Statistics Calculator: Mean, Median, Mode, Standard Deviation, Variance & Probability

• The five fundamental statistics every dataset needs: Mean (average), Median (middle value), Mode (most frequent), Variance (spread squared), Standard Deviation (spread in original units). They tell you where data clusters and how spread out it is.
• Standard deviation is the most important single statistics number: a small SD means data is tightly clustered around the mean; a large SD means high variability. In normal distributions, 68% of data falls within 1 SD, 95% within 2 SD.
• P-value is the probability of observing your result (or more extreme) if the null hypothesis is true. P < 0.05 is the conventional threshold for "statistically significant." P-value is about probability of data given hypothesis — NOT the probability that your hypothesis is correct.
• For Class 11–12 statistics: Focus on mean/median/mode, standard deviation from frequency tables, correlation, and regression. For JEE: Statistics carries approximately 2–3% weightage — high return for preparation time.

Use our free Statistics Calculator to compute descriptive statistics, probability distributions, and regression from any dataset.

MODULE 1: MEASURES OF CENTRAL TENDENCY

Mean (Arithmetic Average)

Mean (x̄) = Σx / n = (x₁ + x₂ + ... + xₙ) / n

Example: Data: 4, 7, 2, 9, 5, 8, 3, 6 x̄ = (4+7+2+9+5+8+3+6) / 8 = 44/8 = 5.5

Weighted mean: x̄_w = Σ(wᵢxᵢ) / Σwᵢ

Example: Exam scores: 70 (weightage 2), 85 (weightage 3), 90 (weightage 1) Weighted mean = (70×2 + 85×3 + 90×1) / (2+3+1) = (140+255+90)/6 = 485/6 = 80.83

Median (Middle Value)

For ungrouped data: 1. Sort data in ascending order 2. If n is odd: Median = value at position (n+1)/2 3. If n is even: Median = average of values at positions n/2 and n/2 + 1

Example (odd n): 3, 5, 7, 9, 11 → n = 5, median at position 3 → Median = 7

Example (even n): 2, 4, 6, 8, 10, 12 → n = 6 Median = (6th/2 + 6/2+1th value) / 2 = (6 + 8)/2 = 7

For grouped data (frequency distribution): Median = L + [(n/2 − cf) / f] × h Where: L = lower boundary of median class, n = total frequency, cf = cumulative frequency before median class, f = frequency of median class, h = class width

Mode (Most Frequent Value)

For ungrouped data: The value that appears most often.

Example: 3, 5, 3, 7, 3, 8, 5 → Mode = 3 (appears 3 times)

Bimodal: 2, 3, 3, 5, 7, 7, 9 → Mode = 3 and 7

For grouped data: Mode = L + [(f₁ − f₀) / (2f₁ − f₀ − f₂)] × h Where: L = lower boundary of modal class (highest frequency), f₁ = frequency of modal class, f₀ = frequency of pre-modal class, f₂ = frequency of post-modal class, h = class width

Empirical relationship: Mode ≈ 3 Median − 2 Mean (for moderately skewed distributions)

MODULE 2: MEASURES OF DISPERSION

Range

Range = Maximum value − Minimum value

Simple but sensitive to outliers.

Variance and Standard Deviation

Population Variance: σ² = Σ(xᵢ − μ)² / N Population SD: σ = √(σ²)

Sample Variance: s² = Σ(xᵢ − x̄)² / (n−1) ← use n−1 (Bessel's correction) Sample SD: s = √(s²)

Computational shortcut (easier for hand calculation): σ² = (Σx²/n) − (x̄)²

Worked Example: Data: 2, 4, 4, 4, 5, 5, 7, 9 (n=8) x̄ = 40/8 = 5 Σ(xᵢ − x̄)²: (2−5)²+(4−5)²+(4−5)²+(4−5)²+(5−5)²+(5−5)²+(7−5)²+(9−5)² = 9+1+1+1+0+0+4+16 = 32

Population variance σ² = 32/8 = 4 Population SD σ = √4 = 2

Standard Deviation from Frequency Table

x̄ = 545/25 = 21.8 σ² = 2225/25 = 89 σ = √89 ≈ 9.43

Coefficient of Variation (CV)

Used to compare variability between datasets with different means.

CV = (σ / x̄) × 100%

Higher CV = more relative variability Example: Dataset A (mean=100, σ=15): CV = 15% Dataset B (mean=20, σ=5): CV = 25% → Dataset B is relatively more variable despite smaller absolute SD

MODULE 3: PROBABILITY

Basic Probability Rules

Worked Example: A bag has 4 red and 6 blue balls. Two drawn without replacement. P(both red)? P(1st red) = 4/10 P(2nd red | 1st red) = 3/9 P(both red) = 4/10 × 3/9 = 12/90 = 2/15

Probability Distributions

#### Binomial Distribution

For n independent trials, each with probability p of success:

P(X = r) = ⁿCᵣ × pʳ × (1−p)^(n−r)

Mean = np | Variance = np(1−p) | SD = √(np(1−p))

Example: Toss a fair coin 10 times. P(exactly 6 heads)? P(X=6) = ¹⁰C₆ × (0.5)⁶ × (0.5)⁴ = 210 × (0.5)¹⁰ = 210/1024 ≈ 0.205

#### Normal Distribution

Bell-shaped, symmetric. Defined by mean (μ) and SD (σ).

Empirical Rule (68-95-99.7):

• 68% of data falls within μ ± 1σ
• 95% within μ ± 2σ
• 99.7% within μ ± 3σ

Z-score (standardisation): Z = (X − μ) / σ Z-score tells you how many standard deviations X is from the mean.

MODULE 4: CORRELATION AND REGRESSION

Pearson Correlation Coefficient

r = Σ[(xᵢ − x̄)(yᵢ − ȳ)] / √[Σ(xᵢ − x̄)² × Σ(yᵢ − ȳ)²]

Shortcut formula: r = [nΣxy − ΣxΣy] / √{[nΣx² − (Σx)²][nΣy² − (Σy)²]}

Linear Regression

Regression line of y on x: y = a + bx

Slope: b = [nΣxy − ΣxΣy] / [nΣx² − (Σx)²]

Intercept: a = ȳ − b x̄

MODULE 5: QUICK REFERENCE FORMULAS

Permutations and Combinations

Important Statistical Tests (Brief)