CUET UG Geography Booster Test 1-Advanced Mean and Median -

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CUET UG Geography Booster Test 1-Advanced Mean and Median

⏱️ 30 Minutes

❓ 20 Questions

📌 Answers are locked once submitted — results and explanations appear at the end.

CUET UG Geography Booster Test 1-Advanced Mean and Median

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QUESTION 1 OF 20

Consider the following statements regarding the grouped indirect method:
Statement 1: The indirect method helps in reducing the values of observations to smaller numbers by subtracting a constant value.
Statement 2: The principles of this formula for grouped data are similar to that of the indirect method given for ungrouped data.

Answer Review

1 Consider the following statements regarding the grouped indirect method:
Statement 1: The indirect method helps in reducing the values of observations to smaller numbers by subtracting a constant value.
Statement 2: The principles of this formula for grouped data are similar to that of the indirect method given for ungrouped data.

A Only Statement 1 is true

B Only Statement 2 is true

C Both statements are true

D Both statements are false

💡 Short Explanation

The indirect method scales data values down by subtracting a fixed baseline value. This subtraction simplifies large measurements, making manual arithmetic less error-prone. The underlying logic of working with differences remains the same for lists and tables.

✅ Correct Answer Explanation

Both statements are true. Statement 1 accurately states that the indirect method reduces raw observation values to smaller, more manageable numbers by subtracting a constant value (the assumed mean, A). This calculation step is known as coding. Statement 2 is also true because the core mathematical principle of using deviations to find the central point is identical in both ungrouped and grouped workflows. The only operational difference is that the grouped formula includes an extra step to multiply these deviations by their corresponding class row frequencies (Σfd).

❓ Why Other Options Are Incorrect

Option A: This option is incorrect because it labels Statement 2 as false, failing to recognize that the grouped indirect method adapts its core logic directly from the ungrouped format.
Option B: This option is incorrect because it labels Statement 1 as false, ignoring the primary definition and function of the indirect method as a tool for scaling numbers down.
Option D: This option is incorrect because it rejects both statements, which accurately describe the functionality and structural similarities of indirect statistical calculations.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Cross-referencing both statements with standard textbook definitions shows that the indirect method uses subtraction to scale data and follows the same principles across both data layouts.

Final Logic: Since both statements provide accurate descriptions of the method's mechanics and background, Option C is the correct choice.

🧠 Memory Trick

Subtracting a baseline scales numbers down (Statement 1 is true) + Lists and tables share the same core shortcut logic (Statement 2 is true).

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean of Grouped Data

2 If a class distribution spans from 50 to 150, why is the class 90-110 preferred over 50-70 as the assumed mean group?

A Because 90-110 has the highest absolute frequency

B Because it is near the middle of the series, which minimises the magnitude of computation

C Because 50-70 cannot generate positive step deviations

D Because the assumed mean must perfectly equal the true mean

💡 Short Explanation

Choosing a baseline value near the extreme edges of a table results in large differences. Picking a baseline row near the center of a distribution keeps these differences small. These small values combine to keep the final calculation numbers manageable.

✅ Correct Answer Explanation

The class row 90-110 is preferred because it sits near the middle of the 50-150 range, which minimizes the overall magnitude of computation. Selecting a center baseline row ensures that the resulting deviations (d = x - A) are split between small positive numbers for higher rows and small negative numbers for lower rows. When these values are added together, they partially cancel each other out, keeping the final column sum (Σfd) small and minimizing manual calculation errors.

❓ Why Other Options Are Incorrect

Option A: Selecting a baseline class based entirely on high frequency ignores its position in the range, which can result in large calculations if that class sits at an extreme edge.
Option C: The class 50-70 sits at the bottom edge of the range, meaning it would generate large positive deviations for all rows above it rather than failing to generate them.
Option D: The assumed mean is a temporary guess used as a shortcut; it does not need to equal the true mean because the formula includes a correction step to find the exact center.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Connecting the choice of a middle baseline with the goal of keeping calculation sizes small helps identify Option B as the correct answer.

Final Logic: Choosing an assumed mean near the center of a distribution minimizes calculation size by balancing positive and negative differences.

