the sum of deviations from the mean is zero, the sum of Z-scores is zero. A is true.

Step 2: Evaluate Reason (R). By definition of the mean, $\sum (x_{i} - μ) = \sum x_{i} - n μ = n μ - n μ = 0$ . R is true.

Step 3: Check the link. The fact that $\sum (x_{i} - μ) = 0$ is the exact mathematical reason why $\sum Z_{i} = 0$ . R correctly and fully explains A.

Answer: Both A and R are true, and R is the correct explanation of A.

Question 4 · Quantitative Aptitude MCQ

If a data set is stated to have a "single mode" of 25, what must be true about the frequency of 25?

A.
It is equal to the frequencies of all other values.
B.
It is strictly greater than the frequency of any other value.
C.
It is the only value that appears exactly once.
D.
It appears exactly twice in the data set.

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition question about the unique mode condition, recognisable because it specifies a "single mode".

Step 1: Recall the definition of a mode. The mode is the value that appears most frequently in a data set.

Step 2: Understand the "single mode" constraint. If there is a single mode, it means exactly one value has the highest frequency, and no other value shares this highest frequency.

Step 3: Apply this to the value 25. Since 25 is the single mode, its frequency must be strictly greater than the frequency of any other value in the data set.

Answer: It is strictly greater than the frequency of any other value.

Question 5 · Quantitative Aptitude MCQ

A dataset has a mean of $50$ and a standard deviation of $10$ . What is the Z-score of a raw value $70$ ?

A.
0.5
B.
2.0
C.
-2.0
D.
0.2

Correct Answer:

Step-by-Step Solution

Key idea: The Z-score measures how many standard deviations a value is from the mean.

Step 1: Identify the given parameters: mean $μ = 50$ , standard deviation $σ = 10$ , and raw value $x = 70$ .

Step 2: Apply the Z-score formula: $Z = \frac{x - μ}{σ}$ .

Step 3: Substitute the values: $Z = \frac{70 - 50}{10} = \frac{20}{10} = 2.0$ .

Answer: The Z-score is $2.0$ .

Question 6 · Quantitative Aptitude MCQ

How many of the following properties are ALWAYS true for a dataset after Z-score normalization?
1. The mean of the normalized values is

0

.
2. The standard deviation of the normalized values is

1

.
3. The minimum value of the normalized dataset is always

- 3

.
4. The normalized values are unitless.

A.
1
B.
2
C.
3
D.
4

Correct Answer:

Step-by-Step Solution

Key idea: Recall the fundamental properties of Z-score normalization.

Step 1: Property 1: True. The transformation centers the data at 0.

Step 2: Property 2: True. The transformation scales the data to have a standard deviation of 1.

Step 3: Property 3: False. The minimum Z-score depends on the original data's minimum value and can be any negative number, not always -3.

Step 4: Property 4: True. Since we divide by the standard deviation (which has the same units as the data), the units cancel out.

Step 5: Count the true properties: 1, 2, and 4 are true. Total = 3.

Answer: 3 properties are always true.

Question 7 · Quantitative Aptitude MCQ

The Z-scores of four observations in a dataset are $- 3.0, - 1.0, 2.0$ , and $4.0$ . If the dataset has a mean of $100$ and a standard deviation of $10$ , which of the following is the maximum raw value among these observations?

A.
60
B.
70
C.
120
D.
140

Correct Answer:

Step-by-Step Solution

Key idea: This is a contradiction question where the maximum magnitude of a Z-score contradicts the maximum algebraic Z-score. The maximum raw value corresponds to the maximum algebraic Z-score.

Step 1: Identify the maximum algebraic Z-score from the given values: $- 3.0, - 1.0, 2.0, 4.0$ . The maximum is $4.0$ .

Step 2: Use the Z-score formula to find the raw value: $x = μ + Z σ$ .

Step 3: Substitute the values: $x = 100 + (4.0 \times 10) = 100 + 40 = 140$ .

Answer: The maximum raw value is $140$ .

Question 8 · Quantitative Aptitude MCQ

Which measure of central tendency is most influenced by extreme outlier values in a data set?

A.
Mode
B.
Median
C.
Mean
D.
Range

Correct Answer:

Step-by-Step Solution

Key idea: This is a conceptual question about the interplay of central tendencies, recognisable because it asks about the effect of outliers on a specific measure.

Step 1: Recall the properties of the three main measures of central tendency: mean, median, and mode.

Step 2: Understand how each measure is calculated. The mean is the arithmetic average, which includes every value in the sum. The median is the middle value, and the mode is the most frequent value.

Step 3: Determine the effect of an extreme outlier. Since the mean includes every value in its calculation, an extreme outlier will pull the mean towards it. The median and mode are resistant to outliers.

Answer: Mean.

Question 9 · Quantitative Aptitude MCQ

When using the "Fixed Sum and Middle Element" pattern to solve for unknown integers, what is the primary purpose of subtracting the median from the total sum?

A.
To find the sum of the remaining elements.
B.
To find the mode of the data set.
C.
To calculate the mean of the extreme values.
D.
To determine the range of the data set.

Correct Answer:

Step-by-Step Solution

Key idea: This is a pattern recognition question, recognisable because it asks about the specific algebraic step in the "Fixed Sum and Middle Element" method.

Step 1: Recall the "Fixed Sum and Middle Element" pattern. This pattern is used when you know the total sum of a data set and the value of the median (the middle element).

Step 2: Understand the algebraic goal. The total sum includes all elements. By subtracting the median (the middle element), you isolate the sum of all the other elements.

Step 3: Identify the purpose. This isolated sum is then used to solve for the remaining unknown integers.

Answer: To find the sum of the remaining elements.

Question 10 · Quantitative Aptitude MCQ

When treating mean, median, and mode as a system of equations to lock down possible values, what does the mode specifically tell you?

A.
The total sum of the data set
B.
The spread of the data around the mean
C.
The exact middle value in sorted order
D.
The value that appears most frequently

Correct Answer:

Step-by-Step Solution

Key idea: This is a definition recall question, recognisable because it asks for the specific constraint imposed by the mode.

Step 1: Recall the definitions of the three measures of central tendency.

Step 2: The mean fixes the total sum. The median fixes the exact middle value in sorted order.

Step 3: The mode specifically identifies the value that appears most frequently in the data set.

Answer: The value that appears most frequently.