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The table gives the distribution of marks obtained in a test by 80 students.

| Mark | [tex]$10-29$[/tex] | [tex]$30-39$[/tex] | [tex]$40-49$[/tex] | [tex]$50-59$[/tex] | [tex]$60-69$[/tex] | [tex]$70-79$[/tex] | [tex]$80-89$[/tex] |
|------------|---------|---------|---------|---------|---------|---------|---------|
| No. of Students | 2 | 8 | 19 | 14 | 28 | 10 | 4 |

Tasks:
a. Draw the histogram of the distribution.
b. Use your histogram to estimate the mode mark.



Answer :

GinnyAnswer
To solve this problem, let's go through a step-by-step approach to draw the histogram and estimate the modal mark from the histogram.

### Step-by-Step Solution

#### Part (a): Drawing the Histogram of the Distribution

To draw a histogram of the distribution of marks obtained by the students, follow these steps:

1. Identify the Class Intervals and Frequencies:
- The class intervals (marks ranges) are: 10-29, 30-39, 40-49, 50-59, 60-69, 70-79, 80-89
- The corresponding frequencies (number of students) are: 2, 8, 19, 14, 28, 10, 4

2. Determine the Midpoints of Each Class Interval:
To effectively place each bar of the histogram, calculate the midpoint of each class interval:
- Midpoint of 10-29 = (10 + 29) / 2 = 19.5
- Midpoint of 30-39 = (30 + 39) / 2 = 34.5
- Midpoint of 40-49 = (40 + 49) / 2 = 44.5
- Midpoint of 50-59 = (50 + 59) / 2 = 54.5
- Midpoint of 60-69 = (60 + 69) / 2 = 64.5
- Midpoint of 70-79 = (70 + 79) / 2 = 74.5
- Midpoint of 80-89 = (80 + 89) / 2 = 84.5

3. Create the Histogram:
- On the horizontal axis (x-axis), plot the class intervals using the midpoints: 19.5, 34.5, 44.5, 54.5, 64.5, 74.5, 84.5.
- On the vertical axis (y-axis), plot the frequencies (number of students): 2, 8, 19, 14, 28, 10, 4.
- Draw bars for each class interval where the height of the bar corresponds to the frequency of the interval.

#### Part (b): Estimating the Modal Mark

The modal mark is the mark that corresponds to the highest frequency in the distribution.

1. Identify the Class Interval with the Highest Frequency:
- The highest frequency is 28, which corresponds to the class interval 60-69.

2. Estimate the Modal Mark:
- Since the modal class is 60-69, we estimate the modal mark to be around the midpoint of this class interval.
- Midpoint of 60-69 = (60 + 69) / 2 = 64.5

Therefore, the modal mark is estimated to be 64.5.

### Summary:

(a) Draw the histogram as described, plotting class intervals 10-29, 30-39, 40-49, 50-59, 60-69, 70-79, and 80-89, and their corresponding frequencies.

(b) The modal mark, which is the mark that occurred most frequently, is estimated to be 64.5 based on the histogram.