Introduction to Python Modules Class 11: import, math, random and statistics

What Introduction to Python Modules is about

Weightage: This chapter is part of Unit II, Computational Thinking and Programming - 1, which carries 45 of the 70 theory marks in the school annual exam together with Introduction to Problem Solving, Getting Started with Python, Flow of Control, Strings, Lists and Tuples and Dictionaries.Source: CBSE Computer Science (083) syllabus 2026-27, Class XI distribution of marks, Unit II Computational Thinking and Programming - 1

Key concepts

Modules and the import statement

NCERT section NCERT 7.5.1, 7.5.2 (A)

Python's standard library has built-in functions that are always available (print, len, abs, max, min, sum, round, int ...) and modules: .py files that group related functions and constants. A module's functions become available only after it is imported.

import modulename (several can be listed: import math, random) loads the whole module. Each function is then called with the module name and a dot: math.sqrt(25). Using sqrt(25) alone gives a NameError, and calling a module before importing it also gives a NameError.

Points to remember: module names are lowercase (Python is case-sensitive: import Math fails with ModuleNotFoundError); a module needs to be imported only once in a program; the import is usually written at the top. A program can also import a module under a shorter name with import statistics as st (then st.mean(...)).

Worked example

Find the error and correct it:
import math
print(sqrt(49) + math.pi)

Try it first, then show the solutionSolution

1. After import math, functions must be called with the prefix math.
2. sqrt alone is not defined: NameError.
3. Correct: print(math.sqrt(49) + math.pi).

Answer: Use math.sqrt(49); the output is then 10.141592653589793.

The from statement

NCERT section NCERT 7.5.2 (B)

from modulename import name1, name2 brings only the named functions or constants into the program, and they are then used without the module prefix: from math import sqrt, pi; print(sqrt(16), pi). Only the names imported can be used; math.ceil would fail here because the name math itself was not imported.

from modulename import * brings in every public name. It saves typing but can clash with names in your own program (for example, a variable called e would be overwritten by math.e), so it is better avoided in large programs.

Statement	How to call	What is available
import math	math.sqrt(x)	the whole module, through the prefix
from math import sqrt	sqrt(x)	only sqrt
from math import *	sqrt(x), floor(x) ...	all names, no prefix

Worked example

Which line fails, and why?
from math import ceil, pi
print(ceil(pi))
print(math.floor(pi))

Try it first, then show the solutionSolution

1. ceil and pi were imported by name: ceil(3.14159...) = 4 prints fine.
2. The name math was never imported, so math.floor raises NameError.

Answer: The third line fails with NameError; the second prints 4.

math constants and square root, power, absolute value

NCERT section NCERT 7.5.2, Table 7.2

After import math:

• math.pi = 3.141592653589793; math.e = 2.718281828459045 (constants, no brackets).
• math.sqrt(x): square root as a float: math.sqrt(49) = 7.0. A negative x raises ValueError.
• math.pow(x, y): x raised to y, always a float: math.pow(2, 10) = 1024.0. (The built-in pow(2, 10) and 2 ** 10 give the int 1024.)
• math.fabs(x): absolute value as a float: math.fabs(-7) = 7.0. (The built-in abs(-7) gives the int 7.)

Most math functions return floats, which is a common trap in output questions.

Worked example

Give the output:

import math
print(math.sqrt(81), math.pow(3, 3), math.fabs(-12), abs(-12))
print(round(math.pi, 3), round(math.e, 2))

Try it first, then show the solutionSolution

1. sqrt gives 9.0; math.pow gives 27.0; fabs gives 12.0; built-in abs keeps the int 12.
2. pi to 3 places is 3.142; e to 2 places is 2.72.

Answer: 9.0 27.0 12.0 12
3.142 2.72

ceil() and floor()

NCERT section NCERT Table 7.2

• math.ceil(x): the smallest integer greater than or equal to x (rounds up towards plus infinity).
• math.floor(x): the largest integer less than or equal to x (rounds down towards minus infinity).

Both return an int in Python 3. Careful with negatives: ceil(-4.2) = -4 (up is towards zero here) and floor(-4.2) = -5. For a whole number both return the same value: ceil(6) = floor(6) = 6.

These differ from int(), which simply cuts off the fraction (int(-4.7) = -4), and from round(), which goes to the nearest integer.

Worked example

Give math.ceil and math.floor of 7.01, -7.01, 7.99 and -7.0, and compare with int().

