Interpret Grouped Frequency Tables

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Question 1 of 6

1. Question

A bus depot conducted a survey on how many passengers were on each bus through an `8` hour period. How many buses had less than `20` passengers?

`\text(Class)`	`\text(Class Centre) (c.c.)`	`\text(Frequency) (f)`	`f⋅c.c.`
`0-4`	`2`	`5`	`10`
`5-9`	`7`	`9`	`63`
`10-14`	`12`	`13`	`156`
`15-19`	`17`	`10`	`170`
`20-24`	`22`	`7`	`154`
`25-29`	`27`	`2`	`54`

(37) `\text(buses)`

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A grouped frequency distribution table is used for larger data sets and uses classes with intervals to group the scores.

First, identify which classes has less than `20` passengers.

Class	Class Centre `(c.c.)`	Frequency `(f)`	`f⋅c.c.`
0-4	2	5	10
5-9	7	9	63
10-14	12	13	156
15-19	17	10	170
20-24	22	7	154
25-29	27	2	54

Finally, add up the frequencies of the highlighted classes to get the total number of buses.

Class	Class Centre `(c.c.)`	Frequency `(f)`	`f⋅c.c.`
0-4	2	5	10
5-9	7	9	63
10-14	12	13	156
15-19	17	10	170
20-24	22	7	154
25-29	27	2	54

`\text(Total Buses)`	`=`	`5+9+13+10`
	`=`	`37`

`37 \text(buses had less than) 20 \text(passengers)`

Question 2 of 6

2. Question

A bus depot conducted a survey on how many passengers were on each bus through an `8` hour period. Find the modal class.

`\text(Class)`	`\text(Class Centre) (c.c.)`	`\text(Frequency) (f)`	`f⋅c.c.`
`0-4`	`2`	`5`	`10`
`5-9`	`7`	`9`	`63`
`10-14`	`12`	`13`	`156`
`15-19`	`17`	`10`	`170`
`20-24`	`22`	`7`	`154`
`25-29`	`27`	`2`	`54`

`\text(Modal Class )=` (10)`-` (14)

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A modal class is the group of scores with the highest frequency.

Notice that the highest value in the Frequency column is `13` and it corresponds to `10-14`.

Class	Class Centre `(c.c.)`	Frequency `(f)`	`f⋅c.c.`
0-4	2	5	10
5-9	7	9	63
10-14	12	13	156
15-19	17	10	170
20-24	22	7	154
25-29	27	2	54

In other words, the class `10-14` occurs the most frequently, and is therefore the modal class.

`\text(Modal Class)=10-14`

Question 3 of 6

3. Question

A bus depot conducted a survey on how many passengers were on each bus through an `8` hour period. Find the mean.

`\text(Class)`	`\text(Class Centre) (c.c.)`	`\text(Frequency) (f)`	`f⋅c.c.`
`0-4`	`2`	`5`	`10`
`5-9`	`7`	`9`	`63`
`10-14`	`12`	`13`	`156`
`15-19`	`17`	`10`	`170`
`20-24`	`22`	`7`	`154`
`25-29`	`27`	`2`	`54`

Round off your answer to one decimal place.

`\text(Mean )=` (13.2)

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Mean Formula

$$\text{Mean}=\frac{\color{#314EC4}{\sum f⋅c.c.}}{\color{#9202AA}{\sum f}}$$

Remember

A grouped frequency distribution table is used for larger data sets and uses classes with intervals to group the scores.

First, find the sum of both the Frequency and `f⋅c.c.` columns.

`sum f`	`=`	`5+9+13+10+7+2`
	`=`	`46`

`sum f⋅x`	`=`	`10+63+156+170+154+54`
	`=`	`607`

Class	Class Centre `(c.c.)`	Frequency `(f)`	`f⋅c.c`
0-4	2	5	10
5-9	7	9	63
10-14	12	13	156
15-19	17	10	170
20-24	22	7	154
25-29	27	2	54
		`\text(Total) =46`	`\text(Total) =607`

Finally, use the formula to compute for the mean.

`\text(Mean)`	`=`	$$\frac{\color{#314EC4}{\sum f⋅c.c.}}{\color{#9202AA}{\sum f}}$$	Mean Formula

`\text(Mean)`	`=`	$$\frac{\color{#314EC4}{607}}{\color{#9202AA}{46}}$$	Substitute values

`\text(Mean)`	`=`	`13.2`	Rounded to one decimal place

`\text(Mean)=13.2`

Question 4 of 6

4. Question

The mass of `32` students was measured and the results are shown. Complete the grouped frequency distribution table and find the mean.

Class	Class Centre `(c.c.)`	Frequency `(f)`
51-55	53	1
56-60	58	4
61-65	63	8
66-70	68	10
71-75	73	6
76-80	78	2
81-85	83	1
		`\text(Total) =32`

Round off your answer to one decimal place.

`\text(Mean )=` (67.1)

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Fantastic!

Incorrect

Mean Formula

$$\text{Mean}=\frac{\color{#314EC4}{\sum f⋅c.c.}}{\color{#9202AA}{\sum f}}$$

Remember

A grouped frequency distribution table is used for larger data sets and uses classes with intervals to group the scores.

