The problem provides a frequency distribution table of marks obtained by 50 students. It asks us to: (a) Construct the cumulative frequency table and draw the cumulative frequency curve. (b) Use the cumulative frequency curve to find the median and the 70th percentile. (c) Determine the mode of the distribution.

1. Problem Description

The problem provides a frequency distribution table of marks obtained by 50 students. It asks us to:

(a) Construct the cumulative frequency table and draw the cumulative frequency curve.

(b) Use the cumulative frequency curve to find the median and the 70th percentile.

(c) Determine the mode of the distribution.

2. Solution Steps

(a) Constructing the cumulative frequency table:

| Mark | Frequency | Cumulative Frequency |

|-----------|-----------|----------------------|

| 40-44 | 3 | 3 |

| 45-49 | 5 | 3 + 5 = 8 |

| 50-54 | 8 | 8 + 8 = 16 |

| 55-59 | 11 | 16 + 11 = 27 |

| 60-64 | 9 | 27 + 9 = 36 |

| 65-69 | 7 | 36 + 7 = 43 |

| 70-74 | 5 | 43 + 5 = 48 |

| 75-79 | 2 | 48 + 2 = 50 |

Drawing the cumulative frequency curve (Ogive):

The cumulative frequency curve is a graph with the upper class limits on the x-axis (44, 49, 54, 59, 64, 69, 74, 79) and the cumulative frequencies on the y-axis. Plot the points (44, 3), (49, 8), (54, 16), (59, 27), (64, 36), (69, 43), (74, 48), (79, 50) and connect them with a smooth curve. I cannot actually draw it in this text format.

(b) Finding the median and the 70th percentile:

The median is the value at the 50th percentile. Since there are 50 students, the median corresponds to the value at the 50/2 = 25th position. Find 25 on the y-axis of the cumulative frequency curve, draw a horizontal line to the curve, and then draw a vertical line down to the x-axis. Estimate the value on the x-axis. Based on the cumulative frequencies, the median should be in the class 55-

5

9. Looking at the graph the median is approximately

5

8.

The 70th percentile corresponds to 70% of the total number of students, which is $0.70 \times 50 = 35$. Find 35 on the y-axis of the cumulative frequency curve, draw a horizontal line to the curve, and then draw a vertical line down to the x-axis. Estimate the value on the x-axis. Based on the cumulative frequencies, the 70th percentile should be in the class 60-

6

4. Looking at the graph the 70th percentile is approximately

6

4.

(c) Determining the mode:

The mode is the class with the highest frequency. From the frequency distribution table, the class 55-59 has the highest frequency of

1

1. Therefore, the modal class is 55-

5

9. To estimate the mode, one common way is to use the formula:

$Mode = L + \frac{f_m - f_1}{(f_m - f_1) + (f_m - f_2)} \times w$

Where:

$L$ = Lower limit of the modal class = 55

$f_m$ = Frequency of the modal class = 11

$f_1$ = Frequency of the class preceding the modal class = 8

$f_2$ = Frequency of the class succeeding the modal class = 9

$w$ = Class width = 5

$Mode = 55 + \frac{11 - 8}{(11 - 8) + (11 - 9)} \times 5 = 55 + \frac{3}{3 + 2} \times 5 = 55 + \frac{3}{5} \times 5 = 55 + 3 = 58$

3. Final Answer

(a) Cumulative frequency table:

| Mark | Frequency | Cumulative Frequency |

|-----------|-----------|----------------------|

| 40-44 | 3 | 3 |

| 45-49 | 5 | 8 |

| 50-54 | 8 | 16 |

| 55-59 | 11 | 27 |

| 60-64 | 9 | 36 |

| 65-69 | 7 | 43 |

| 70-74 | 5 | 48 |

| 75-79 | 2 | 50 |

Cumulative frequency curve: (Description provided above)

(b) Using the cumulative frequency curve:

Median: Approximately 58

70th percentile: Approximately 64

(c) Mode: 58