🧠 Memory Trick

Center baseline = Balanced positive and negative differences = Minimizes calculation sizes.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean

3 In the grouped indirect method formula X̄ = A ± (Σfd ÷ N), the value 'd' represents the deviation of the midpoint of each class from the midpoint of the ________ group.

A modal

B cumulative

C pre-median

D assumed mean

💡 Short Explanation

The shortcut formula for grouped tables tracks how rows drift from a baseline point. This baseline point is selected by choosing an assumed mean row from the table. The variable d tracks the difference between each class midpoint and this baseline midpoint.

✅ Correct Answer Explanation

In the grouped indirect mean formula, the variable d represents the deviation of each class midpoint from the midpoint of the assumed mean group (A). To find these deviations, a geographer selects a representative baseline row near the center of the table and subtracts its midpoint from every other class midpoint (d = x - A). The resulting column tracks how far each category drifts from your initial guess.

❓ Why Other Options Are Incorrect

Option A: The modal group is the row that holds the single highest frequency count, which defines the mode rather than serving as the baseline for this mean formula.
Option B: Cumulative groups are a running total column used to find positional midpoints for the median, not for calculating mean deviations.
Option C: The pre-median group is the row that sits right before the median class line, which is an element used exclusively in the grouped median formula.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Matching the algebraic definition of d (d = x - A) with its components shows that deviations are measured from the assumed mean (A), pointing directly to Option D.

Final Logic: The variable d measures how far class midpoints drift from the chosen assumed mean baseline.

🧠 Memory Trick

The deviation column d tracks the distance of each row midpoint from the Assumed mean baseline.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean of Grouped Data

4 Match the components of the grouped mean dataset with their appropriate statistical representation:

List I	List II
1. Sum of cases	A. N or Σf
2. Frequency	B. f
3. Sum of coded scores	C. Σfd
4. Assumed mean	D. A

A 1-B, 2-A, 3-D, 4-C

B 1-A, 2-B, 3-C, 4-D

C 1-C, 2-B, 3-A, 4-D

D 1-A, 2-C, 3-B, 4-D

💡 Short Explanation

Statistical formulas use standard symbols to represent different dataset components. The symbol f represents the frequency of a class interval. N or Σf represents the total number of observations. Σfd and A are used in the grouped indirect mean formula.

✅ Correct Answer Explanation

This matching question connects the components of a grouped mean dataset with their standard statistical symbols. Sum of cases (1) is represented by N or Σf (A) because it denotes the total frequency. Frequency (2) is represented by f (B), indicating the number of observations in each class interval. Sum of coded scores (3) is represented by Σfd (C), which is obtained by adding the products of frequencies and deviations. Assumed mean (4) is represented by A (D), which serves as the reference value for calculating deviations. Therefore, the correct matching is 1-A, 2-B, 3-C, 4-D.

❓ Why Other Options Are Incorrect

Option A: This option incorrectly exchanges the symbols for sum of cases and frequency, and also swaps Σfd with the assumed mean.
Option C: This option incorrectly identifies Σfd as the symbol for sum of cases and N or Σf as the symbol for sum of coded scores.
Option D: This option wrongly matches frequency with Σfd and sum of coded scores with f, leading to incorrect statistical representations.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Match each dataset component with its universally accepted statistical symbol used in the grouped indirect mean formula.

Final Logic: The standard statistical symbols correctly pair as Sum of cases → N or Σf, Frequency → f, Sum of coded scores → Σfd, and Assumed mean → A, confirming Option B.

🧠 Memory Trick

Sum of cases = N or Σf (1-A); Frequency = f (2-B); Sum of coded scores = Σfd (3-C); Assumed mean = A (4-D).

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean of Grouped Data

5 If a grouped wage rate class is defined as 110-130, what will be the numerical value of the interval width (i) applied in indirect statistical formulas?

A 10

B 20

C 110

D 130

💡 Short Explanation

Grouped tables arrange data using structured categories called class intervals. Each row covers a specific range of values across the dataset. This range is found by subtracting the lower boundary value from the upper boundary value.

