Probability Calculator

This calculator computes fundamental descriptive statistics — mean, median, mode, variance, standard deviation — along with basic probability for user-defined events.

Mean (Arithmetic Average):

x̄ = Σxᵢ / n

Variance (Population):

σ² = Σ(xᵢ − x̄)² / n

Standard Deviation:

σ = √(Variance)

For sample data (n − 1 denominator, Bessel’s correction):

s = √[Σ(xᵢ − x̄)² / (n − 1)]

Basic Probability:

P(A) = Favorable Outcomes / Total Outcomes

For independent events: P(A and B) = P(A) × P(B)

For mutually exclusive events: P(A or B) = P(A) + P(B)

Worked Example — Dataset: {4, 7, 7, 9, 13}:

• n = 5
• Mean: (4+7+7+9+13)/5 = 40/5 = 8.0
• Median (middle value): 7
• Mode (most frequent): 7
• Variance (population): [(4−8)²+(7−8)²+(7−8)²+(9−8)²+(13−8)²]/5 = [16+1+1+1+25]/5 = 44/5 = 8.8
• Std Dev: √8.8 = ≈ 2.97

Probability Example: A bag has 3 red and 7 blue marbles. P(red) = 3/10 = 30%. P(red on two draws with replacement) = 0.30 × 0.30 = 9%.

Key Reference: In a normal distribution, 68% of values fall within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ — the empirical (68-95-99.7) rule, also called the three-sigma rule.

Z-Score (standardized value):

z = (x − x̄) / s

A z-score tells you how many standard deviations a value is from the mean. A z-score of +2.0 means the value is 2 standard deviations above average — placing it in approximately the 97.7th percentile of a normal distribution. This is widely used in testing, quality control, and academic grading.