aqa mod1 histograms
TRANSCRIPT
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≤h<160 4
160≤h<165 9
165≤h<170 9
170≤h<185 11
Source: censusatschool.org.uk
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...
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...
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
To draw a histogram for data with different class widths we
need to calculate the frequency density for each class.
To calculate frequency density...
FrequencyClass width
Work out the frequency density column...
Height Frequency Class widthFrequency 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
Giving...
Height Frequency Class widthFrequency 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
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)
Fre
quency D
ensit
y
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.
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)
Fre
quency D
ensit
y
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)
Fre
quency D
ensit
y
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.
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)
Fre
quency D
ensit
y
width
length
For each histogram bar the width is theclass width and the
length is the frequency density.
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)
Fre
quency D
ensit
y
width
length
With any rectangle if we do width x length we get the area of
the rectangle.
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)
Fre
quency D
ensit
y
width
length
And for a histogram the area of the bar is
equal to the frequency for that
class.
How does this work?
FrequencyDensity
=FrequencyClass width
can be rearranged to...
FrequencyDensity
=FrequencyClass width
How does this work?
Frequency =
FrequencyDensity
× Class width
can be rearranged to...
Frequency =
FrequencyDensity
× 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 frequency data histogram is equal to the frequency
•
FrequencyDensity
=FrequencyClass width
Frequency =
FrequencyDensity
× Class width