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MathsMediumClass 11

Mean, Median, Mode — Grouped and ungrouped data

Statistics

6

JEE Qs

8%

Hard

75

min

Master the correct application of all formulas for both grouped and ungrouped data, paying close attention to data interpretation and arithmetic accuracy.

🧮 Key Formulas

Mean (Ungrouped data): x̄ = (Σx_i) / n
Median (Ungrouped data, n is odd): ((n+1)/2)-th observation after sorting
Median (Ungrouped data, n is even): Average of (n/2)-th and ((n/2)+1)-th observations after sorting
Mode (Ungrouped data): Observation with the highest frequency
Mean (Grouped data - Direct Method): x̄ = (Σf_i * x_i) / (Σf_i)
Mean (Grouped data - Assumed Mean Method): x̄ = A + [(Σf_i * d_i) / (Σf_i)], where d_i = x_i - A
Mean (Grouped data - Step Deviation Method): x̄ = A + [(Σf_i * u_i) / (Σf_i)] * h, where u_i = (x_i - A) / h
Median (Grouped data): L + [(n/2 - cf) / f] * h, where L = lower limit of median class, n = Σf_i, cf = cumulative frequency of class preceding median class, f = frequency of median class, h = class size
Mode (Grouped data): L + [(f_m - f_1) / (2f_m - f_1 - f_2)] * h, where L = lower limit of modal class, f_m = frequency of modal class, f_1 = frequency of class preceding modal class, f_2 = frequency of class succeeding modal class, h = class size
Empirical relation: Mode ≈ 3 Median - 2 Mean

✅ Key Points for JEE

  • 1For ungrouped data median, observations must always be arranged in ascending or descending order before identification.
  • 2The empirical relation between Mean, Median, and Mode (Mode ≈ 3 Median - 2 Mean) is useful for estimating one measure if the other two are known, especially in moderately skewed distributions.
  • 3For grouped data, the class mark (midpoint) is crucial for calculating the mean. Ensure class intervals are continuous; if not, adjust them.
  • 4Carefully identify the 'median class' (class containing n/2-th observation based on cumulative frequency) and 'modal class' (class with highest frequency) before applying the respective formulas for grouped data.
  • 5Mean is sensitive to extreme values (outliers), while Median is more robust as it represents the middle value.

⚠️ Common Mistakes

  • ✕Not sorting ungrouped data before finding the median, or misidentifying the (n/2)-th position for even 'n'.
  • ✕Incorrectly calculating cumulative frequencies (cf) or class marks (x_i) for grouped data, leading to errors in mean/median calculations.
  • ✕Confusing the frequencies (f_m, f_1, f_2) or cumulative frequency (cf) in the grouped median/mode formulas.
  • ✕Failing to adjust class intervals to be continuous (exclusive) when provided in inclusive format for grouped data.

📝 Practice Questions

See all

Q1. Let x1, x2, … , x10 be ten observations such that ∑10i=1 (xi −2) = 30, ∑10i=1 (xi −β)2 = 98, β > 2, and their variance is 4 . If μ and σ2 are respectively the mean and the variance of 2 (x1 −1) + 4β , 5 2 (x2 −1) + 4β, … . , 2 (x10 −1) + 4β , then βμσ2 is equal to : (1) 100 (2) 120 (3) 110 (4) 90

2025·MCQMedium

Q11.The area of the region {(x, y) : x2 + 4x + 2 ≤y ≤|x + 2|} is equal to (1) 7 (2) 5 (3) 24/5 (4) 20/3

2025·MCQMedium

Q5. Marks obtains by all the students of class 12 are presented in a freqency distribution with classes of equal width. Let the median of this grouped data be 14 with median class interval 12-18 and median class frequency 12 . If the number of students whose marks are less than 12 is 18 , then the total number of students is (1) 52 (2) 48 (3) 44 (4) 40

2025·MCQMedium

Q86. X α 1 0 −3 Let the mean and the standard deviation of the probability distribution be μ and σ, P(X) 31 K 16 41 respectively. If σ −μ = 2, then σ + μ is equal to________ JEE Main 2024 (05 Apr Shift 2) JEE Main Previous Year Paper

2024·NumericalMedium

Q69.If the mean of the following probability distribution of a random variable X : X 0 2 4 6 8 46 is , then the variance of the distribution is P(X) a 2a a + b 2b 3b 9 (1) 173 (2) 566 27 81 (3) 151 (4) 581 27 81

2024·MCQMedium

Q69.Consider 10 observation 𝑥1, 𝑥2, . .. 𝑥10, such that ∑𝑖=10 1 𝑥𝑖−𝛼= 2 and ∑𝑖=10 1 𝑥𝑖−𝛽2 = 40, where 𝛼, 𝛽 are 6 84 𝛽 positive integers. Let the mean and the variance of the observations be and respectively. The is equal to: 5 25 𝛼 (1) 2 (2) 3 2 (3) 5 (4) 1 2

2024·MCQMedium

NCERT Chapters

  • Class 9 Maths Ch 14: Statistics
  • Class 10 Maths Ch 14: Statistics
  • Class 11 Maths Ch 15: Statistics