Univariate least-squares fit

Calculation contract

slope = sample_cov(x,y)/sample_var(x); intercept = mean(y) - slope*mean(x)

Method and interpretation

The fixture pairs 1,2,3,4 with 2,4,5,7. The fitted slope is 1.6 and the intercept is 0.5, with residual sum of squares 0.2. These numbers describe the supplied pairs; they do not identify a causal mechanism or establish a forecasting relationship.

The implementation uses the pinned Simple Statistics least-squares function and separately calculates residual fit. Independent Python checks use centered sums, providing a second arithmetic path. Constant x is rejected because no slope is identified; constant y retains a valid fit but withholds R-squared.

Where the reference stops

No p-values, robust standard errors, confidence intervals, multiple regressors or out-of-sample predictions are implied. Large offsets with very small variation can be numerically ill-conditioned; rescale before interpreting. This bounded teaching implementation is not a statistical certification.

Published synthetic inputs

{
  "x": [
    1,
    2,
    3,
    4
  ],
  "y": [
    2,
    4,
    5,
    7
  ]
}

Frozen calculation output

{
  "slope": 1.6,
  "intercept": 0.5,
  "r_squared": 0.9846153846153847,
  "residual_sum_squares": 0.1999999999999999,
  "observations": 4
}

Univariate in-sample least squares with an intercept. R-squared is withheld for constant response; no inference or out-of-sample accuracy is supplied.

Verification coverage

• Known line, residual sums, constant-response and constant-explanatory cases.
• Input alignment and missing-value rejection tested.

Unit and independent-reference checks test arithmetic and rejection rules, not future investment performance.

Method references

• SciPy: least-squares regression
• Simple Statistics: reference implementation and API

Read the assumptions in context