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About Statistics — Class 9 Olympiad

Compute mean, median, mode for grouped and ungrouped data; solve data interpretation problems. This topic is part of the Olympiad Class 9 mathematics syllabus (chapter: Module 11). On this page you can practice 55 questions across three difficulty levels — 20 easy, 20 medium, and 15 hard — each with a visual step-by-step solution, plus a timed 32-question mastery test with a live leaderboard. Worked examples and sample questions from the topic are below.

Statistics — solved examples for Class 9 Olympiad

Example 1easy

A school conducted a survey among its students. They collected data on: (i) the number of siblings each student has, (ii) the height of each student, and (iii) the favorite color of each student. Which of these collected data sets represents 'discrete data'?
  1. A)(i) only
  2. B)(i) and (iii) only
  3. C)(ii) only
  4. D)(i), (ii) and (iii)

Step-by-step solution

  1. Data (i) 'number of siblings' is discrete because you can count siblings (0, 1, 2, ...), and it takes whole number values.
  2. Data (ii) 'height of each student' is continuous because height can take any value within a range (e.g., 150.5 cm, 150.55 cm, etc.), and it is measured.
  3. Data (iii) 'favorite color' is also discrete (categorical discrete data) as it consists of distinct categories (e.g., red, blue, green), which can be counted as frequencies. It does not fall on a continuous scale.
  4. Therefore, (i) and (iii) represent discrete data.

Answer: (i) and (iii) only

Example 2medium

A group of 'n' students initially had an average score of 'M' in a test. When a student scoring 'X' marks was added to the group, the new average score became 'M''. What is the initial number of students, 'n', in terms of M, M', and X?
  1. A)(X - M) / (M' - M) - 1
  2. B)(X + M') / (M' - M)
  3. C)(X - M') / (M' - M)
  4. D)(X - M') / (M - M')

Step-by-step solution

  1. Let the initial sum of scores for 'n' students be S. So, S = n × M.
  2. When a student scoring X marks is added, the new number of students is (n+1), and the new sum of scores is S + X.
  3. The new average M' is given by M' = (S + X) / (n + 1).
  4. Substitute S = nM: M' = (nM + X) / (n + 1). Rearranging gives M'(n + 1) = nM + X, which simplifies to M'n + M' = nM + X. Grouping terms with 'n': n(M' - M) = X - M'. Therefore, n = (X - M') / (M' - M).

Answer: (X - M') / (M' - M)

Example 3hard

The mean of N observations was calculated as 50. It was later found that two observations, which were actually 20 and 30, were mistakenly recorded as 70 and 80 respectively. If the correct mean of the N observations is 48, find N.
  1. A)40
  2. B)50
  3. C)60
  4. D)70

Step-by-step solution

  1. Let the sum of the N observations, based on the incorrect readings, be S_incorrect. Given the incorrect mean is 50, S_incorrect = 50N.
  2. The actual observations were 20 and 30 (sum = 50), but they were recorded as 70 and 80 (sum = 150).
  3. The correct sum, S_correct, can be found by subtracting the sum of incorrect readings and adding the sum of correct readings: S_correct = S_incorrect - 150 + 50 = S_incorrect - 100.
  4. We are given that the correct mean is 48, so S_correct = 48N. Substituting this into the equation: 48N = 50N - 100. This simplifies to 2N = 100, which gives N = 50.

Answer: 50

Practice questions on Statistics

  1. Q1.easy

    Consider the following data set: 4, 7, 5, 9, 3, 7, 4, 7, 5, 8. If 'X' represents the frequency of the most occurring number and 'Y' represents the frequency of the least occurring number(s) in this data set, what is the value of X - Y?
    1. A)1
    2. B)2
    3. C)3
    4. D)4
    Show answer

    Answer: 2

    Hint: First, count the frequency of each distinct number in the data set. Then, identify the highest and lowest frequencies.