Try it first, then show the solutionSolution

1. 7.01: ceil 8, floor 7. -7.01: ceil -7, floor -8.
2. 7.99: ceil 8, floor 7. -7.0: both -7.
3. int() cuts off: int(7.99) = 7, int(-7.01) = -7.

Answer: ceil: 8, -7, 8, -7; floor: 7, -8, 7, -7.

sin(), cos() and tan()

NCERT section NCERT Table 7.2

math.sin(x), math.cos(x) and math.tan(x) take the angle x in radians, not degrees. To use degrees, convert first: radians = degrees × π / 180 (for example, 30° = math.pi / 6). The results are floats and may show tiny rounding errors, such as math.sin(math.pi) = 1.2246467991473532e-16 instead of exactly 0, so round() them when displaying.

Worked example

Find sin 30°, cos 60° and tan 45° using the math module, rounded to 2 decimal places.

Try it first, then show the solutionSolution

1. import math
   for deg, f in [(30, math.sin), (60, math.cos), (45, math.tan)]:
       rad = deg * math.pi / 180
       print(deg, round(f(rad), 2))

2. Convert each angle to radians before calling the function.

Answer: 30 0.5
60 0.5
45 1.0

The random module

NCERT section NCERT Table 7.3

After import random:

• random.random(): a random float with 0.0 ≤ value < 1.0 (it can be 0.0 but never 1.0).
• random.randint(a, b): a random integer with a ≤ value ≤ b: both ends included.
• random.randrange(stop): a random integer from 0 to stop - 1.
• random.randrange(start, stop[, step]): a random integer from the values that range(start, stop, step) would give, so stop is excluded: randrange(1, 10, 2) gives one of 1, 3, 5, 7, 9.

Because the output changes every run, exam questions ask for the possible outputs or the minimum and maximum values a variable can take. To scale random(): int(random.random() * 6) + 1 gives an integer from 1 to 6, like a dice.

Worked example

Give the smallest and largest possible values of: (a) random.randint(3, 8) (b) random.randrange(3, 8) (c) random.randrange(10, 50, 10) (d) int(random.random() * 5) + 2

Try it first, then show the solutionSolution

1. (a) both ends included: 3 to 8.
2. (b) stop excluded: 3 to 7.
3. (c) values 10, 20, 30, 40: 10 to 40.
4. (d) random() * 5 is 0 to just below 5; int gives 0 to 4; +2 gives 2 to 6.

Answer: (a) 3 and 8 (b) 3 and 7 (c) 10 and 40 (d) 2 and 6

The statistics module

NCERT section NCERT Table 7.4

After import statistics:

• statistics.mean(data): the arithmetic mean (sum ÷ count).
• statistics.median(data): the middle value of the sorted data; for an even number of values, the average of the two middle values (so the result may be a float not in the data).
• statistics.mode(data): the most frequent value. It works on numbers and on non-numeric data such as strings. If several values tie for most frequent, Python 3.8 and later return the first of them met in the data.

The data may be a list or a tuple; an empty sequence raises an error (StatisticsError).

Worked example

For marks = [56, 78, 91, 78, 64, 85], find the mean, median and mode with the statistics module.

Try it first, then show the solutionSolution

1. import statistics as st
   marks = [56, 78, 91, 78, 64, 85]
   print(round(st.mean(marks), 2), st.median(marks), st.mode(marks))

2. Mean = 452 / 6 = 75.33. Sorted: 56, 64, 78, 78, 85, 91; middle two 78 and 78 → median 78.0. 78 occurs twice: mode 78.

Answer: 75.33 78.0 78

Introduction to Python Modules in real life

Each concept, met in an everyday situation. Read the situation, answer the question in your own words, then compare with the model answer. The one-line hook is what to remember.

Topics in this chapter

• The math Module: Functions, Constants and Import Forms
• The random and statistics Modules: Possible Output Questions

Introduction to Python Modules: syntax and key facts

Import forms

1. 1.import m
2. 2.m.f(); from m import f
3. 3.f(); from m import *
4. 4.all names without prefix

import m as x → x.f().

math constants

math.pi ≈ 3.141592653589793; math.e ≈ 2.718281828459045

No brackets: they are values, not functions.

Rounding functions

ceil(x): smallest int ≥ x; floor(x): largest int ≤ x

ceil(-2.5) = -2; floor(-2.5) = -3.

Float results

math.sqrt, math.pow, math.fabs return floats

abs() and pow()/** on ints return ints.

Random ranges

random(): [0.0, 1.0); randint(a, b): a..b; randrange(a, b): a..b-1

randrange(a, b, s) follows range(a, b, s).

