To solve this problem and estimate the mode of the given data, follow these steps:
### Step 1: Understand the Data
The data table provided gives the number of runs scored by 50 batsmen in different intervals and the number of batsmen in each interval:
- Runs Scored: [tex]\((3000-4000), (4000-5000), (5000-6000), (6000-7000), (7000-8000), (8000-9000), (9000-10000)\)[/tex]
- Number of Batsmen: [tex]\(4, 18, 9, 6, 7, 2, 4\)[/tex]
### Step 2: Find the Midpoints of Each Interval
To draw the histogram, we will use the midpoints of the intervals. The midpoint of an interval is calculated as:
[tex]\[ \text{Midpoint} = \frac{\text{Lower Limit} + \text{Upper Limit}}{2} \][/tex]
- Midpoint of [tex]\(3000-4000\)[/tex]: [tex]\(\frac{3000 + 4000}{2} = 3500\)[/tex]
- Midpoint of [tex]\(4000-5000\)[/tex]: [tex]\(\frac{4000 + 5000}{2} = 4500\)[/tex]
- Midpoint of [tex]\(5000-6000\)[/tex]: [tex]\(\frac{5000 + 6000}{2} = 5500\)[/tex]
- Midpoint of [tex]\(6000-7000\)[/tex]: [tex]\(\frac{6000 + 7000}{2} = 6500\)[/tex]
- Midpoint of [tex]\(7000-8000\)[/tex]: [tex]\(\frac{7000 + 8000}{2} = 7500\)[/tex]
- Midpoint of [tex]\(8000-9000\)[/tex]: [tex]\(\frac{8000 + 9000}{2} = 8500\)[/tex]
- Midpoint of [tex]\(9000-10000\)[/tex]: [tex]\(\frac{9000 + 10000}{2} = 9500\)[/tex]
### Step 3: Draw the Histogram
1. Axes: Use a graph paper and draw the x-axis and y-axis. Label the x-axis as "Runs Scored" and the y-axis as "Number of Batsmen."
2. Intervals on x-axis: Mark the midpoints calculated above (3500, 4500, 5500, 6500, 7500, 8500, 9500).
3. Number of Batsmen on y-axis: Mark the range based on the maximum value in the given data (0 to 20 should suffice as the peak number of batsmen is 18).
For each interval, draw a bar whose height corresponds to the number of batsmen in that interval:
- From 3500-4500, height = 4
- From 4500-5500, height = 18
- From 5500-6500, height = 9
- From 6500-7500, height = 6
- From 7500-8500, height = 7
- From 8500-9500, height = 2
- From 9500-10500, height = 4
### Step 4: Estimate the Mode
The mode of the histogram is the interval with the highest bar, indicating the highest frequency. From the data provided, the highest bar is for the interval [tex]\(4000-5000\)[/tex] with 18 batsmen.
To estimate the mode more precisely, you may use the midpoint of this interval, as it represents the highest frequency:
[tex]\[ \text{Mode Estimate} = \text{Midpoint of } 4000-5000 = 4500 \][/tex]
### Concluding Remarks
Thus, the mode of the data, based on the histogram, is estimated to be around 4500 runs.
