Histograms: Unequal class Intervals In a histogram the heights of the bars give the frequency. We could if we wished to, use the area of the bars to represent frequency, since the height of a bar is proportional to its area. Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 The area of each bar is class width x height 90 80 Speed (s) Frequency 0 < s < 10 2 10 < s < 20 6 20 < s < 30 16 30 < s < 40 32 40 < s < 50 50 50 < s < 60 78 60 < s < 70 96 70 < s < 80 82 80 < s < 90 36 90 < s < 100 14 100 < s < 110 8 110 < s < 120 70 This second bar has twice the height and therefore twice the area of the first. 60 10 x 32 frequency 50 40 10 x 16 320 30 20 160 10 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals It is sometimes desirable to combine two or more bars into a single, larger, class interval. Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 We are going to combine the first 3 bars of this histogram into a single bar with a (0 - 30) class interval When we do this we can no longer use the height of the bar to represent frequency. We must use its AREA 90 80 Speed (s) Frequency 0 < s < 10 2 10 < s < 20 6 20 < s < 30 16 30 < s < 40 32 40 < s < 50 50 50 < s < 60 78 60 < s < 70 96 70 < s < 80 82 80 < s < 90 36 90 < s < 100 14 100 < s < 110 8 110 < s < 120 70 60 Total area of bars = 240 Our single bar must also have this area. frequency 50 These two bars are not that clear so we could combine them with the 3rd . 40 240 30 ? 30 20 160 10 60 20 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals It is sometimes desirable to combine two or more bars into a single, larger, class interval. Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 We are going to combine the first 3 bars of this histogram into a single bar with a (0 - 30) class interval When we do this we can no longer use the height of the bar to represent frequency. We must use its AREA 90 80 Speed (s) Frequency 0 < s < 10 2 10 < s < 20 6 20 < s < 30 16 30 < s < 40 32 40 < s < 50 50 50 < s < 60 78 60 < s < 70 96 70 < s < 80 82 80 < s < 90 36 90 < s < 100 14 100 < s < 110 8 110 < s < 120 70 60 Total area of bars = 240 Our single bar must also have this area. frequency 50 40 240 30 ? 30 20 10 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals It is sometimes desirable to combine two or more class intervals to give a clearer picture. This is usually done with low frequency adjacent bars. Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 Repeating this for the last 4 bars gives: 90 80 Speed (s) Frequency 0 < s < 10 2 10 < s < 20 6 20 < s < 30 16 30 < s < 40 32 40 < s < 50 50 50 < s < 60 78 60 < s < 70 96 70 < s < 80 82 80 < s < 90 36 90 < s < 100 14 100 < s < 110 8 110 < s < 120 70 Total area of bars = 640 60 Our single bar must also have this area. frequency 50 40 360 640 80 120 ? 30 20 140 80 10 60 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals It is sometimes desirable to combine two or more class intervals to give a clearer picture. This is usually done with low frequency adjacent bars. Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 Repeating this for the last 3 bars gives: 90 80 Speed (s) Frequency 0 < s < 10 2 10 < s < 20 6 20 < s < 30 16 30 < s < 40 32 40 < s < 50 50 50 < s < 60 78 60 < s < 70 96 70 < s < 80 82 80 < s < 90 36 90 < s < 100 14 100 < s < 110 8 110 < s < 120 70 Total area of bars = 640 60 Our single bar must also have this area. frequency 50 40 640 80 120 ? 30 20 10 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR, Not the height Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 90 80 Speed (s) Frequency 0 < s < 10 2 10 < s < 20 6 20 < s < 30 16 30 < s < 40 32 40 < s < 50 50 50 < s < 60 78 60 < s < 70 96 70 < s < 80 82 80 < s < 90 36 90 < s < 100 14 100 < s < 110 8 110 < s < 120 The frequency table needs sorting out! 70 This has to go We now give the vertical axis a new label called frequency density. frequency 60 frequency 50 frequency density 40 30 20 10 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR Not the height Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 90 80 Speed (s) Frequency 0 < s < 30 240 30 < s < 40 320 40 < s < 50 500 50 < s < 60 780 60 < s < 70 960 70 < s < 80 820 80 < s < 120 640 Speed (s) Frequency 0 < s < 30 240 30 < s < 40 40 < s < 50 50 < s < 60 60 < s < 70 70 < s < 80 80 < s < 90 640 70 60 50 frequency density 40 320 500 780 960 820 30 20 10 640 240 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR Not the height Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 90 80 Speed (s) Frequency 0 < s < 30 240 30 < s < 40 320 40 < s < 50 500 50 < s < 60 780 60 < s < 70 960 70 < s < 80 820 80 < s < 120 640 This needs tidying up and we need a column for frequency density 70 60 50 frequency density Which bar has the higher frequency? 40 320 500 780 960 820 30 20 10 640 240 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
Histograms: Unequal class Intervals Remember: When dealing with a histogram with unequal class intervals, frequency is represented by the AREA of the BAR Not the height Histograms: Unequal class Intervals Recorded Speeds of Cars on a Motorway 100 90 80 Speed f fd 0 - 30 240 8 30 - 40 320 32 40 - 50 500 50 50 - 60 780 78 60 - 70 960 96 70 - 80 820 82 80 - 120 640 16 Why is it called frequency density? The situation is similar to the normal use of the term density. 70 60 50 In this case its how much area you get per standard class width. frequency density 40 320 500 780 960 820 30 16 Remember: frequency density is just the height of each bar. 20 8 10 640 240 10 20 30 40 50 60 70 80 90 100 110 120 Speed mph
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