histograms objectives: a grade construct and interpret a histogram with unequal class intervals...
TRANSCRIPT
HistogramsHistograms
Objectives:
A Grade Construct and interpret a histogram with unequal class intervals
Prior knowledge:
Draw a frequency diagram
Rearrange equations
Be able to calculate the area of a rectangle
HistogramsHistograms
Summary of column graphs so far
0
20
40
60
80
100
1st Qtr 2nd Qtr 3rd Qtr 4th Qtr
A Bar ChartSales of Paddling Pools per quarter
A Frequency Diagram
0 2 4 6 8 10
Time taken to complete a Table Test
Time in Minutes
1
2
3
4
5
6
Fre
quen
cySpot the differences from a bar chart to a frequency diagram:• There are no gaps between the bars• The horizontal axis has a continuous scale as it represents continuous data• The width of the bars is the full width of the class interval• The class intervals (the width of the bars) is the same for all bars
HistogramsHistograms
A histogram is a frequency diagram with unequal class intervals.
The area of the bar represents the frequency
Why have different class intervals?
Sometimes data will be collected where there may be very littledata in a class interval, and yet in another class interval of the same width.
e.g. a survey was conducted to find the ages of surfers
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e.g. a survey was conducted to find the ages of 60 surfers, the resultsare in the table below:
Age Frequency 0 < x ≤
154
15 < x ≤ 20
14
20 < x ≤ 25
12
25 < x ≤ 30
8
30 < x ≤ 40
12
40 < x ≤ 70
10
Notice how the class intervals are different
If you kept the class interval of 5 (as for ages 15 to 30) you mayfind several empty class intervals between 40 and 70
because the class width is different if we used frequency for the y-axis the sizes of the bars would represent the frequenciesdisproportionately.
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We use frequency density to show the height.
The frequency is the area of the bar
The width of the bar is the class width
Fre
quen
cy d
ensi
ty
Class width
Frequency=
AREA
Area of a rectangle = length × width
frequency = frequency density × class width
rearranging this equation
frequency density = frequency class width
HistogramsHistograms
Age Frequency 0 < x ≤
155
15 < x ≤ 20
15
20 < x ≤ 25
10
25 < x ≤ 30
8
30 < x ≤ 40
12
40 < x ≤ 70
10
Draw a histogram to represent this dataClass Width Frequency Density
15555
1030
13
32
1.61.2
13
0 10 20 30 40 50 60 70
3
2
1
Fre
quen
cy d
ensi
ty
Age (years)
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From this histogram you could be asked further questions:
0 10 20 30 40 50 60 70
3
2
1
Fre
quen
cy d
ensi
ty
Age (years)
Estimate how many people between the ages of 35 and 55 surf.
Identify the area of the bars to which this applies.
35 55
Remember:frequency is the area
Find the area separately of each bar 5
1.2
15
13
5 × 1.2 = 6 15 × 1 = = 5 3
Estimate of the frequency = 6 + 5 = 11
HistogramsHistogramsDraw Histograms for these tables:
Temperature (nearest oC)
FrequencyClass
IntervalFrequency
Density
10 ≤ t < 11 5
11 ≤ t < 12 7
12 ≤ t < 15 18
15 ≤ t < 16 4
16 ≤ t < 17 3
17 ≤ t < 20 6
Height (cm) FrequencyClass
IntervalFrequency
Density
151 ≤ h < 153 64
153 ≤ h < 154 43
154 ≤ h < 155 47
155 ≤ h < 159 96
159 ≤ h < 160 12
1
1
3
1
1
3
5
7
6
4
3
2
Temperature oC
Freq
uenc
y D
ensi
ty
10
8
6
4
2
010 11 12 13 14 15 16 17 18 19 20
Height (cm)
Freq
uenc
y D
ensi
ty
50
40
30
20
10
0 151 152 153 154 155 156 157 158 159 160
2
1
1
4
1
32
43
47
24
12
HistogramsHistogramsDraw Histograms for these tables:
Time (nearest minute)
FrequencyClass
IntervalFrequency
Density
0 ≤ t < 8 72
9 ≤ t < 12 84
12 ≤ t < 15 54
15 ≤ t < 20 35
Age (nearest year)
FrequencyClass
IntervalFrequency
Density
11 ≤ a < 14 51
14 ≤ a < 16 36
16 ≤ a < 17 12
17 ≤ a < 20 20
8
4
3
5
9
21
18
7
Time (mins)
Freq
uenc
y D
ensi
ty
25
20
15
10
5
0 0 2 4 6 8 10 12 14 16 18 20
3
2
1
3
17
18
12
6.7
Age (years)
Freq
uenc
y D
ensi
ty
25
20
15
10
5
0 11 12 13 14 15 16 17 18 19 20
HistogramsHistogramsDraw Histograms for these tables:
Temperature (nearest oC)
FrequencyClass
IntervalFrequency
Density
10 ≤ t < 11 5
11 ≤ t < 12 7
12 ≤ t < 15 18
15 ≤ t < 16 4
16 ≤ t < 17 3
17 ≤ t < 20 6
Height (cm) FrequencyClass
IntervalFrequency
Density
152 ≤ h < 153 64
153 ≤ h < 154 43
154 ≤ h < 155 47
155 ≤ h < 159 96
159 ≤ h < 160 12
HistogramsHistogramsDraw Histograms for these tables:
Time (nearest minute)
FrequencyClass
IntervalFrequency
Density
0 ≤ t < 8 72
9 ≤ t < 12 84
12 ≤ t < 15 54
15 ≤ t < 20 35
Age (nearest year)
FrequencyClass
IntervalFrequency
Density
11 ≤ a < 14 51
14 ≤ a < 16 36
16 ≤ a < 17 12
17 ≤ a < 20 20