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03 // 双向 STEM 与金融计算器 // 统计推断、概率分布与风险指标计算器b₁ = r × (s_y / s_x)

最小二乘线性回归斜率计算器 (b₁ = r × s_y / s_x)

根据 b₁ = r × (s_y / s_x) 求解回归直线斜率 b₁、皮尔逊相关系数 r、因变量标准差 s_y 或自变量标准差 s_x。

b₁ = r × (s_y / s_x)
选择要计算的未知量（或直接在任意输入框中输入数值）：当前求解: b₁ — OLS Regression Slope (Δy / Δx)
计算结果

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
计算结果 · OLS Regression Slope (Δy / Δx) (b₁)
1.6

分步代入计算过程

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).

常见参考标准值

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 行数值对照表

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 当前1.6 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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常见问题解答 (FAQ)

如何使用在线 最小二乘线性回归斜率计算器 (b₁ = r × s_y / s_x) 工具？+

输入已知参数或拖动滑块，即可在浏览器中实时计算并可视化 最小二乘线性回归斜率计算器 (b₁ = r × s_y / s_x)（基于关系式 b₁ = r × (s_y / s_x)）。根据 b₁ = r × (s_y / s_x) 求解回归直线斜率 b₁、皮尔逊相关系数 r、因变量标准差 s_y 或自变量标准差 s_x。

可以导出或打印 最小二乘线性回归斜率计算器 (b₁ = r × s_y / s_x) 的结果吗？+

可以。支持一键复制数据、打印计算工作表或导出高清 SVG、PNG 和 CSV 文件。

我可以清空 最小二乘线性回归斜率计算器 (b₁ = r × s_y / s_x) 中的示例数据并从空白状态开始吗？+

可以。点击工具顶部的“清空 / 空白开始”按钮即可重置所有输入项，随时也可一键重新加载示例数据。