Q.All the branches of XYZ school conducted an aptitude test for all the students in the age group 14 - 16. There were a total of n students. The marks of n students are stored in a list. Write a program using a user defined function that accepts a list of marks as an argument and calculates the 'xth' percentile (where x is any number between 0 and 100).You are required to perform the following steps to be able to calculate the 'xth' percentile.
Note: Percentile is a measure of relative performance i.e. It is calculated based on a candidate's performance with respect to others. For example : If a candidate's score is in the 90th percentile, that means she/he scored better than 90% of people who took the test.
Steps to calculate the xth percentile:
I. Order all the values in the data set from smallest to largest using Selection Sort. In general any of the sorting methods can be used.
II. Calculate index by multiplying x percent by the total number of values, n. For example: to find 90th percentile for 120 students: 0.90*120 = 108
III. Ensure that the index is a whole number by using math.round()
IV. Display the value at the index obtained in Step 3.
The corresponding value in the list is the xth percentile.

import math

def percentile(marks, x):
    data = marks[:]
    n = len(data)
    # Step I - selection sort (ascending)
    for i in range(n - 1):
        min_index = i
        for j in range(i + 1, n):
            if data[j] < data[min_index]:
                min_index = j
        data[i], data[min_index] = data[min_index], data[i]

    # Step II & III - index as a whole number
    index = round(x / 100 * n)
    if index < 1:
        index = 1

    # Step IV - the value at that position (list is 0-based)
    return data[index - 1]

marks = [55, 72, 48, 90, 61, 83, 79, 66, 95, 58]
print('Sorted :', sorted(marks))
print('90th percentile :', percentile(marks, 90))
print('50th percentile :', percentile(marks, 50))

Output

Sorted : [48, 55, 58, 61, 66, 72, 79, 83, 90, 95]
90th percentile : 90
50th percentile : 66
``` …

Concept understanding

The mean tells you the average score; a percentile tells you the rank. Saying a candidate is at the 90th percentile means roughly 90% of the candidates scored at or below that value. So the value we want is not computed arithmetically from all the marks — it is selected from the ordered list.

That is why Step I is a sort: until the marks are in order, "the 90% position" has no meaning. Once sorted, the xth percentile is simply the mark standing at position x% of n.

The four steps

Program

def selection_sort(data):
    '''Step I - sort the marks in ascending order.'''
    n = len(data)
    for i in range(n - 1):
        min_index = i
        for j in range(i + 1, n):
            if data[j] < data[min_index]:
                min_index = j
        if min_index != i:
            data[i], data[min_index] = data[min_index], data[i]
    return data


def percentile(marks, x):
    '''Return the xth percentile of the list 'marks' (0 <= x <= 100).'''
    if not marks:
        return None
    if x < 0 or x > 100:
        print('x must lie between 0 and 100')
        return None

    data = selection_sort(marks[:])      # work on a copy - Step I
    n = len(data)

    index = round(x / 100 * n)           # Steps II and III
    if index < 1:                        # guard for very small x
        index = 1
    if index > n:                         # guard for x = 100
        index = n

    return data[index - 1]               # Step IV (1-based -> 0-based)


# ---- driver ----
marks = [55, 72, 48, 90, 61, 83, 79, 66, 95, 58]   # n = 10 students
print('Marks         :', marks)
print('Sorted marks  :', selection_sort(marks[:]))
for x in (25, 50, 90, 100):
    print('{}th percentile : {}'.format(x, percentile(marks, x)))

Output

Marks         : [55, 72, 48, 90, 61, 83, 79, 66, 95, 58]
Sorted marks  : [48, 55, 58, 61, 66, 72, 79, 83, 90, 95]
25th percentile : 61
50th percentile : 66
90th percentile : 90
100th percentile : 95

Dry run — the 90th percentile of the sample list

Step I — selection sort (n = 10)