✅ Correct Answer Explanation

The interval width (i) for the wage rate class 110-130 is 20. In statistics, the interval width measures the numerical range covered by a single class row. It is calculated by finding the difference between the upper limit and the lower limit of that category (130 − 110 = 20). This value is placed into formulas to scale adjusted step deviations back to match the real-world scale of the table.

❓ Why Other Options Are Incorrect

Option A: A value of 10 represents only half of the range covered by this class row, which would distort the formula if used.
Option C: The value 110 represents the lower boundary limit of the class row, which tracks where the category begins rather than its width.
Option D: The value 130 represents the upper boundary limit of the class row, which tracks where the category ends.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Applying the interval width formula (Upper Limit − Lower Limit) to the given values (130 - 110) leads directly to 20, confirming Option B.

Final Logic: The class interval width is found by calculating the difference between the upper and lower limits of the row.

🧠 Memory Trick

Interval width i = Upper limit minus Lower limit (130 − 110 = 20).

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean of Grouped Data

6 In indirect computational methods, subtracting an assumed mean from class midpoints is a coding operation that primarily serves to:

A Identify the exact mode of the distribution

B Skew the dataset to a positive distribution

C Reduce the values of observations to smaller numbers to minimize computational magnitude

D Alter the intrinsic value of the raw dataset

💡 Short Explanation

Adding up long columns of large numbers manually can become tedious and prone to errors. The indirect method scales numbers down by subtracting a fixed baseline value. Working with these smaller differences makes manual addition easier and reduces errors.

✅ Correct Answer Explanation

Subtracting an assumed mean from class midpoints is a coding operation designed to reduce the values of observations to smaller numbers to minimize computational magnitude. When working with large numbers, direct addition can become slow and prone to manual errors. Shifting the dataset down by subtracting a fixed baseline constant scales the values down into smaller numbers, making manual calculation faster and more accurate.

❓ Why Other Options Are Incorrect

Option A: Finding the exact mode requires identifying the single category with the highest frequency count, which is separate from scaling numbers down.
Option B: Coding is an algebraic translation that shifts the scale of the dataset without distorting its internal shape or distribution properties.
Option D: This operation changes the size of the numbers to make calculations easier, but it does not alter the underlying values because a correction step adds the baseline back at the end.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Identifying the primary purpose of the indirect shortcut method (simplifying calculations) points directly to the option that mentions reducing computational sizes.

Final Logic: Subtracting an assumed mean scales large numbers down to simplify manual arithmetic.

🧠 Memory Trick

Coding scales numbers down to make manual addition easier and minimize calculation sizes.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean

7 To calculate step deviations systematically, one must list the deviations of the ________ of each class interval from the assumed mean group.

A lower limit

B upper limit

C midpoint

D total frequency

💡 Short Explanation

Grouped tables sort individual data points into category rows. Because individual values are hidden, each row needs a single representative value. This representative value is the class midpoint, which is used to calculate row deviations.

✅ Correct Answer Explanation

To calculate step deviations, you must measure the deviations of the midpoint (X) of each class interval from the midpoint of the assumed mean group (A). Because individual data points are hidden inside category rows, calculations run on grouped tables must use a single representative value for each row. This representative value is the class midpoint, found by averaging the upper and lower limits of the row.

❓ Why Other Options Are Incorrect

Option A: The lower limit tracks where a category row begins, which is only one part of finding the representative midpoint.
Option B: The upper limit tracks where a category row ends, which cannot represent the entire row on its own.
Option D: The total frequency counts how many items fall into the row, which measures the size of a category rather than its data value.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Remembering that class rows use midpoints as proxy values for calculations helps you identify that deviations must be measured between midpoints.

Final Logic: Grouped calculations use class midpoints to calculate deviations across table rows.

🧠 Memory Trick

Grouped calculations hide individual numbers → Always use class Midpoints to represent rows and calculate deviations.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean of Grouped Data

8 Arrange the logical progression for evaluating Σfd in the grouped indirect method:
1. Find the absolute difference of the separately added positive and negative sums.
2. Replace the ± sign in the final formula following A with the sign attached to the absolute difference.
3. Multiply each frequency (f) by its deviation (d) to generate fd values.
4. Add positive and negative values of fd separately.

