Question

The score of the student at which position is taken as the median ?

Answer 1

The question asks for the position of the student whose score represents the median of the entire group.

The median is the value of the middle observation in a dataset. For a set of N observations, the position of the median is given by the ((N+1)/(2))-th observation if N is odd. If N is even, it is the average of the (N)/(2)-th and ((N)/(2)+1)-th observations. In the context of grouped data, we first find the position (N)/(2).

The total number of students is given as N = 45.
Since we are dealing with individual students lined up, we can consider the discrete positions. The middle position for an odd number of items is found by (N+1)/(2).
Median Position = (45+1)/(2) = (46)/(2) = 23 This means the score of the 23rd student, when they are arranged in order of their scores, is the median score.

The score of the student at the 23rd position is taken as the median.

Answer 2

This question asks for the theoretical score of the 20th student based on the standard assumptions for calculating the median from grouped data. This is a step towards finding the median value itself. First, we need to locate the class interval where the 20th student falls.

We need to create a cumulative frequency table to find the class containing the 20th student. The assumption for median calculation is that the data is uniformly distributed within the median class.
The formula to find a specific value within a class is similar to the median formula structure:
Value = l + ( position - cf ) × h
where l is the lower limit, c is the cumulative frequency of the preceding class, f is the frequency of the class, and h is the class width.

Let's construct the cumulative frequency (cf) table:
\includegraphics[width=0.8\linewidth]13ii.png
The 20th student falls in the class interval 60-70, as the cumulative frequency up to 60 is 19. So students from 20th to 29th position are in the 60-70 class.
Lower limit of this class, l = 60.
Cumulative frequency of the class preceding this class, c = 19.
Frequency of this class, f = 10.
Class width, h = 70 - 60 = 10.
The position we are interested in is the 20th student.
Score of 20th student = l + ( (20 - c)/(f) ) × h
Score = 60 + ( (20 - 19)/(10) ) × 10 Score = 60 + ( (1)/(10) ) × 10 = 60 + 1 = 61

Answer 3

We need to calculate the median score for the given grouped frequency distribution.

The formula for the median of grouped data is:
Median = l + ( (N)/(2) - cf ) × h where:
l = lower class boundary of the median class.
N = total frequency.
c = cumulative frequency of the class preceding the median class.
f = frequency of the median class.
h = class width.

First, we use the cumulative frequency table from the previous part.
\includegraphics[width=0.8\linewidth]13iii.png
1. Find the median position: (N)/(2) = (45)/(2) = 22.5.
2. Identify the median class: The first class with a cumulative frequency greater than 22.5 is 60-70 (cf=29). So, the median class is 60-70.
3. Identify the values for the formula:
- Lower limit of the median class, l = 60.
- Total frequency, N = 45.
- Cumulative frequency of the class preceding the median class, c = 19.
- Frequency of the median class, f = 10.
- Class width, h = 70 - 60 = 10.
4. Substitute the values into the formula:
Median = 60 + ( (22.5 - 19)/(10) ) × 10 Median = 60 + ( (3.5)/(10) ) × 10 Median = 60 + 3.5 Median = 63.5 The median score is 63.5.