MathsMediumClass 11

Variance & Standard Deviation

Statistics

JEE Qs

Hard

min

Master the direct formulas and the properties related to change of origin and scale, as these are frequently tested for quick calculation and conceptual understanding.

🧮 Key Formulas

Mean (for individual data): x_bar = (sum(x_i)) / n

Mean (for discrete frequency distribution): x_bar = (sum(f_i * x_i)) / (sum(f_i))

Mean (for grouped frequency distribution): x_bar = (sum(f_i * x_i)) / (sum(f_i)) (where x_i are midpoints)

Variance (sigma^2, for individual data): (sum((x_i - x_bar)^2)) / n

Variance (sigma^2, for individual data, alternative): (sum(x_i^2) / n) - (x_bar)^2

Variance (sigma^2, for discrete/grouped frequency distribution): (sum(f_i * (x_i - x_bar)^2)) / (sum(f_i))

Variance (sigma^2, for discrete/grouped frequency distribution, alternative): (sum(f_i * x_i^2) / sum(f_i)) - (x_bar)^2

Standard Deviation (sigma): sqrt(Variance)

Effect of change of origin (y_i = x_i + a): Variance(Y) = Variance(X), SD(Y) = SD(X)

Effect of change of scale (y_i = b * x_i): Variance(Y) = b^2 * Variance(X), SD(Y) = |b| * SD(X)

Effect of change of origin and scale (y_i = a + b * x_i): Variance(Y) = b^2 * Variance(X), SD(Y) = |b| * SD(X)

Coefficient of Variation (CV): (Standard Deviation / Mean) * 100

Combined Variance (for two datasets): sigma_combined^2 = (n1*sigma1^2 + n2*sigma2^2 + n1*d1^2 + n2*d2^2) / (n1+n2) where d1 = x_bar1 - x_bar_combined, d2 = x_bar2 - x_bar_combined

✅ Key Points for JEE

1Variance measures the average squared deviation from the mean, while standard deviation is its square root, making it interpretable in the original units of data.
2Variance and Standard Deviation are independent of change of origin (adding/subtracting a constant) but dependent on change of scale (multiplying/dividing by a constant). If each observation is multiplied by 'k', variance becomes k^2 times and standard deviation becomes |k| times.
3The alternative formulas for variance (using sum of squares of observations) are often more computationally efficient, especially for large datasets or when calculating manually.
4Coefficient of Variation (CV) is a unitless measure used to compare the relative variability or consistency between two or more datasets, even if they have different means or units.
5Be mindful of the calculation for combined variance; it involves individual variances, means, and the overall combined mean. Ensure you calculate the combined mean correctly first.

⚠️ Common Mistakes

✕Incorrectly calculating the mean before computing deviations, leading to errors in variance and standard deviation.
✕Forgetting to take the square root when converting variance to standard deviation, or vice-versa, confusing the two measures.
✕Failing to apply the correct transformations for variance and standard deviation when data is scaled (e.g., if observations are multiplied by 'k', variance changes by k^2, not k).
✕Errors in sum of squares calculations, especially for grouped data or when using the alternative formula sum(x_i^2) or sum(f_i*x_i^2).

📝 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 11 Maths Ch 15: Statistics

🚀 This Unlocks

Coefficient of Variation