A 1, 2, 3, 4

B 4, 3, 2, 1

C 3, 4, 1, 2

D 3, 1, 4, 2

💡 Short Explanation

Combining a mix of positive and negative values follows a specific algebraic sequence. The workflow begins by multiplying row counts by row deviations to find individual values. Next, you group and add the positive values and negative values separately. Finally, you find the net difference between those two sums and attach the correct sign to the result.

✅ Correct Answer Explanation

Evaluating the total sum column (Σfd) follows a specific mathematical sequence. First, you calculate the row values by multiplying each frequency by its deviation to generate the fd values (3). Second, you group your entries by adding the positive values and negative values separately (4). Third, you find the net total by calculating the absolute difference between those two sums (1). Fourth, you finalize the equation by replacing the ± sign after A with the positive or negative sign attached to that net difference (2). This establishes 3, 4, 1, 2 as the correct order.

❓ Why Other Options Are Incorrect

Option A: This option reverses the workflow, trying to find the final net difference (step 1) before calculating the row values or adding them up.
Option B: This sequence suggests grouping and adding values (step 4) before you have performed the multiplication step (step 3) to create those values.
Option D: This sequence suggests finding the final net difference (step 1) before grouping and adding the separate positive and negative values (step 4).

🎯 Strategy

used

Option Grouping

Application: Knowing that multiplying frequencies by deviations (step 3) must be the absolute first step narrows your choices down to Option C or D.

Final Logic: Since you must add the positive and negative groups separately (step 4) before you can find the net difference between them (step 1), the correct sequence must be 3-4-1-2.

🧠 Memory Trick

Multiply columns (fd) (3) → Separate and add positive/negative rows (4) → Find the net difference (1) → Attach the final sign (2).

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean of Grouped Data

📄 Passage

"Mean is the simple arithmetic average of the different values of a variable. For ungrouped and grouped data, the methods for calculating mean are necessarily different. Mean can be calculated by direct or indirect methods, for both grouped and ungrouped data. Note that the mean value comes the same when computed either of the two methods. Furthermore, the Indirect mean method will work for both equal and unequal class intervals."

Based on the passage, how do the statistical results of the direct method compare to those of the indirect method?

A The direct method yields a noticeably higher mean value.

B The mean value comes the same for both methods.

C The indirect method yields a slightly higher mean value.

D The results vary depending entirely on the class interval size.

💡 Short Explanation

The provided text evaluates direct and indirect calculation methods. The passage explicitly states how the answers from both methods compare. The fourth sentence provides a direct, word-for-word comparison.

✅ Correct Answer Explanation

The provided passage explicitly answers this comparison. The fourth sentence states: "Note that the mean value comes the same when computed either of the two methods." This matches Option B word-for-word, confirming that direct summation and indirect deviation shortcuts are simply two different algebraic paths that lead to the exact same statistical center.

❓ Why Other Options Are Incorrect

Option A: The direct method does not inflate values or yield a higher mean because both methods follow exact mathematical rules.
Option C: The indirect method scales numbers down using a baseline, but its correction step adds that value back, preventing it from yielding a higher mean.
Option D: The text notes that the indirect method works for any interval style, meaning results are not distorted by class sizes.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Scanning the fourth sentence of the passage for the phrase "computed either of the two methods" leads directly to Option B as the correct choice.

Final Logic: The passage explicitly states that the calculated mean value remains identical across both methods.

🧠 Memory Trick

Trust the text: The passage explicitly states that "the mean value comes the same when computed either of the two methods."

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean

📄 Passage

According to the passage, the indirect mean method is mathematically versatile enough to calculate the mean for which types of class intervals?

A Only strictly equal intervals

B Only completely unequal intervals

C Only continuous overlapping intervals

D Both equal and unequal class intervals

💡 Short Explanation

The provided text evaluates the capabilities of the indirect method. The passage explicitly outlines how the formula handles table formatting. The final sentence provides a direct, word-for-word confirmation.

✅ Correct Answer Explanation

The final sentence of the provided passage explicitly defines the versatility of this formula, stating: "Furthermore, the Indirect mean method will work for both equal and unequal class intervals." This matches Option D word-for-word, confirming that because deviations are calculated row-by-row based on exact midpoints, variations in row widths do not distort the calculation.

