Histograms - Higher tier
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A histogram looks like a A type of graph showing values that are represented by rectangular bars., except the area of the bar, and not the height, shows the frequency of the Values, typically letters or numbers.. The vertical axis shows the frequency density.
Histograms are typically used when the data is recorded in classes of unequal width.
frequency density = \(\frac{frequency}{class~width}\)
Example
The table below shows the lengths of 40 babies at birth. Columns for class width and frequency density have then been added.
| Length (cm) | Frequency | Class width | Frequency density |
| \(30 \textless l \leq 35\) | 5 | 5 | \(5\div5=1\) |
| \(35 \textless l \leq 40\) | 10 | 5 | \(10 \div 5 = 2\) |
| \(40 \textless l \leq 42\) | 8 | 2 | \(8 \div 2 = 4\) |
| \(42 \textless l \leq 44\) | 7 | 2 | \(6 \div 3 = 3.5\) |
| \(44 \textless l \leq 46\) | 4 | 2 | \(4 \div 2 = 2\) |
| \(46 \textless l \leq 54\) | 4 | 8 | \(4 \div 8 = 0.5\) |
| Length (cm) | \(30 \textless l \leq 35\) |
|---|---|
| Frequency | 5 |
| Class width | 5 |
| Frequency density | \(5\div5=1\) |
| Length (cm) | \(35 \textless l \leq 40\) |
|---|---|
| Frequency | 10 |
| Class width | 5 |
| Frequency density | \(10 \div 5 = 2\) |
| Length (cm) | \(40 \textless l \leq 42\) |
|---|---|
| Frequency | 8 |
| Class width | 2 |
| Frequency density | \(8 \div 2 = 4\) |
| Length (cm) | \(42 \textless l \leq 44\) |
|---|---|
| Frequency | 7 |
| Class width | 2 |
| Frequency density | \(6 \div 3 = 3.5\) |
| Length (cm) | \(44 \textless l \leq 46\) |
|---|---|
| Frequency | 4 |
| Class width | 2 |
| Frequency density | \(4 \div 2 = 2\) |
| Length (cm) | \(46 \textless l \leq 54\) |
|---|---|
| Frequency | 4 |
| Class width | 8 |
| Frequency density | \(4 \div 8 = 0.5\) |
The histogram can now be drawn.
Using a histogram to estimate frequencies
The area represents frequency.
\(frequency = frequency~density \times class~width\)
Example
To estimate the number of babies whose length lies in the interval \( 33 \textless{l}\leq{41}\), find the sum of the areas of the three rectangles, 33 to 35, 35 to 40 and 40 to 41:
(frequency density × width) gives \((1 \times 2) + (2 \times 5) + (4 \times 1) = 2 + 10 + 4 = 16\).
