Question 1208080: Marks of 75 students are summarized in the following frequency distribution,
Marks Number of students
40-44 7
45-49 10
50-54 20
55-59 f4
60-64 f5
65-69 6
70-74 3
If 20% of the students have marks between 55 and 59
i. Find the missing frequencies f4 and 15.
ii. Find the mean.
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Part (i)
Original table
| Marks | Number of students | | 40-44 | 7 | | 45-49 | 10 | | 50-54 | 20 | | 55-59 | f4 | | 60-64 | f5 | | 65-69 | 6 | | 70-74 | 3 |
Spreadsheet software is strongly recommended.
20% of 75 = 0.20*75 = 15 students have marks between 55 and 59.
This means f4 = 15.
Add up the frequencies in the 2nd column.
Set this sum equal to 75 so we can determine the value of f5.
7+10+20+f4+f5+6+3 = 75
7+10+20+15+f5+6+3 = 75
61+f5 = 75
f5 = 75-61
f5 = 14
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Part (ii)
In the previous part we found that
f4 = 15 and f5 = 14
After replacing f4 and f5 with those values, we now have this grouped frequency table.
| Marks | Number of students | | 40-44 | 7 | | 45-49 | 10 | | 50-54 | 20 | | 55-59 | 15 | | 60-64 | 14 | | 65-69 | 6 | | 70-74 | 3 |
Let's introduce a new column which I'll refer to as column m.
m = midpoint of the corresponding class interval
To find the midpoint, add the endpoints and divide by 2.
Example: m = 42 for the first class since (40+44)/2 = 42
Here's what the table looks like now
| Marks | Number of students | m | | 40-44 | 7 | 42 | | 45-49 | 10 | 47 | | 50-54 | 20 | 52 | | 55-59 | 15 | 57 | | 60-64 | 14 | 62 | | 65-69 | 6 | 67 | | 70-74 | 3 | 72 |
The midpoint is the best representative mark from each class interval.
Multiply the frequency value (f) with its corresponding midpoint (m).
This will form a new column which I'll label as f*m.
For example, f*m = 7*42 = 294 is the first item in this new column.
| Marks | Number of students | M = midpoint | f*m | | 40-44 | 7 | 42 | 294 | | 45-49 | 10 | 47 | 470 | | 50-54 | 20 | 52 | 1040 | | 55-59 | 15 | 57 | 855 | | 60-64 | 14 | 62 | 868 | | 65-69 | 6 | 67 | 402 | | 70-74 | 3 | 72 | 216 |
Add up the values in this new column to get
294+470+1040+855+868+402+216 = 4145
Then divide this over the total number of people (75) to get 4145/75 = 55.266667 which is the approximate mean.
The 6's go on forever but we have to round at some point.
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Answers:- f4 = 15 and f5 = 14
- mean = 55.266667 approximately
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