❓ Why Other Options Are Incorrect

Option A: Limiting the formula to strictly equal intervals ignores the versatility explicitly highlighted in the final sentence of the text.
Option B: Claiming the formula only works for unequal intervals is incorrect because it is fully compatible with standard, uniform tables.
Option C: Continuous overlapping intervals are a common data layout, but the text focuses on its flexibility across equal and unequal class widths.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Scanning the final sentence of the passage for the phrase "class intervals" leads directly to "both equal and unequal," confirming Option D.

Final Logic: The text explicitly states that the indirect mean method works for both equal and unequal class intervals.

🧠 Memory Trick

Follow the text: The final sentence explicitly notes that the method "will work for both equal and unequal class intervals."

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean of Grouped Data

11 Consider the following analytical statements about the median:
Statement 1: The median is defined as the point in a distribution with an equal number of cases on each side of it.
Statement 2: It is highly dependent on the actual extreme values within the dataset.

A Both statements are correct

B Only Statement 2 is correct

C Only Statement 1 is correct

D Neither statement is correct

💡 Short Explanation

The median represents the physical center of a sorted dataset. This midpoint splits the observations into two equal halves. Because it focuses on position, altering the extreme outer values does not shift the center slot.

✅ Correct Answer Explanation

Statement 1 is entirely correct because the median is a positional average defined as the physical midpoint that splits a sorted dataset into two equal halves, ensuring 50% of the cases sit on each side. Statement 2 is incorrect because the median is a positional metric that is not affected by extreme outliers. For example, in the sorted list [10, 20, 30, 40, 5000], the median is 30. Changing the extreme value from 5000 to 50 leaves the center slot unchanged, demonstrating that the median does not depend on extreme values.

❓ Why Other Options Are Incorrect

Option A: This option is incorrect because it accepts Statement 2, failing to recognize that the median is a positional value that remains unaffected by extreme outliers.
Option B: This option is incorrect because it validates Statement 2, which wrongly attributes the outlier sensitivity of the mean to the median.
Option D: This option is incorrect because it rejects Statement 1, which provides the foundational statistical definition of the median.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Remembering that the median tracks the physical center slot of a sorted list helps you recognize that it is unaffected by extreme outliers, leaving Statement 1 as the correct choice.

Final Logic: Since the median splits a dataset into equal halves and is unaffected by outliers, only Statement 1 is correct.

🧠 Memory Trick

The median splits the dataset down the middle (1 is true) + Changing the outer edges does not shift the center slot (2 is false).

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median

12 Match the statistical measure with its standard symbolic representation from the dataset:

List I	List II
1. Median	A. M
2. Lower limit of the median class	B. l
3. Assumed mean constant	C. A
4. Class interval width	D. i

A 1-C, 2-B, 3-A, 4-D

B 1-B, 2-A, 3-D, 4-C

C 1-A, 2-B, 3-C, 4-D

D 1-D, 2-C, 3-B, 4-A

💡 Short Explanation

Statistical formulas use standard symbols to represent different variables. M represents the median value, while l represents the lower limit of the median class. A denotes the assumed mean used in mean calculations. i represents the class interval width used in grouped data formulas.

✅ Correct Answer Explanation

This matching question connects statistical measures with their standard formula symbols. The Median (1) is represented by M (A). The Lower limit of the median class (2) is represented by l (B) because it marks the starting value of the median class interval. The Assumed mean constant (3) is represented by A (C) and is used as the reference value in the indirect method of calculating the mean. The Class interval width (4) is represented by i (D), indicating the numerical width of each class interval. Therefore, the correct matching is 1-A, 2-B, 3-C, 4-D.

❓ Why Other Options Are Incorrect

Option A: This option incorrectly identifies A as the symbol for Median and M as the symbol for Assumed mean, resulting in incorrect pairings.
Option B: This option incorrectly exchanges the symbols for Median and Lower limit, and also swaps Class interval width with the Assumed mean.
Option D: This option incorrectly matches Median with i, Lower limit with A, and Assumed mean with l, which do not represent the correct statistical symbols.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Match each statistical measure with its standard symbol commonly used in arithmetic mean and median formulas.

