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03 // Bidirectional STEM Calculators // STATISTICSb₁ = r × (s_y / s_x)

OLS Linear Regression Slope Calculator (b₁ = r × s_y / s_x)

In Ordinary Least Squares (OLS) simple linear regression, the best-fit slope b₁ equals the Pearson correlation coefficient r scaled by the ratio of vertical to horizontal standard deviations: b₁ = r × (s_y / s_x).

b₁ = r × (s_y / s_x)
Pick which value you want to calculate (or type into any field):Solving for: b₁ — OLS Regression Slope (Δy / Δx)
Calculated

Expected change in y per 1-unit increase in x

-100100
unit

Linear correlation between x and y (-1 to +1)

-0.9990.999
unit

Standard deviation of outcome variable y

0.0110,000.00
unit

Standard deviation of explanatory variable x

0.0110,000.00
unit
Calculated Result · OLS Regression Slope (Δy / Δx) (b₁)
1.6

Step-by-Step Calculation

01Canonical Law: b₁ = r × (s_y / s_x)
02Given Inputs (Base Units): r = 0.8, s_y = 20, s_x = 10
03Solve for b₁ (OLS Regression Slope (Δy / Δx)): b₁ = 1.6

In Ordinary Least Squares (OLS) simple linear regression, the best-fit slope b₁ equals the Pearson correlation coefficient r scaled by the ratio of vertical to horizontal standard deviations: b₁ = r × (s_y / s_x).

Real-World Reference Values

Coefficient of Determination (R²)R² = r² = 0.640 (64% Variance)

Share of variance in y explained by linear predictor x

Regression to the Mean|r| < 1 ⇒ |z_y| < |z_x|

Predicted standardized y is always closer to the mean than x

OLS Intercept Formula (b₀)b₀ = ȳ - b₁ × x̄

Regression line always passes through the centroid (x̄, ȳ)

20-Row Comparison Table

OLS Regression Slope (Δy / Δx) (b₁) · Pearson Correlation Coefficient (r)

OLS Linear Regression Slope Calculator (b₁ = r × s_y / s_x) — Pearson Correlation Coefficient → OLS Regression Slope (Δy / Δx)
#Pearson Correlation Coefficient (r)OLS Regression Slope (Δy / Δx) (b₁)Δ %
010.296 0.592 -63.00%
020.354947 0.709895 -55.63%
030.413895 0.827789 -48.26%
040.472842 0.945684 -40.89%
050.531789 1.0636 -33.53%
060.590737 1.1815 -26.16%
070.649684 1.2994 -18.79%
080.708632 1.4173 -11.42%
090.767579 1.5352 -4.05%
100.8 Current1.6 0.00%
110.885474 1.7709 +10.68%
120.944421 1.8888 +18.05%
131.0034 2.0067 +25.42%
141.0623 2.1246 +32.79%
151.1213 2.2425 +40.16%
161.1802 2.3604 +47.53%
171.2392 2.4783 +54.89%
181.2981 2.5962 +62.26%
191.3571 2.7141 +69.63%
201.416 2.832 +77.00%
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Frequently Asked Questions

How is Pearson correlation r different from regression slope b₁?+

Correlation r is dimensionless and symmetric (-1 ≤ r ≤ 1), whereas slope b₁ carries physical units of (units of y per unit of x).

What is R-squared (R²) in simple linear regression?+

In bivariate OLS regression, the coefficient of determination R² is simply the square of Pearson's r (at r = 0.80, R² = 0.64 or 64%).

How do I recover Pearson's r from a published regression slope?+

Click SOLVE FOR on r and enter the reported slope b₁ along with the standard deviations s_x and s_y.