  2. Q2.easy

    In a grouped frequency distribution, the class marks are given as 15, 25, 35, 45, 55. What are the lower and upper limits of the fourth class interval?
    1. A)40 and 50
    2. B)39.5 and 49.5
    3. C)40.5 and 50.5
    4. D)44.5 and 54.5
    Show answer

    Answer: 40 and 50

    Hint: The class size is the difference between consecutive class marks. The class mark is the midpoint of its class interval.

  3. Q3.easy

    A researcher wants to graphically represent the distribution of heights (in cm) of 50 students in a class. Which type of graph would be most appropriate for this data, and why?
    1. A)Bar graph, because it shows frequency for discrete categories.
    2. B)Histogram, because it displays frequency for continuous class intervals.
    3. C)Frequency polygon, because it highlights trends in data over time.
    4. D)Pictograph, because it uses images to represent data counts simply.
    Show answer

    Answer: Histogram, because it displays frequency for continuous class intervals.

    Hint: Consider whether the data is discrete (countable categories) or continuous (measurable range) and how each graph type typically handles such data.

  4. Q4.medium

    A set of 5 distinct positive integers `S = {a, b, c, d, e}` is arranged in ascending order such that `a < b < c < d < e`. If the median of `S` is 15, the mean of `S` is 16, and the smallest integer `a = 10`, what is the minimum possible value of `e` (the largest integer)?
    1. A)20
    2. B)22
    3. C)23
    4. D)21
    Show answer

    Answer: 21

    Hint: Use the given information to set up equations for the mean and median, then consider the distinct integer conditions to find the bounds for 'e'.

  5. Q5.medium

    The scores of 10 students in a test are: `12, 15, 12, 18, x, 15, 12, y, 18, 15`. If the mode of these scores is unique and is 15, and `x` and `y` are distinct integers, with all scores (including x and y) being between 10 and 20 inclusive, what is the maximum possible value of `x+y`?
    1. A)35
    2. B)34
    3. C)33
    4. D)36
    Show answer

    Answer: 35

    Hint: First, count the frequencies of the known scores. Then, use the condition of a unique mode (15) to determine the values of x and y, respecting their distinctness and range.

  6. Q6.medium

    There are three groups of students, P, Q, and R. The average age of students in group P is 15 years, in group Q is 18 years, and in group R is 16 years. If the number of students in group P is `n`, in group Q is `2n`, and in group R is `3n`, what is the overall average age of all the students combined?
    1. A)16 years
    2. B)17 years
    3. C)16.5 years
    4. D)17.5 years
    Show answer

    Answer: 16.5 years

    Hint: Calculate the total age for each group first, then sum the total ages and divide by the total number of students.

  7. Q7.hard

    The numbers 2, 5, 8, x, 12, 15, 18 are arranged in ascending order. If the median of these 7 numbers is twice the mode of the numbers 1, 2, 2, 3, x, 5, 6, 6 (where x is an integer), find the value of x.
    1. A)8
    2. B)10
    3. C)12
    4. D)15
    Show answer

    Answer: 12

    Hint: First, determine the median of the first set and the possible modes of the second set, considering the constraints on 'x'. Look for a consistent value of 'x' that satisfies all conditions.

  8. Q8.hard

    A set of 10 positive integers has a unique mode. If the smallest number in the set is 5 and the largest is 20, what is the minimum possible value of this unique mode?
    1. A)5
    2. B)6
    3. C)7
    4. D)8
    Show answer

    Answer: 5

    Hint: To minimize the unique mode, choose the smallest possible number as the mode. Ensure its frequency is uniquely highest, and the smallest and largest numbers are represented.

  9. Q9.hard

    Consider a set of N distinct positive integers in arithmetic progression. The mean is M, and the median is P. Which of the following statements is always true for any such set?
    1. A)M < P
    2. B)M > P
    3. C)M = P
    4. D)M and P cannot be determined without specific values.
    Show answer

    Answer: M = P

    Hint: Recall the properties of measures of central tendency for symmetric distributions. An arithmetic progression has a perfectly symmetric distribution.

These are 9 of the 55 questions available for Statistics. Start practicing above to unlock visual solutions, AI coaching, and the topic leaderboard — completely free.