Final Logic: The accepted statistical symbols are Median → M, Lower limit → l, Assumed mean → A, and Class interval width → i, making Option C the correct answer.

🧠 Memory Trick

Median = M (1-A); Lower limit = l (2-B); Assumed mean = A (3-C); Class interval width = i (4-D).

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Arithmetic Mean and Median Formulas

13 If a researcher needs to calculate the median for the scores: 61, 10, 88, 37, 61, what is the mandatory initial action required?

A Dividing the sum of all scores by 5

B Finding the score with maximum occurrences directly

C Arranging the scores as 10, 37, 61, 61, 88

D Taking the assumed mean as 61

💡 Short Explanation

The median represents the physical center of a sorted dataset. Locating this value requires counting into the dataset from the outer edges. To ensure this count is accurate, you must sort the jumbled numbers by size first.

✅ Correct Answer Explanation

The mandatory initial action required to calculate the median is arranging the jumbled scores in ascending order: 10, 37, 61, 61, 88. Because the median is a positional average that identifies the physical center of a distribution, the dataset must be sorted by size before you can locate the center slot. Running a position formula on an unsorted list would lead to a random, incorrect number.

❓ Why Other Options Are Incorrect

Option A: Dividing the sum of all scores by 5 is the method used to find the arithmetic mean, which is a separate calculation path.
Option B: Finding the score with the maximum occurrences identifies the mode of the dataset (which is 61 here), rather than locating its physical center.
Option D: Choosing an assumed mean baseline is a step used exclusively in the shortcut indirect method for calculating the mean.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Remembering that the median requires a sorted series helps you identify that arranging the numbers from smallest to largest is the necessary first step.

Final Logic: Calculating a positional median requires sorting the raw dataset by size before locating the center slot.

🧠 Memory Trick

Median = Physical center slot → You must always Sort the jumbled list from smallest to largest first.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median

14 Once the ungrouped data series is ordered, the median is effectively found by locating the ________ observation in the arranged series.

A highest

B central

C cumulative

D lowest

💡 Short Explanation

The median represents the physical center of a sorted dataset. This midpoint splits the observations into two equal halves. Once the list is sorted, you locate the median by finding the central slot.

✅ Correct Answer Explanation

Once an ungrouped data series is ordered by size, the median is found by locating the central observation in the arranged series. As a positional average, the median is defined as the exact middle value of a sorted list. This center value balances the distribution, ensuring that an equal number of observations sit above it and below it in the sequence.

❓ Why Other Options Are Incorrect

Option A: The highest observation tracks the maximum value or upper extreme of the dataset, which sits at the end of an ascending list.
Option C: Cumulative counts are a running total column used to find positions within grouped tables, rather than an observation type in a raw list.
Option D: The lowest observation tracks the minimum value or lower extreme of the dataset, which sits at the start of an ascending list.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Aligning the description of the median as a middle-ranking value helps you select "central" as the term that completes the definition.

Final Logic: Once a dataset is sorted by size, the median is identified by locating the central observation value in the sequence.

🧠 Memory Trick

The median always represents the exact Middle or Central slot of a sorted dataset.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median

15 For 7 Himalayan mountain peaks, calculating the median height requires evaluating the positional formula ((N + 1) ÷ 2)th item. This mathematical operation indicates the median is the value of the ________ item in the series.

A 3rd

B 4th

C 5th

D 7th

💡 Short Explanation

An odd-sized dataset has a single observation sitting right in the center slot. The position formula ((N + 1) ÷ 2) calculates the location of this center slot. Placing the count of 7 peaks into this formula determines the middle rank.

✅ Correct Answer Explanation

Placing the count of 7 peaks into the ungrouped median position formula determines that the median is the value of the 4th item in the series. The formula is calculated as follows: Position = ((N + 1) ÷ 2) = ((7 + 1) ÷ 2) = (8 ÷ 2) = 4th item Once the 7 mountain peaks are sorted by height, counting in from either the highest or lowest end leads to the 4th peak, which serves as the physical center of the dataset.

