CUET UG Geography Booster Test 2-Skewness and Comparative Measures

A normal distribution forms a symmetrical bell-shaped curve because values fall evenly on both sides of the center. When extreme high or low values occur more frequently than expected, they accumulate on one side of the dataset. This creates an unbalance that stretches the graph's tail in one direction, breaking its natural symmetry and turning it into a distorted, skewed distribution.

The three metrics separate because they use different mathematical approaches. The mode represents the most common score, keeping it anchored directly under the highest peak of the graph. The arithmetic mean is highly sensitive to extreme outliers because its formula sums every value, dragging it out toward the long tail. The median tracks positions, allowing it to resist outliers and sit between the other two metrics.

Because most districts recorded low rainfall, the highest concentration of data points sits at the lower end of the scale. On a frequency graph, the vertical axis tracks the number of occurrences. This means the highest peak or "hump"—which represents the mode—will sit on the left side of the graph over the lower numerical values, while a few high flood-level records stretch a long tail out to the right.

Reading a positively skewed curve from left to right follows an increasing numerical order along the X-axis. In this distribution, the peak sits on the left, meaning the mode (2) has the lowest numerical value. Extreme high outliers stretch a long tail to the right, dragging the sensitive arithmetic mean (3) to the far right. The median (1) remains in the middle as a positional midpoint, establishing 2, 1, 3 as the correct sequence.

This matching question connects different distribution shapes with the data conditions responsible for them. A Negative Skew (1) occurs when extreme low values stretch the distribution towards the left tail (B). A Positive Skew (2) occurs when extreme high values stretch the distribution towards the right tail (A). A Symmetrical Distribution (3) has values distributed equally on both sides of the centre (C). A No Skew (4) is characterised by the mean, median, and mode coinciding at the centre of the distribution (D). Therefore, the correct matching is 1-B, 2-A, 3-C, 4-D.

In a negatively skewed distribution, the mean is numerically lesser than the median. A negative skew features a long tail extending toward the lower end of the scale on the left. Because the arithmetic mean is highly sensitive to outliers, these low values drag it down. The median resists these outliers and stays closer to the center peak, leaving the mean with a lower numerical value than the median (Mean < Median)

Statement II is correct. The defining advantage of the mode is its direct connection to the tallest peak of a frequency curve. Because the mode represents the score with the highest frequency, it always sits directly beneath the highest point of the graph, allowing a researcher to find it instantly without looking at calculations. Statement I is incorrect because the mean shifts toward extreme outliers and cannot be found simply by looking at the peak.

Finding the mode follows a logical sequence. First, the researcher should sort the raw dataset into an ascending or descending sequence to group identical values together (2). Next, the researcher scans the sorted list to identify which value has the maximum occurrence or repetition count (1). Finally, the researcher declares this most frequent score as the mode of the dataset (3). This establishes 2, 1, 3 as the correct sequence.

The median remains 30 because it is a positional metric that relies on rank rather than a sum of values. In both provided datasets, the numbers are already sorted in ascending order, and the total count (N=5) stays the same. The position of the midpoint is always the third item in the list. Changing the highest score from 50 to 1000 alters the total sum but does not shift the center position, leaving the median unchanged.

This matching question connects the variables in the grouped median formula with their respective meanings. The term N ÷ 2 (1) represents the value used to locate the central median position (B). The variable c (2) represents the cumulative frequency of the class preceding the median class (D). The variable f (3) represents the actual frequency of the median class (A). The variable l (4) represents the lower limit of the median class (C). Therefore, the correct matching is 1-B, 2-D, 3-A, 4-C.

The formula Σfx ÷ N ensures that every data point influences the final average. In a grouped table, individual scores are hidden within class intervals. To calculate a mean, the midpoint (x) of each interval acts as the representative value for that group. Multiplying this midpoint by the group's frequency count (f) reconstructs an estimate of the total sum for those observations, ensuring every data point is accounted for in the calculation.

The assumed mean is preferably chosen from a class near the middle of the series. The indirect method simplifies calculations by subtracting a baseline constant (A) to create smaller deviation numbers (d). Choosing a baseline near the physical center of the series ensures that some deviations are positive and others are negative. These values help cancel each other out when added together, keeping the total sum small and minimizing the manual arithmetic required.

The core assumption of a bell-shaped curve is that extreme scores are rare, while most observations cluster around the center. When measuring natural traits across a large population, most individuals score near the average, while exceptionally high or low scores occur infrequently. When plotted, this pattern forms a symmetrical curve that peaks in the middle and slopes downward toward both edges, creating a classic bell shape.

Statement I is correct because comparing the positions of the mean, median, and mode shows whether a dataset is symmetrical or pulled to one side. Statement II is incorrect because the sequence Mode < Median < Mean indicates a positive skew, not a negative one. In a positive skew, extreme high values drag the arithmetic mean to the far right, giving it the largest value, while the mode stays at the left peak with the smallest value.

The formula Σx ÷ N is used for ungrouped data. This basic direct formula calculates an average by summing individual scores (x) and dividing by the total count (N). This method works when dealing with a raw list where every score is displayed distinctly. Once data is organized into class intervals, individual values lose their identity, requiring the grouped formula Σfx ÷ N instead.

Selecting a central tendency metric follows a step-by-step workflow. First, the researcher looks at the dataset structure to check its distribution type, noting whether it is formatted as a grouped table or an ungrouped raw list (2). Next, the researcher checks the values to identify any extreme outliers that could skew specific averages (1). Finally, based on these conditions, the researcher selects the mean for mathematical inclusion or the median to resist outliers (3). This establishes 2, 1, 3 as the correct sequence.

This matching question connects the variables from the Himalayan peak example with their calculated values. N (1) represents the total number of observations, which is 7 (B). Applying the formula ((N + 1) ÷ 2) gives 4, identifying the 4th item (A) as the median position. The value at this position is 8,172 m (C), which is the final median. The Median Position (4) refers to the central observation in the ordered dataset (D). Therefore, the correct matching is 1-B, 2-A, 3-C, 4-D.

The statement is true. When calculating an indirect mean for a grouped table, selecting an assumed mean (A) near the physical center of the series is a standard efficiency step. In the textbook example, choosing the midpoint of the central 90–110 bracket (A=100) ensures that deviations balance out as small positive and negative numbers, reducing the size of the values and minimizing the manual arithmetic required.

The correct term to complete the statement is "alternatives." Standard multiple-choice questions provide a question stem followed by a fixed set of choices. The textbook assessment framework uses a four-alternative format for its objective questions, requiring students to evaluate the choices carefully and select the correct answer based on their recall of the concepts.

The exercise framework includes multiple question formats to evaluate a range of skills. Multiple-choice questions check quick recall of facts. Short answers (30 words) require students to provide brief, precise definitions of core concepts. Long answers (125 words) give students the space needed to explain complex statistical processes and geographical applications, providing a thorough evaluation of their understanding.