Statistics

mean = sum/count; median = middle of sorted data (average of two middles if even); mode = most frequent

mode returns the first of tied values (Python 3.8+).

How the school exam asks Introduction to Python Modules

The question patterns to expect, and the method that works for each.

Tricks to remember Introduction to Python Modules

Short memory aids. Say each one aloud once, then use it on a question the same day.

• Dot after import, no dot after from

  import math → math.sqrt; from math import sqrt → sqrt.

  Use it when: Choosing the right call style.

• Ceiling is above, floor is below

  Picture a number line: ceil goes up to the ceiling, floor goes down to the floor, even for negatives.

  Use it when: ceil/floor of negative numbers.

• randINT is INclusive

  randint includes both ends; randrange behaves like range and stops short.

  Use it when: Possible-output questions.

• math gives floats

  sqrt, pow, fabs: the 'f' in fabs reminds you of float.

  Use it when: Output format of math functions.

• Trig wants radians

  Multiply degrees by pi/180 before sin, cos or tan.

  Use it when: Trigonometric outputs.

Introduction to Python Modules: exam-level solved questions

Ordered from exam level to HOTS. Cover the solution, write your own, then compare step by step.

import math
print(math.ceil(-2.5), math.floor(-2.5), math.ceil(2.5), math.floor(2.5))
print(math.pow(4, 0.5), math.sqrt(2.25), math.fabs(-0.5))

from statistics import mean, median, mode
t = (12, 7, 3, 7, 10, 3, 7, 15)
print(mean(t), median(t), mode(t))

Case-study practice (Section E style)

Read the situation, then answer every part. In a school exam set on the board pattern this is a 4-mark question split into 1 + 1 + 1 + 1 marks.

Where the marks are

How a typical answer is split into step marks. Show every step: a right final answer alone rarely earns full marks.

Where students lose marks in Introduction to Python Modules

The slips examiners see most often, and what to write instead.

• ✗Mistake: Writing math.sqrt(16) = 4

  sqrt returns a float.

  Wrong working

  print(math.sqrt(16)) → 4

  Right working

  print(math.sqrt(16)) → 4.0

  ✓Do this: Show .0 for sqrt, pow and fabs results.

  Usually costs: Half to 1 mark

• ✗Mistake: Rounding negatives the wrong way with ceil and floor

  ceil goes towards plus infinity, floor towards minus infinity.

  Wrong working

  math.floor(-3.2) = -3

  Right working

  math.floor(-3.2) = -4; math.ceil(-3.2) = -3

  ✓Do this: Mark the number on a number line.

  Usually costs: 1 mark

• ✗Mistake: Including stop in randrange

  randrange(a, b) can never return b.

  Wrong working

  randrange(1, 6) can give 6

  Right working

  randrange(1, 6) gives 1 to 5; randint(1, 6) gives 1 to 6

  ✓Do this: Treat randrange like range().

  Usually costs: 1 mark

• ✗Mistake: Calling with the prefix after a from import

  from math import sqrt does not create the name math.

  Wrong working

  from math import sqrt
  print(math.sqrt(9))

  Right working

  from math import sqrt
  print(sqrt(9))

  ✓Do this: Match the call to the import form.

  Usually costs: 1 mark

• ✗Mistake: Passing degrees to sin()

  math.sin expects radians.

  Wrong working

  math.sin(30) → 0.5

  Right working

  math.sin(30) ≈ -0.988 (30 radians); math.sin(math.pi / 6) ≈ 0.5

  ✓Do this: Convert with degrees × pi / 180.

  Usually costs: 1 mark

NCERT exercise guide: Introduction to Python Modules

What each exercise in the NCERT chapter tests, how to approach it, and a question of the same kind solved step by step. Questions here are written in our own words.

Practice questions on Introduction to Python Modules

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

1. Q1 · 1 mark · MCQExam level

   Which module must be imported to use median()? (a) math (b) random (c) statistics (d) none

   Hint

   Average-type functions.

   Answer

   (c) statistics

2. Q2 · 1 mark · Very short answerHarder exam level

   Give the output: import math; print(math.floor(-0.5), math.ceil(-0.5), math.sqrt(0.25))

   Hint

   Negative numbers round towards the stated side.

   Answer

   -1 0 0.5 (ceil(-0.5) is shown as 0)

3. Q3 · 1 mark · Very short answerHarder exam level

   What is the smallest and largest value of y? import random; y = random.randrange(5, 25, 5) * 2

   Hint

   randrange gives 5, 10, 15, 20.

   Answer

   Smallest 10, largest 40.