❓ Why Other Options Are Incorrect

Option A: The 3rd item sits to the left of the center, leaving 2 items below it and 4 items above it, which fails to split the dataset evenly.
Option C: The 5th item sits to the right of the center, leaving 4 items below it and 2 items above it, which is unbalanced.
Option D: The 7th item represents the absolute end of the sorted series, tracking either the maximum or minimum value rather than the center.

🎯 Strategy

Used

Contextual/Tonal Matching

Application: Running the calculation ((7 + 1) ÷ 2 = 4) maps the dataset size directly to the 4th position, pointing to Option B.

Final Logic: Evaluating the ungrouped median position formula for a dataset of 7 items identifies the 4th item as the physical center.

🧠 Memory Trick

Add 1 to the total count and divide by 2: (7 + 1 = 8), then (8 ÷ 2 = 4th item).

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median

16 Consider evaluating a dataset possessing 8 ungrouped observations. How is the final median mathematically derived?

A It is explicitly the value of the 4th item.

B It is explicitly the value of the 5th item.

C It is derived by taking the average of the two middle-ranking values.

D A dataset with an even number of items lacks a median.

💡 Short Explanation

An odd-sized dataset has a single observation sitting right in the center slot. An even-sized dataset splits down the middle, leaving two numbers sharing the center space. To find a single physical center, you calculate the average of those two middle numbers.

✅ Correct Answer Explanation

When a dataset contains an even number of observations (such as N = 8), it does not have a single middle slot. Instead, the dataset splits evenly into two halves, leaving the 4th and 5th items sharing the center space. To find the median, you calculate the average of these two middle-ranking values by adding them together and dividing by 2 ((Value₄ + Value₅) ÷ 2), creating a balanced central reference point.

❓ Why Other Options Are Incorrect

Option A: Selecting only the 4th item leaves out the upper center value, shifting the center point downward and distorting the result.
Option B: Selecting only the 5th item leaves out the lower center value, shifting the center point upward.
Option D: Even-sized datasets can always be solved for a median by calculating the average of their two center values.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Remembering how an even-sized dataset splits down the middle helps you identify the option that describes averaging the two center values.

Final Logic: The median of an even-sized dataset is calculated by finding the average of its two center values.

🧠 Memory Trick

Even counts split evenly down the middle → Add the two center numbers and divide by 2 to find their average.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median

17 For grouped distributions, calculating N/2 is crucial to determine the specific class interval where an individual observation is ________ located.

A centrally

B vertically

C symmetrically

D sequentially

💡 Short Explanation

Sorting data points into summary tables hides the exact values of individual numbers. The median represents the physical midpoint of a distribution. Finding the median from a table requires locating the center point of the dataset.

✅ Correct Answer Explanation

Calculating N/2 is crucial to determine the specific class interval where an individual observation is centrally located. In a grouped table, individual data values are hidden inside category rows. To find the median, you must locate the middle slot of the dataset by dividing the total number of cases by 2 (N/2). This target number is used to scan the cumulative frequency column and identify the median class row where the physical center of the distribution sits.

❓ Why Other Options Are Incorrect

Option B: Vertical layouts describe how data columns are formatted in a table, which is separate from calculating a data position.
Option C: Symmetrical layouts describe balanced data distributions, whereas this calculation step applies to any table regardless of its shape.
Option D: Sequential order describes sorting data rows from top to bottom, which is a basic formatting step rather than the goal of locating a center.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Aligning the sentence structure with standard textbook definitions ensures that "centrally" is selected as the term that completes the definition of a grouped median.

Final Logic: The target value N/2 is calculated to identify the class row where the median observation sits centrally within the distribution.

🧠 Memory Trick

The median is a measure of central tendency → The calculation helps you find where data points sit Centrally.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median of Grouped Data

18 If N = 50, rendering N/2 = 25, and the cumulative frequencies are 3, 10, 21, 37, 45, the median group is found by stopping at the cumulative frequency that is next greater than 25. Which cumulative frequency identifies the median class?

A 10

B 21

C 37

D 45

💡 Short Explanation

Finding the median row in a table uses a running total column called cumulative frequency. The target position for the median is found by calculating N/2. You locate the median row by finding the first running total that clears this target.

