To present the data in the form of a frequency polygon using a histogram, follow these detailed steps:
### Step 1: Organize the Data
First, take note of the class intervals and the number of workers in each class:
- Class Intervals (Daily Wages in ₹): [tex]\(60-80\)[/tex], [tex]\(80-100\)[/tex], [tex]\(100-120\)[/tex], [tex]\(120-140\)[/tex], [tex]\(140-160\)[/tex], [tex]\(160-180\)[/tex], [tex]\(180-200\)[/tex]
- Number of Workers: 3, 5, 10, 15, 7, 4, 2
### Step 2: Calculate the Midpoints of Each Class Interval
To create a frequency polygon, you use the midpoint of each class interval. The midpoint (or class mark) is the average of the lower and upper limits of the class interval.
For the class interval [tex]\(60-80\)[/tex]:
[tex]\[
\text{Midpoint} = \frac{60 + 80}{2} = 70.0
\][/tex]
Repeating this for each class interval:
- [tex]\(80-100\)[/tex]: [tex]\[
\frac{80 + 100}{2} = 90.0
\][/tex]
- [tex]\(100-120\)[/tex]: [tex]\[
\frac{100 + 120}{2} = 110.0
\][/tex]
- [tex]\(120-140\)[/tex]: [tex]\[
\frac{120 + 140}{2} = 130.0
\][/tex]
- [tex]\(140-160\)[/tex]: [tex]\[
\frac{140 + 160}{2} = 150.0
\][/tex]
- [tex]\(160-180\)[/tex]: [tex]\[
\frac{160 + 180}{2} = 170.0
\][/tex]
- [tex]\(180-200\)[/tex]: [tex]\[
\frac{180 + 200}{2} = 190.0
\][/tex]
So, the midpoints for the respective intervals are:
[tex]\[
[70.0, 90.0, 110.0, 130.0, 150.0, 170.0, 190.0]
\][/tex]
### Step 3: Prepare Frequency Polygon Data Points
Pair each midpoint with the corresponding number of workers to form data points for the frequency polygon:
[tex]\[
(70.0, 3), (90.0, 5), (110.0, 10), (130.0, 15), (150.0, 7), (170.0, 4), (190.0, 2)
\][/tex]
### Step 4: Plot the Histogram and Frequency Polygon
Construct the histogram by plotting the class intervals [tex]\(60-80\)[/tex], [tex]\(80-100\)[/tex], [tex]\(100-120\)[/tex], [tex]\(120-140\)[/tex], [tex]\(140-160\)[/tex], [tex]\(160-180\)[/tex], [tex]\(180-200\)[/tex] on the x-axis and the number of workers on the y-axis.
(Here, visualize the bars corresponding to each class interval. The height of each bar represents the number of workers in that interval.)
### Step 5: Draw the Frequency Polygon
To draw the frequency polygon:
1. Plot each midpoint on the x-axis against the number of workers (as listed in Step 3).
2. Connect these plotted points with straight lines to form the frequency polygon.
Points to be plotted:
- (70.0, 3), (90.0, 5), (110.0, 10), (130.0, 15), (150.0, 7), (170.0, 4), (190.0, 2)
These points when connected will form the frequency polygon on top of the histogram bars.
### Conclusion
You've organized the data, calculated midpoints for the class intervals, and paired them with the frequencies to create coordinates for the frequency polygon. By plotting these points and connecting them, you successfully represent the data through a frequency polygon overlaying a histogram.