First, to complete the table, fill in the `f⋅c.c` column by multiplying `f` and `c.c.` on each row.

Class	Class Centre `(c.c.)`	Frequency `(f)`	`f⋅c.c`
51-55	53	1	53
56-60	58	4	232
61-65	63	8	504
66-70	68	10	680
71-75	73	6	438
76-80	78	2	156
81-85	83	1	83
		`\text(Total) =32`

Next, find the sum of `f⋅c.c.` column.

`sum f⋅x`	`=`	`53+232+504+680+438+156+83`
	`=`	`2146`

Class	Class Centre `(c.c.)`	Frequency `(f)`	`f⋅c.c`
51-55	53	1	53
56-60	58	4	232
61-65	63	8	504
66-70	68	10	680
71-75	73	6	438
76-80	78	2	156
81-85	83	1	83
		`\text(Total) =32`	`\text(Total) =2146`

Finally, use the formula to compute for the mean.

`\text(Mean)`	`=`	$$\frac{\color{#314EC4}{\sum f⋅c.c.}}{\color{#9202AA}{\sum f}}$$	Mean Formula

`\text(Mean)`	`=`	$$\frac{\color{#314EC4}{2146}}{\color{#9202AA}{32}}$$	Substitute values

`\text(Mean)`	`=`	`67.1`	Rounded to one decimal place

`\text(Mean)=67.1`

Question 5 of 6

5. Question

A bus depot conducted a survey on how many passengers were on each bus through an `8` hour period. Find the median class.

`\text(Class)`	`\text(Class Centre) (c.c.)`	`\text(Frequency) (f)`
`0-4`	`2`	`5`
`5-9`	`7`	`9`
`10-14`	`12`	`13`
`15-19`	`17`	`10`
`20-24`	`22`	`7`
`25-29`	`27`	`2`
		`\text(Total) =46`

`\text(Median Class )=` (10)- (14)

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The class where a data set’s median is located is called the median class.

Add a Cumulative Frequency column to the table.

To find the median, fill in the Cumulative Frequency column by adding the frequency, as seen below.

Class	Class Centre `(c.c.)`	Frequency `(f)`	Cumulative Frequency
0-4	2	5	5
5-9	7	9	5+9=14
10-14	12	13	14+13=27
15-19	17	10	27+10=37
20-24	22	7	37+7=44
25-29	27	2	44+2=46
		`\text(Total) =46`

To check, the last entry in the Cumulative Frequency column must be equal to the
Total Frequency.

Since the total number of scores is `46`, the middle score should be the average of the `23`rd and `24`th scores.

Find the row that includes the `23`rd and `24`th scores.

Class	Class Centre `(c.c.)`	Frequency `(f)`	Cumulative Frequency
0-4	2	5	5
5-9	7	9	5+9=14
10-14	12	13	14+13=27
15-19	17	10	27+10=37
20-24	22	7	37+7=44
25-29	27	2	44+2=46
		`\text(Total) =46`

The third row covers all scores between the `15`th and `27`th position — this includes the `23`rd and `24`th.

Since we are looking for the median class, it will be the class where the two scores are located, which is `10-14`.

`\text(Median Class)=10-14`

Question 6 of 6

6. Question

The mass of `32` students was measured and the results are shown. Find the median class.

`\text(Class)`	`\text(Class Centre) (c.c.)`	`\text(Frequency) (f)`
`51-55`	`53`	`1`
`56-60`	`58`	`4`
`61-65`	`63`	`8`
`66-70`	`68`	`10`
`71-75`	`73`	`6`
`76-80`	`78`	`2`
`81-85`	`83`	`1`
		`\text(Total) =32`

`\text(Median Class )=` (66)- (70)

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Incorrect

The class where a data set’s median is located is called the median class.

First, add a Cumulative Frequency column to the table.

To find the median, fill in the Cumulative Frequency column by adding the frequency, as seen below.

Class	Class Centre `(c.c.)`	Frequency `(f)`	Cumulative Frequency
51-55	53	1	1
56-60	58	4	1+4=5
61-65	63	8	5+8=13
66-70	68	10	13+10=23
71-75	73	6	23+6=29
76-80	78	2	29+2=31
81-85	83	1	31+1=32
		`\text(Total) =32`

To check, the last entry in the Cumulative Frequency column must be equal to the
Total Frequency.

Since the total number of scores is `32`, the middle score should be the average of the `16`th and `17`th scores.

Find the row that includes the `16`rd and `17`th scores.

Class	Class Centre `(c.c.)`	Frequency `(f)`	Cumulative Frequency
51-55	53	1	1
56-60	58	4	1+4=5
61-65	63	8	5+8=13
66-70	68	10	13+10=23
71-75	73	6	23+6=29
76-80	78	2	29+2=31
81-85	83	1	31+1=32
		`\text(Total) =32`

The fourth row covers all scores between the `14`th and `23`rd position — this includes the `16`th and `17`th.

Since we are looking for the median class, it will be the class where the two scores are located, which is `66-70`.

`\text(Median Class)=66-70`

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