Seven numbers define every regression output. Learn what each one means, where it comes from, and how to compute it yourself.

Example Regression Output

This is a simplified regression output for a model predicting Sales from three marketing channels. Every tab on this page references this table.

Variable Coefficient Std Error T Stat P-Value Significant? Lower 95% Upper 95%
Intercept 5000 1200 — — — — —
TV Spend 3.20 0.80 — — — — —
Search Spend 5.50 1.10
Social Spend 1.40 1.50 — — — — —
Model Details
Observations (n): 52 weekly data points  |  Predictors (k): 3  |  Degrees of Freedom (df): n − k − 1 = 48  |  Alpha (α): 0.05

The Coefficient and Std Error columns come directly from the regression. Everything else—T Stat, P-Value, Significance, and the Confidence Interval bounds—is derived from those two numbers (plus the degrees of freedom). The remaining tabs show you exactly how.

As you visit each tab, the computed values will fill in automatically in this table. Come back here anytime to see the full picture.

T-Statistics & P-Values

The T-Statistic

The t-statistic measures how many standard errors the coefficient is away from zero. Larger absolute values mean the coefficient is more distinguishable from noise.

Formula t = Coefficient / Standard Error

For TV Spend: t = 3.20 / 0.80 = 4.00. For Social Spend: t = 1.40 / 1.50 = 0.93. The bigger the t-stat, the stronger the evidence that the coefficient is not zero.

From T-Stat to P-Value

The p-value answers: If the true coefficient were zero, how likely would we see a t-stat this extreme (or more)? It uses the two-tailed t-distribution with df degrees of freedom.

Two-Tailed P-Value p = 2 × P(T > |t|), where T ~ t-distribution(df)

Interactive: T-Distribution Explorer

Drag the slider to change the t-statistic. Watch the shaded tail areas (the p-value) grow and shrink in real time.

2.00
2.00
T-Statistic
0.0512
P-Value (two-tailed)
—
Significant at α = 0.05?

Statistical Significance

Significance is a yes/no decision: compare the p-value to your chosen threshold (alpha, typically 0.05).

Decision Rule If p-value ≤ α → Significant (reject H₀) If p-value > α → Not Significant (fail to reject H₀)

Applied to Our Regression

—
Intercept
—
TV Spend
—
Search Spend
—
Social Spend
Important
“Not significant” does not mean the effect is zero. It means we don’t have enough evidence to confidently say it’s different from zero at this sample size and alpha level. A larger dataset or a different alpha could change the verdict.
What This Means in Practice
For TV Spend and Search Spend, the data provide strong evidence that these channels affect Sales. For Social Spend, the coefficient is too imprecise relative to its size—we can’t distinguish it from noise at α = 0.05.

Confidence Intervals

Margin of Error

The margin of error tells you how far the true coefficient could plausibly be from the estimate. It uses the critical t-value at your chosen confidence level.

Margin of Error MoE = t_critical(α, df) × Std Error

In Excel: =TINV(0.05, 48) gives the two-tailed critical value. At df=48 and α=0.05, t_critical ≈ 2.011.

Confidence Interval

Lower & Upper Bounds Lower 95% CI = Coefficient − MoE Upper 95% CI = Coefficient + MoE
Percentage Margin of Error %MoE = (MoE / |Coefficient|) × 100

Applied to Our Regression

Interactive: Confidence Interval Visualizer

Each variable’s coefficient and 95% CI plotted on a number line. If the CI crosses zero (the dashed line), the variable is not significant.

Reading the Plot
The dot is the coefficient estimate. The horizontal bar is the 95% confidence interval. If the bar crosses the dashed zero line, we cannot rule out that the true effect is zero—the variable is not significant at α = 0.05.