The following table shows the marks of 90 students in a test of 80 marks: Marks Number of Students 1 – 10 5 11 – 20 8 21 – 30 10 31 – 40 13 41 – 50 18 51 – 60 17 61 – 70 12 71 – 80 7
The percentage of students who have obtained less than or equal to 50% marks is
40%
Calculating Percentage of Students from Marks Distribution Table
This question asks us to find the percentage of students who scored less than or equal to 50% marks in a test that was conducted for a total of 80 marks. To solve this, we first need to determine the mark value that corresponds to 50% of the total marks and then use the provided frequency distribution table to count the number of students who scored at or below this mark value.
Step 1: Determine the Mark Value for 50%
The total marks for the test are 80. To find 50% of 80 marks, we perform the following calculation:
\(50\% \text{ of } 80 = \frac{50}{100} \times 80\)
\(= 0.50 \times 80\)
\(= 40\)
So, students who obtained less than or equal to 50% marks are those who scored 40 marks or less.
Step 2: Identify Students Scoring Less Than or Equal to 40 Marks Using the Table
The provided table shows the number of students in different mark ranges (class intervals). We need to find the number of students in the intervals where the upper limit is less than or equal to 40. Let's look at the table:
| Marks | Number of Students |
| 1 – 10 | 5 |
| 11 – 20 | 8 |
| 21 – 30 | 10 |
| 31 – 40 | 13 |
| 41 – 50 | 18 |
| 51 – 60 | 17 |
| 61 – 70 | 12 |
| 71 – 80 | 7 |
We are interested in students who scored less than or equal to 40 marks. This includes students in the following mark intervals:
- 1 – 10 marks: 5 students
- 11 – 20 marks: 8 students
- 21 – 30 marks: 10 students
- 31 – 40 marks: 13 students
Step 3: Calculate the Total Number of Students Scoring Less Than or Equal to 40 Marks
To find the total number of students who scored 40 marks or less, we sum the number of students in the intervals identified in Step 2:
Total students with marks \(\le 40 = 5 + 8 + 10 + 13\)
\(= 36\) students
Step 4: Calculate the Percentage of Students
The total number of students who took the test is given as 90 (also confirmed by summing the 'Number of Students' column: \(5+8+10+13+18+17+12+7 = 90\)).
The percentage of students who obtained less than or equal to 50% marks (i.e., \(\le 40\) marks) is calculated as:
Percentage \( = \left( \frac{\text{Number of students with marks } \le 40}{\text{Total number of students}} \right) \times 100\)
Percentage \( = \left( \frac{36}{90} \right) \times 100\)
Percentage \( = \left( \frac{36 \div 18}{90 \div 18} \right) \times 100\)
Percentage \( = \left( \frac{2}{5} \right) \times 100\)
Percentage \( = 0.4 \times 100\)
Percentage \( = 40\%\)
Therefore, the percentage of students who have obtained less than or equal to 50% marks is 40%.
Revision Table: Key Concepts in Student Marks Analysis
| Concept | Description | How it Applies Here |
| Total Marks | Maximum possible score in a test. | Used to determine the threshold for 50% marks (80 marks). |
| Percentage Marks | Score expressed as a fraction of total marks, multiplied by 100. | We calculated 50% of total marks to set the scoring threshold. |
| Frequency Distribution Table | A table showing how often each value or range of values occurs in a dataset. | Used to find the number of students within specific mark ranges. |
| Class Interval | A range of values within a frequency distribution table. | The table uses intervals like 1-10, 11-20, etc. |
| Cumulative Frequency | The total frequency up to the upper limit of a class interval (sum of frequencies up to that class). | Implicitly used by summing frequencies of intervals below the threshold. |
Additional Information: Analyzing Grouped Student Data
When student marks are presented in grouped frequency tables like the one in the question, we are dealing with grouped data. We don't know the exact score of each student, only which range their score falls into. However, for questions involving thresholds like "less than or equal to a certain mark", we can sum the frequencies of all class intervals whose upper limit is below or at the specified threshold. This allows us to determine the number of students meeting that criteria.
Understanding cumulative frequency is also helpful here. The cumulative frequency for the 31-40 interval is the total number of students scoring up to 40 marks. This is the sum of frequencies for 1-10, 11-20, 21-30, and 31-40 intervals.
Calculating percentages from grouped data involves finding the number of items (students) in the relevant groups and dividing by the total number of items, then multiplying by 100. This is a fundamental skill in data analysis and statistics.
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