SOLUTION: 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

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.

--------------------------------------------------------------------------

Answers:

f4 = 15 and f5 = 14
mean = 55.266667 approximately

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