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Histograms
This table shows
the height of 50
random
secondary
school students
in the UK
Height Frequency
140≤h<150 8
150≤h<155 9
155≤...
Drawing a histogram
Height Frequency
140≤h<150 8
150≤h<155 9
155≤h<160 4
160≤h<165 9
165≤h<170 9
170≤h<185 11
We need to a...
Drawing a histogram
Height Frequency Class width
140≤h<150 8
150≤h<155 9
155≤h<160 4
160≤h<165 9
165≤h<170 9
170≤h<185 11
...
Drawing a histogram
Giving...
Height Frequency Class width
140≤h<150 8 10
150≤h<155 9 5
155≤h<160 4 5
160≤h<165 9 5
165≤h<...
To draw a histogram for data
with different class widths we
need to calculate the frequency
density for each class.
To calculate
frequency density...
Frequency
Class width
Work out the frequency
density column...
Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<...
Giving...
Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5 9 ÷ 5 = 1.8
155≤h<160 4...
185140 145 150 155 160 165 170 175 180
2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
Heigh...
Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5 9 ÷ 5 = 1.8
155≤h<160 4 5 4 ÷ 5 =...
Height Frequency Class width
Frequency
Density
140≤h<150 8 10 8 ÷ 10 = 0.8
150≤h<155 9 5 9 ÷ 5 = 1.8
155≤h<160 4 5 4 ÷ 5 =...
What, if anything, can we see
from this histogram?
Why?
If we look a bit closer there a
few other useful things about a
histogram.
140 145 150 155 160 165 170 175 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
FrequencyDensity
width
length
For ...
140 145 150 155 160 165 170 175 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
FrequencyDensity
width
length
With...
140 145 150 155 160 165 170 175 180
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
FrequencyDensity
width
length
And ...
How does this work?
Frequency
Density
=
Frequency
Class width
can be rearranged to...
Frequency
Density
=
Frequency
Class width
How does this work?
Frequency =
Frequency
Density
× Class width
can be rearrange...
Frequency =
Frequency
Density
× Class width
width of the bar
length of the bar
area of the bar
KEY POINTS
• To draw a histogram for unequal class widths
you must calculate frequency density
•
• The area of a bar on a ...
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Aqa Mod1 Histograms

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Aqa Mod1 Histograms

  1. 1. Histograms
  2. 2. This table shows the height of 50 random secondary school students in the UK Height Frequency 140≤h<150 8 150≤h<155 9 155≤h<160 4 160≤h<165 9 165≤h<170 9 170≤h<185 11 Source: censusatschool.org.uk
  3. 3. Drawing a histogram Height Frequency 140≤h<150 8 150≤h<155 9 155≤h<160 4 160≤h<165 9 165≤h<170 9 170≤h<185 11 We need to add a class width column to do this...
  4. 4. Drawing a histogram Height Frequency Class width 140≤h<150 8 150≤h<155 9 155≤h<160 4 160≤h<165 9 165≤h<170 9 170≤h<185 11 Like so...
  5. 5. Drawing a histogram Giving... Height Frequency Class width 140≤h<150 8 10 150≤h<155 9 5 155≤h<160 4 5 160≤h<165 9 5 165≤h<170 9 5 170≤h<185 11 15
  6. 6. To draw a histogram for data with different class widths we need to calculate the frequency density for each class.
  7. 7. To calculate frequency density... Frequency Class width
  8. 8. Work out the frequency density column... Height Frequency Class width Frequency Density 140≤h<150 8 10 8 ÷ 10 = 0.8 150≤h<155 9 5 155≤h<160 4 5 160≤h<165 9 5 165≤h<170 9 5 170≤h<185 11 15
  9. 9. Giving... Height Frequency Class width Frequency Density 140≤h<150 8 10 8 ÷ 10 = 0.8 150≤h<155 9 5 9 ÷ 5 = 1.8 155≤h<160 4 5 4 ÷ 5 = 0.8 160≤h<165 9 5 9 ÷ 5 = 1.8 165≤h<170 9 5 9 ÷ 5 = 1.8 170≤h<185 11 15 11 ÷ 15 = 0.73
  10. 10. 185140 145 150 155 160 165 170 175 180 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Height (cm) FrequencyDensity Height Frequency Class width Frequency Density 140≤h<150 8 10 8 ÷ 10 = 0.8 150≤h<155 9 5 9 ÷ 5 = 1.8 155≤h<160 4 5 4 ÷ 5 = 0.8 160≤h<165 9 5 9 ÷ 5 = 1.8 165≤h<170 9 5 9 ÷ 5 = 1.8 170≤h<185 11 15 11 ÷ 15 = 0.73 Now use your values to draw the histogram... remember not to leave any gaps between the bars.
  11. 11. Height Frequency Class width Frequency Density 140≤h<150 8 10 8 ÷ 10 = 0.8 150≤h<155 9 5 9 ÷ 5 = 1.8 155≤h<160 4 5 4 ÷ 5 = 0.8 160≤h<165 9 5 9 ÷ 5 = 1.8 165≤h<170 9 5 9 ÷ 5 = 1.8 170≤h<185 11 15 11 ÷ 15 = 0.73 Now use your values to draw the histogram... remember not to leave any gaps between the bars. 185140 145 150 155 160 165 170 175 180 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Height (cm) FrequencyDensity
  12. 12. Height Frequency Class width Frequency Density 140≤h<150 8 10 8 ÷ 10 = 0.8 150≤h<155 9 5 9 ÷ 5 = 1.8 155≤h<160 4 5 4 ÷ 5 = 0.8 160≤h<165 9 5 9 ÷ 5 = 1.8 165≤h<170 9 5 9 ÷ 5 = 1.8 170≤h<185 11 15 11 ÷ 15 = 0.73 Now use your values to draw the histogram... remember not to leave any gaps between the bars. 185140 145 150 155 160 165 170 175 180 2 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 Height (cm) FrequencyDensity
  13. 13. What, if anything, can we see from this histogram? Why?
  14. 14. If we look a bit closer there a few other useful things about a histogram.
  15. 15. 140 145 150 155 160 165 170 175 180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 FrequencyDensity width length For each histogram bar the width is the class width and the length is the frequency density.
  16. 16. 140 145 150 155 160 165 170 175 180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 FrequencyDensity width length With any rectangle if we do width x length we get the area of the rectangle.
  17. 17. 140 145 150 155 160 165 170 175 180 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 FrequencyDensity width length And for a histogram the area of the bar is equal to the frequency for that class.
  18. 18. How does this work? Frequency Density = Frequency Class width can be rearranged to...
  19. 19. Frequency Density = Frequency Class width How does this work? Frequency = Frequency Density × Class width can be rearranged to...
  20. 20. Frequency = Frequency Density × Class width width of the bar length of the bar area of the bar
  21. 21. KEY POINTS • To draw a histogram for unequal class widths you must calculate frequency density • • The area of a bar on a frequency data histogram is equal to the frequency • Frequency Density = Frequency Class width Frequency = Frequency Density × Class width

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