To construct a histogram from the given data, we need to follow a series of steps. Here is a detailed, step-by-step solution for drawing a histogram:
### Step 1: Organize the Data
The numbers represent the miles biked last week by 18 cyclists:
[tex]\[ 11, 7, 5, 9, 11, 24, 4, 23, 10, 18, 4, 4, 6, 20, 23, 21, 13, 12 \][/tex]
### Step 2: Determine the Range of Data
Find the minimum and maximum values in the data set.
- Minimum value: [tex]\(4\)[/tex]
- Maximum value: \ [tex]\(24\)[/tex]
### Step 3: Determine the Number of Bins
A common choice for the number of bins ([tex]\(k\)[/tex]) is the square root of the number of data points rounded to the nearest whole number.
[tex]\[ k \approx \sqrt{18} \approx 4.2 \][/tex]
We can either round this to 4 or 5 bins. Let's use [tex]\(5\)[/tex] bins for this example.
### Step 4: Calculate the Bin Width
The bin width can be calculated using the formula:
[tex]\[ \text{Bin width} = \frac{\text{Range}}{\text{Number of bins}} = \frac{24 - 4}{5} = \frac{20}{5} = 4 \][/tex]
### Step 5: Create Bins
Based on the bin width of 4, we can create the following bins:
1. [tex]\(4 - 7.9\)[/tex]
2. [tex]\(8 - 11.9\)[/tex]
3. [tex]\(12 - 15.9\)[/tex]
4. [tex]\(16 - 19.9\)[/tex]
5. [tex]\(20 - 23.9\)[/tex]
6. [tex]\(24 - 27.9\)[/tex]
### Step 6: Tally the Data into Bins
Count how many data points fall into each bin:
1. [tex]\(4 - 7.9\)[/tex]: [tex]\(4, 4, 4, 5, 6, 7\)[/tex] [tex]\(\rightarrow 6\)[/tex]
2. [tex]\(8 - 11.9\)[/tex]: [tex]\(9, 10, 11, 11\)[/tex] [tex]\(\rightarrow 4\)[/tex]
3. [tex]\(12 - 15.9\)[/tex]: [tex]\(12, 13\)[/tex] [tex]\(\rightarrow 2\)[/tex]
4. [tex]\(16 - 19.9\)[/tex]: [tex]\(18\)[/tex] [tex]\(\rightarrow 1\)[/tex]
5. [tex]\(20 - 23.9\)[/tex]: [tex]\(20, 21, 23, 23\)[/tex] [tex]\(\rightarrow 4\)[/tex]
6. [tex]\(24 - 27.9\)[/tex]: [tex]\(24\)[/tex] [tex]\(\rightarrow 1\)[/tex]
### Step 7: Draw the Histogram
With this information, we can now draw the histogram.
1. Horizontal Axis (x-axis): The number of miles split into the bins.
2. Vertical Axis (y-axis): The frequency of data points in each bin.
The histogram would look as follows:
| Bin | Frequency |
|-----------|------------|
| 4 - 7.9 | 6 |
| 8 - 11.9 | 4 |
|12 - 15.9 | 2 |
|16 - 19.9 | 1 |
|20 - 23.9 | 4 |
|24 - 27.9 | 1 |
### Creating the Histogram:
- For the bin [tex]\(4 - 7.9\)[/tex], draw a bar up to the height of 6.
- For the bin [tex]\(8 - 11.9\)[/tex], draw a bar up to the height of 4.
- For the bin [tex]\(12 - 15.9\)[/tex], draw a bar up to the height of 2.
- For the bin [tex]\(16 - 19.9\)[/tex], draw a bar up to the height of 1.
- For the bin [tex]\(20 - 23.9\)[/tex], draw a bar up to the height of 4.
- For the bin [tex]\(24 - 27.9\)[/tex], draw a bar up to the height of 1.
This completes the construction of the histogram for the given data.
