The marks obtained by 5 students are 21, 27, 19, 26, 32. Later on 5 grace marks are added to each student. What are the average marks of the revised marks of the students?
30
Calculating Average Marks After Adding Grace Marks
The problem asks us to find the average marks of 5 students after adding 5 grace marks to each student's original score. We are given the initial marks obtained by the 5 students.
Understanding the Problem
We are given the initial marks of 5 students:
- Student 1: 21
- Student 2: 27
- Student 3: 19
- Student 4: 26
- Student 5: 32
5 grace marks are added to each student's mark. This means we need to increase each score by 5 and then calculate the average of these new scores.
Step-by-Step Calculation of Revised Marks
First, let's find the revised marks for each student by adding 5 to their original marks:
- Student 1 revised marks: \(21 + 5 = 26\)
- Student 2 revised marks: \(27 + 5 = 32\)
- Student 3 revised marks: \(19 + 5 = 24\)
- Student 4 revised marks: \(26 + 5 = 31\)
- Student 5 revised marks: \(32 + 5 = 37\)
So, the revised marks for the 5 students are 26, 32, 24, 31, and 37.
Calculating the Sum of Revised Marks
To find the average of the revised marks, we first need to find their sum.
\text{Sum of revised marks} = 26 + 32 + 24 + 31 + 37
Let's add these numbers:
\text{Sum} = 58 + 24 + 31 + 37
\text{Sum} = 82 + 31 + 37
\text{Sum} = 113 + 37
\text{Sum} = 150
The sum of the revised marks is 150.
Calculating the Average of Revised Marks
The average is calculated by dividing the sum of the marks by the number of students.
\text{Average} = \frac{\text{Sum of revised marks}}{\text{Number of students}}
Number of students = 5
\text{Average} = \frac{150}{5}
\text{Average} = 30
The average marks of the revised marks of the students is 30.
Alternative Method: Using Properties of Average
There is a useful property of averages: if every value in a dataset is increased by a constant value, the average of the dataset also increases by the same constant value.
Let's calculate the initial average marks first.
Sum of initial marks = \(21 + 27 + 19 + 26 + 32\)
\text{Sum} = \(48 + 19 + 26 + 32\)
\text{Sum} = \(67 + 26 + 32\)
\text{Sum} = \(93 + 32\)
\text{Sum} = 125
Initial average = \(\frac{\text{Sum of initial marks}}{\text{Number of students}} = \frac{125}{5} = 25\)
Since 5 grace marks are added to each student's score, the average of the revised marks will be the initial average plus the grace marks.
Revised average = Initial average + Grace marks per student
Revised average = \(25 + 5\)
Revised average = 30
Both methods give the same result, which is 30.
Revision Table: Key Concepts in Calculating Average Marks
| Concept | Definition | Formula |
|---|---|---|
| Average (Mean) | A measure of central tendency; the sum of all values divided by the number of values. | \(\text{Average} = \frac{\text{Sum of values}}{\text{Number of values}}\) |
| Revised Marks | Original marks adjusted by adding or subtracting a value (in this case, adding grace marks). | Revised Mark = Original Mark + Grace Marks |
| Effect of Constant Addition on Average | If a constant 'k' is added to every value in a dataset, the new average is the original average plus 'k'. | \(\text{New Average} = \text{Original Average} + k\) |
Additional Information: Statistics Basics for Average Marks
Calculating the average, or mean, is a fundamental concept in statistics. It gives us a single value that represents a typical score in a dataset, like student marks.
- The average is sensitive to outliers (extremely high or low values).
- Adding a constant to every data point shifts the entire distribution, including the mean, by that constant amount. This is what happened when grace marks were added.
- Multiplying every data point by a constant 'c' would multiply the average by 'c'.
- These properties are useful shortcuts when dealing with data transformations like adding grace marks or converting units.
Understanding how averages change when data is transformed helps in quickly analyzing datasets without recalculating everything from scratch. In this problem, recognizing the property of adding a constant value to each score allowed for a quicker calculation of the revised average marks.
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