✅ Correct Answer Explanation

The cumulative frequency that identifies the median class is 37. With a dataset size of 50, the center target position is 25 (N/2 = 25). Scanning down the running total column shows that the third row ends at a cumulative total of 21, meaning it does not contain the 25th item. The fourth row covers positions 22 through 37. Since 37 is the first cumulative total that clears your target of 25, this row contains the median value.

❓ Why Other Options Are Incorrect

Option A: A cumulative total of 10 only covers the first 10 items in the table, falling short of your target position of 25.
Option B: A cumulative total of 21 represents the end of the previous row, which falls just short of containing the 25th item.
Option D: A cumulative total of 45 covers positions 38 through 45, which sits further down the table past the target center position.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Comparing the target position (25) against the cumulative totals shows that 37 is the first value that clears the target, confirming Option C.

Final Logic: The median class is identified by locating the first cumulative frequency row that is greater than or equal to the target position N/2.

🧠 Memory Trick

Target position = 25 → Scan the running totals: 3, 10, 21 (too small) → 37 is the first total that Clears the target.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median of Grouped Data

19 In the median calculation formula M = l + (i ÷ f) × ((N ÷ 2) − c), if the identified median group is 80-90, the applied value for the variable 'l' will be:

A 80

B 85

C 90

D 10

💡 Short Explanation

The grouped median formula uses a set of specific row variables. The lowercase letter l represents the starting value or lower limit of the median row. For the class row 80-90, the value where the range begins is 80.

✅ Correct Answer Explanation

For the identified median class row 80-90, the value assigned to the variable l is 80. In the grouped median formula, the lowercase letter l stands for the lower limit of the median class. Because the class row covers the range from 80 to 90, the value where the category begins is 80, which serves as the starting point for the calculation.

❓ Why Other Options Are Incorrect

Option B: The value 85 represents the center midpoint of this class row, which is a variable used in mean formulas rather than this step.
Option C: The value 90 represents the upper limit of the class row, which tracks where the category ends rather than where it begins.
Option D: The value 10 represents the interval width (i) of the class row (90 - 80 = 10), which is a separate variable in the formula.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Identifying the starting boundary value of the class row 80-90 matches the definition of the lower limit (l), pointing directly to 80.

Final Logic: The variable l represents the lower limit where the chosen median class row begins.

🧠 Memory Trick

The variable l stands for the Lower limit of the row → For the class row 80-90, the lower starting number is 80.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median of Grouped Data

20 When assessing the median for the class interval 80-90 with a cumulative frequency of 37, assuming the previous class 70-80 holds a cumulative frequency of 21, what numerical value is assigned to 'c' in the median formula?

A 37

B 21

C 16

D 80

💡 Short Explanation

The grouped median formula uses specific variables to track table positions. The lowercase letter c tracks the running total of items before the median row. This value is found by reading the cumulative frequency of the row right above it.

✅ Correct Answer Explanation

The numerical value assigned to the variable c is 21. In the grouped median formula, the lowercase letter c represents the cumulative frequency of the pre-median class (the row that sits right before the chosen median row). Since the median row is 80-90, the pre-median row is 70-80, which holds a cumulative frequency of 21. This value is subtracted from N/2 to determine how many items into the median row you need to count.

❓ Why Other Options Are Incorrect

Option A: The value 37 represents the cumulative frequency of the median class row itself, which would cause an incorrect calculation if subtracted.
Option C: The value 16 represents the normal frequency (f) of the median class row, calculated by subtracting the two cumulative totals (37 - 21 = 16).
Option D: The value 80 represents the lower limit (l) of the median class row, tracking where the category begins rather than a frequency total.

🎯 Strategy

used

Contextual/Tonal Matching

Application: Matching the definition of c (cumulative frequency of the row right before the median class) with the given data points leads directly to 21, confirming Option B.

Final Logic: The variable c tracks the cumulative frequency total of the class row that sits right before the chosen median class row.

🧠 Memory Trick

The variable c looks backwards → Choose the cumulative frequency of the row sitting right Before the median class.

📚 Class XII – Practical Work in Geography – Chapter: Data Processing – Topic: Median of Grouped Data

CUET UG Geography Booster Test 1-Advanced Mean and Median

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