Statistics Basics
Fundamental statistical measures and concepts.
The basic measures of central tendency, spread and association: mean, median, mode, variance, standard deviation, correlation and the z-score, plus range, interquartile range, covariance, skewness, standard error, coefficient of variation, confidence intervals and the p-value, each with its formula.
Use it when
- You need the definition and formula for a descriptive statistic.
- You are choosing a summary that is robust to outliers.
Watch out for
- Sample and population versions differ (n versus n - 1 in the denominator).
- A p-value is not the probability that a hypothesis is true, and correlation does not establish causation.
The data
15 rows
| Measure | Description | Formula |
|---|---|---|
| Mean (Average) | Sum of all values divided by count. | xbar = Sum(xi) / n |
| Median | Middle value when data is sorted. Robust to outliers. | Middle element (or avg of two) |
| Mode | Most frequently occurring value in the dataset. | argmax frequency |
| Variance | Average squared deviation from the mean. | sigma2 = Sum(xi - xbar)2 / n |
| Standard Deviation | Square root of variance; measures spread in original units. | sigma = sqrt(variance) |
| Correlation | Measure of linear relationship between two variables. | r = Cov(X,Y) / (sigma_x * sigma_y) |
| Z-Score | Number of standard deviations a value is from the mean. | z = (x - xbar) / sigma |
| Range | Difference between the largest and smallest value; very sensitive to outliers. | max - min |
| Interquartile Range | Spread of the middle half of the data; robust to outliers. | Q3 - Q1 |
| Covariance | How two variables vary together; positive when they tend to move in the same direction. | cov(X, Y) = E[(X - mean(X)) (Y - mean(Y))] |
| Skewness | Asymmetry of a distribution; positive when the right tail is longer. | E[((X - mean) / sd)^3] |
| Standard Error of the Mean | Standard deviation of the sample mean across repeated samples. | sd / sqrt(n) |
| Coefficient of Variation | Standard deviation relative to the mean, allowing comparison across different scales. | sd / mean |
| Confidence Interval for a Mean | A range computed so that, over repeated samples, it contains the true mean a stated share of the time. | mean +/- z * sd / sqrt(n) |
| p-value | Probability of data at least as extreme as observed if the null hypothesis were true. | P(data at least this extreme | H0) |
Fields
The shape of this dataset, read from the data itself: every column, its type and a real value. Agents can use this to plan a query without fetching the whole document.
| Section | Field | Type | Example |
|---|---|---|---|
| data | measure | string | Mean (Average) |
| data | description | string | Sum of all values divided by count. |
| data | formula | string | xbar = Sum(xi) / n |
Fetch the same bytes
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curl "https://yjtoon.yunusemrevurgun.com/api/dataset/statistics-basics?format=toon"
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