Dataset Input
Frequency Distribution
Normal Distribution Parameters
Find Probability
Binomial Distribution — B(n, p)
Poisson Distribution — Pois(λ)
Measures of Central Tendency and Spread
Descriptive statistics summarize a dataset's shape, center, and variability:
- Mean (x̄): arithmetic average — Σxᵢ / n. Sensitive to outliers.
- Median: the middle value when data is sorted. Robust to outliers.
- Mode: most frequently occurring value(s). A dataset may be unimodal, bimodal, or multimodal.
- Variance (σ²): average squared deviation from the mean — Σ(xᵢ−x̄)² / n (population) or Σ(xᵢ−x̄)² / (n−1) (sample).
- Standard Deviation: square root of variance — same units as the data.
- IQR: Q3 − Q1 — the middle 50% spread. Used to detect outliers (values below Q1 − 1.5×IQR or above Q3 + 1.5×IQR).
- Skewness: measures asymmetry. Positive = right tail; negative = left tail.
- Kurtosis: measures peak height relative to normal. Excess kurtosis > 0 is leptokurtic (heavy tails); < 0 is platykurtic (light tails).
Understanding the Normal Distribution
The normal (Gaussian) distribution N(μ, σ²) is the bell-shaped curve described by:
f(x) = (1 / (σ√(2π))) × e^(−(x−μ)²/(2σ²))
68-95-99.7 Rule:
P(μ−σ ≤ X ≤ μ+σ) ≈ 68.27%
P(μ−2σ ≤ X ≤ μ+2σ) ≈ 95.45%
P(μ−3σ ≤ X ≤ μ+3σ) ≈ 99.73%
The standard normal N(0,1) with z = (x−μ)/σ is used for probability lookups. The CDF Φ(z) = P(Z ≤ z) gives the area under the curve to the left of z.
When to Use Each Distribution
| Distribution | Use Case | Parameters |
|---|---|---|
| Normal | Continuous data symmetric around a mean; heights, test scores, measurement errors | μ (mean), σ (std dev) |
| Binomial | Fixed n independent trials, each success/fail with constant probability p; coin flips, pass/fail tests | n (trials), p (probability) |
| Poisson | Count of rare events in a fixed interval; calls/hour, defects/unit, arrivals/day | λ (average rate) |
Rule of thumb: Binomial approximates Normal when n is large (n ≥ 30) and neither p nor 1−p is very small. Poisson approximates Binomial when n is large and p is small (λ = np).
Population standard deviation (σ) divides by n and is used when you have data for the entire population. Sample standard deviation (s) divides by n−1 (Bessel's correction) and is used when your data is a sample drawn from a larger population. The subtraction of 1 corrects for the fact that a sample mean slightly underestimates spread — using n−1 makes the sample variance an unbiased estimator of the true population variance.
A z-score (standard score) measures how many standard deviations a data point is from the mean. It is calculated as z = (x − μ) / σ. A z-score of 0 means the value equals the mean; z = 2 means 2 standard deviations above the mean. Z-scores allow comparison across different distributions and are used to find probabilities from the standard normal table. Values with |z| > 2 are sometimes considered unusual; |z| > 3 are outliers in a normal distribution.
Use Binomial when you have a fixed number of independent trials (n) each with the same probability of success (p), and you want to count the number of successes — e.g., 10 coin flips, how many are heads. Use Poisson when counting events that occur at a constant average rate (λ) in a fixed interval of time or space, with no fixed upper bound on the count — e.g., 3 calls per minute, number of website visits per hour. As a rule of thumb, Poisson approximates Binomial well when n ≥ 20 and p ≤ 0.05, with λ = np.
Skewness measures the asymmetry of a distribution about its mean. Zero skewness means the distribution is symmetric (like a normal distribution). Positive skewness (right-skewed) means the right tail is longer — the mean is greater than the median, common in income data or wait times. Negative skewness (left-skewed) means the left tail is longer — the mean is less than the median. In general: |skewness| < 0.5 is roughly symmetric; 0.5–1.0 is moderate skew; > 1.0 is highly skewed.