Descriptive Statistics Calculator

Variance measures the average squared deviation from the mean, expressed in squared units (e.g., dollars², cm²), which can be less intuitive to interpret. Standard deviation is simply the square root of variance and expresses variability in the same units as your original data, making it easier to understand. For example, if you're measuring heights in centimeters with a variance of 100 cm², the standard deviation is 10 cm—meaning heights typically vary by about 10 cm from the average. Both measure spread, but standard deviation is preferred for interpretation because it's in meaningful units. In statistical formulas and calculations, variance is often used because it has nice mathematical properties (e.g., variances add when combining independent random variables).

Skewness measures the asymmetry of a data distribution, indicating whether values are concentrated on one side with a tail extending in the opposite direction. Positive skewness (right-skewed) means the distribution has a longer tail on the right side, with most values clustered on the left and the mean greater than the median—common in income distributions, house prices, and response times. Negative skewness (left-skewed) has a longer left tail with values clustered on the right and mean less than median—seen in test scores with a ceiling effect or age at retirement. Zero or near-zero skewness indicates a symmetric distribution like the normal distribution, where mean ≈ median ≈ mode. Skewness values |skew| < 0.5 are considered fairly symmetric, 0.5-1.0 are moderately skewed, and > 1.0 are highly skewed. Understanding skewness helps choose appropriate statistical methods—for example, the median is often preferred over the mean for skewed data.

Kurtosis measures how heavy or light the tails of a distribution are compared to a normal distribution, indicating the likelihood of extreme values (outliers). A normal distribution has kurtosis = 3 (or excess kurtosis = 0). High kurtosis (> 3, or excess > 0) indicates 'heavy tails' with more extreme values and a sharper peak than normal—common in financial returns, where rare but extreme events occur. Low kurtosis (< 3, or excess < 0) indicates 'light tails' with fewer outliers and a flatter distribution than normal. Platykurtic (low kurtosis) distributions have values clustered near the mean with few extremes, like uniform distributions. Leptokurtic (high kurtosis) distributions have long tails and many outliers, requiring robust statistical methods. In practice, kurtosis helps assess risk—high kurtosis in stock returns means more frequent crashes or booms than a normal model would predict.

Use weighted statistics when different observations contribute unequally to the analysis—i.e., some data points have more importance, frequency, or reliability than others. Common scenarios include: (1) Grade Point Average (GPA): weight grades by credit hours (a 4-credit A counts more than a 1-credit A). (2) Survey data: weight responses by the number of people each respondent represents (e.g., demographic surveys where one respondent represents 1000 people in that group). (3) Investment portfolios: weight returns by the dollar amount invested in each asset. (4) Quality scores: weight ratings by reliability or confidence (e.g., expert opinions weighted higher than novice). (5) Grouped data: when you have frequency counts rather than raw values (e.g., 10 students scored 85, 15 scored 90). The weighted mean formula is Σ(w_i · x_i) / Σw_i, where w_i are the weights. Using regular (unweighted) statistics when weights matter can produce misleading averages that don't reflect the true center of your data.

This tool uses two complementary methods to detect outliers: (1) Z-score method: A value is flagged as an outlier if it's more than 3 standard deviations away from the mean (|z| > 3). This works well for approximately normal distributions and identifies extreme values in terms of standard deviations. For example, if mean = 100, σ = 10, then values below 70 or above 130 are outliers. (2) IQR (Interquartile Range) method: A value is an outlier if it's below Q1 - 1.5×IQR or above Q3 + 1.5×IQR, where IQR = Q3 - Q1. This method is robust to non-normal distributions and doesn't assume any particular shape. The tool reports outliers detected by either method. Not all outliers are errors—some represent genuine extreme cases (e.g., a billionaire in income data, a genius in IQ scores). Investigate outliers to determine if they're measurement errors, data entry mistakes, or valid extreme observations that deserve special attention or separate analysis.

Cohen's d is a standardized effect size that measures the difference between two group means in units of standard deviation: d = (mean₁ - mean₂) / pooled_σ. It's 'standardized' because it's unit-free, making it comparable across different measurements and studies. Interpretation guidelines: |d| < 0.2 = negligible difference (groups are nearly identical), 0.2-0.5 = small effect (noticeable but subtle difference), 0.5-0.8 = medium effect (moderate practical significance), > 0.8 = large effect (substantial difference that's obvious in practice). For example, d = 0.5 means the groups differ by half a standard deviation—a medium effect where about 69% of one group scores above the mean of the other group. Cohen's d is used to assess practical significance beyond statistical significance—a statistically significant p-value with d = 0.1 may not be meaningful in practice, while d = 0.9 indicates a large real-world difference even if the sample is small. In clinical trials, d > 0.5 often indicates clinically significant improvement.