4. Q4 · 3 marks · Short answerExam level

   Write a program to input the three sides of a triangle and print its area using Heron's formula, s = (a + b + c)/2 and area = √(s(s-a)(s-b)(s-c)).

   Hint

   math.sqrt.

   Answer

   import math
   a = float(input()); b = float(input()); c = float(input())
   s = (a + b + c) / 2
   print(math.sqrt(s * (s - a) * (s - b) * (s - c)))
   (For 3, 4, 5 the area is 6.0.)

5. Q5 · 1 mark · MCQHarder exam level

   Assertion (A): math.pow(2, 3) and 2 ** 3 print the same thing. Reason (R): math.pow always returns a float. Choose (a) both true, R explains A (b) both true, R does not explain A (c) A false, R true (d) both false.

   Hint

   8.0 versus 8.

   Answer

   (c) A is false (8.0 and 8 are printed); R is true.

6. Q6 · 2 marks · Short answerHarder exam level

   Give the output:

   from math import ceil, floor
   L = [2.3, -2.3, 5.0]
   print([ceil(x) for x in L], [floor(x) for x in L])

   Hint

   A list built by a loop expression.

   Answer

   [3, -2, 5] [2, -3, 5]

7. Q7 · 2 marks · Short answerHOTS / competency

   import random
   N = random.randint(2, 4)
   for i in range(1, N):
   print(i * '*')
   Which outputs are possible? (i) * then (ii) * then then * (iii) only * (iv) * then then * then **

   Hint

   N is 2, 3 or 4; range(1, N) stops before N.

   Answer

   N = 2 → (iii); N = 3 → (i); N = 4 → (ii). (iv) is impossible.

8. Q8 · 2 marks · Short answerExam level

   Why does print(math.sin(90)) not print 1.0? Correct it.

   Hint

   Units.

   Answer

   sin takes radians; 90 radians is not 90°. Use math.sin(math.radians(90)) or math.sin(90 * math.pi / 180), which gives 1.0.

9. Q9 · 3 marks · Short answerHOTS / competency

   Write a program that generates 5 random marks between 30 and 100 (inclusive) into a list and prints the list with its mean and the highest mark rounded up to the next multiple of 10.

   Hint

   math.ceil(x / 10) * 10.

   Answer

   import random, math, statistics
   m = []
   for i in range(5):
   m.append(random.randint(30, 100))
   print(m, statistics.mean(m), math.ceil(max(m) / 10) * 10)

10. Q10 · 2 marks · Short answerHarder exam level

    Give the output: import statistics as s; print(s.mean([1, 2, 3, 4]), s.median([5, 1, 3]), s.mode('hello'))

    Hint

    A string is a sequence of characters.

    Answer

    2.5 3 l

11. Q11 · 2 marks · Short answerHOTS / competency

    A program has 'from math import *' and later the line e = 5. What happens to the name e (which the import set to Euler's number) in that program? Why is this a risk?

    Hint

    Name clash.

    Answer

    The name e (which from math import * set to 2.718...) is replaced by 5; any later use of e meaning Euler's number is wrong. Importing everything without a prefix can clash with your own names.

12. Q12 · 1 mark · Very short answerExam level

    Name the function: (a) returns the absolute value as a float (b) gives the largest integer not greater than x (c) random float below 1

    Hint

    All three are in the syllabus list.

    Answer

    (a) math.fabs (b) math.floor (c) random.random

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1. 1.Which module has the function median()?Show answer

   statistics.

2. 2.What is math.ceil(4.0001)?Show answer

   5.

3. 3.What is the range of random.random()?Show answer

   0.0 up to but not including 1.0.

4. 4.What does math.fabs(-3) return?Show answer

   3.0.

5. 5.Can random.randint(1, 10) return 10?Show answer

   Yes, both ends are included.

6. 6.How do you import only pi from math?Show answer

   from math import pi.

7. 7.What does statistics.median([4, 1, 3, 2]) return?Show answer

   2.5.

Quick revision checklist

The night before the exam: can you do each of these without looking?

• ☐Module: a file of related functions; must be imported before use.
• ☐import math → math.sqrt(x); from math import sqrt → sqrt(x).
• ☐math.pi, math.e are constants.
• ☐sqrt, pow, fabs return floats; ceil (up) and floor (down) return ints.
• ☐sin, cos, tan take radians.
• ☐random() in [0, 1); randint(a, b) includes b; randrange(a, b) excludes b.
• ☐mean, median (average of the middle two for even counts), mode (first of ties).

Introduction to Python Modules: frequently asked questions