Question

The image shows distribution of marks secured by 2500 students of a particular district in four subjects. Which of the statements must be TRUE?

• A. If 50 marks are required for passing, more students will fail in English than in Hindi
• B. If 75 marks are required for securing a distinction, most distinctions will be awarded in English
• C. The average marks secured in Science is highest compared to other subjects
• D. Students have performed more uniformly in Maths than in any other subject

Hint:

In a normal distribution graph: the peak's location on the x-axis gives the average (mean), and the curve's width indicates the data's spread. A narrow curve means uniform data (low standard deviation), while a wide curve means diverse data (high standard deviation).

Answer 1

The Correct Option is A, D

Step 1: Understanding the Concept:
This question requires interpreting data presented as normal distribution curves. The peak of the curve represents the mean (average), and the width of the curve represents the standard deviation (spread or uniformity of the data). A tall, narrow curve indicates low spread (high uniformity), while a short, wide curve indicates high spread (low uniformity).
Step 2: Detailed Explanation:
Let's analyze the curves for each subject:

English: Mean \(\approx\) 50. Very wide spread (large standard deviation).
Maths: Mean \(\approx\) 66. Very narrow spread (small standard deviation).
Hindi: Mean \(\approx\) 57. Moderate spread.
Science: Mean \(\approx\) 65. Wide spread.
Now let's evaluate the statements:

A. If 50 marks are required for passing, more students will fail in English than in Hindi. For English, the mean is 50. In a normal distribution, 50% of the data is below the mean. So, approximately 50% of students will fail English. For Hindi, the mean is \(\approx\) 57. The passing mark of 50 is significantly below the mean. Therefore, a much smaller percentage (less than 50%) of students will fail Hindi. This statement is TRUE.

B. If 75 marks are required for securing a distinction, most distinctions will be awarded in English. For English, the mean is 50 and the curve is wide. 75 marks is far into the right tail, representing a very small percentage of students. For Science (mean \(\approx\) 65, wide curve) or Maths (mean \(\approx\) 66, narrow curve), the distinction mark of 75 is closer to the mean, and a larger (though still small) percentage of students would likely achieve it compared to English. This statement is FALSE.

C. The average marks secured in Science is highest compared to other subjects. The average mark is the peak of the curve. The peak for Science is at \(\approx\) 65. The peak for Maths is at \(\approx\) 66. Therefore, the average marks in Maths are higher than in Science. This statement is FALSE.

D. Students have performed more uniformly in Maths than in any other subject. Uniform performance corresponds to a small spread or standard deviation. The distribution curve for Maths is significantly taller and narrower than for any other subject, indicating that most students scored close to the average. This represents the most uniform performance. This statement is TRUE.


Step 3: Final Answer:
Statements A and D must be true based on the provided graphs.

Answer 2

Rather than reading off each mean and spread separately, it helps to directly compare subjects in pairs against the specific mark mentioned in each statement.

1. Option A: Since English's curve is centred close to the 50-mark line while Hindi's curve is centred further above it, roughly half of English's students sit at or below 50, whereas only a smaller slice of Hindi's students do. So more students fail English than Hindi. True.
2. Option B: The 75-mark line sits far into the thin, distant tail of English's wide, lower-centred curve, whereas for the taller, higher-centred subjects that same mark is much closer to their peak, putting a larger share of students past it. So distinctions are not concentrated in English. False.
3. Option C: Comparing peak positions directly, the curve centred furthest to the right belongs to a different subject than Science, so Science's average is not the highest of the four. False.
4. Option D: Comparing the width of each curve, the narrowest, most tightly bunched curve belongs to Maths, meaning its students' scores cluster closest to the average, which is the definition of the most uniform performance. True.

Direct pairwise comparisons at each stated mark and each curve's spread again single out A and D as the statements that hold.

So the correct answer is A and D.