Consider the following data of 11 samples on 8 variables by Anscombe,

Question

Consider the following data of 11 samples on 8 variables by Anscombe, Francis J. (1973):

x₁	x₂	x₃	x₄	y₁	y₂	y₃	y₄
10	10	10	8	8.04	9.14	7.46	6.58
8	8	8	8	6.95	8.14	6.77	5.76
13	13	13	8	7.58	8.74	12.74	7.71
9	9	9	8	8.81	8.77	7.11	8.84
11	11	11	8	8.33	9.26	7.81	8.47
14	14	14	8	9.96	8.10	8.84	7.04
6	6	6	8	7.24	6.13	6.08	5.25
4	4	4	19	4.26	3.10	5.39	12.50
12	12	12	8	10.84	9.13	8.15	5.56
7	7	7	8	4.82	7.26	6.42	5.56
5	7	5	8	5.68	4.74	5.73	5.56

If the vector $\underline{x}=\begin{pmatrix}x_1\\x_2\\x_3\\x_4\end{pmatrix}\quad\text{and}\quad\underline{y}=\begin{pmatrix}y_1\\y_2\\y_3\\y_4\end{pmatrix}$ then obtain the sample covariance matrix between $\underline{x}\text{and}\underline{y}$

Source: Anscombe, Francis J. (1973). Graphs in statistical analysis. The American Statistician, 27, 17– 21. doi: 10.2307/2682899.

29 Apr 2025

Answer :

Word Count : 510

We are asked to compute the sample covariance matrix between $\underline{x} = (x_1, x_2, x_3, x_4)^\top$ and $\underline{y} = (y_1, y_2, y_3, y_4)^\top$ for Anscombe's dataset. Let’s solve manually step by step. --- ### Step 1: Recall formula for covariance For two variables $X$ and $Y$ with $n$ observations: $$ \text{Cov}(X, Y) = \frac{1}{n-1} \sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y}) $$ The sample covariance matrix between vectors $\underline{x} = (x_1, x_2, x_3, x_4)$ and $\underline{y} = (y_1, y_2, y_3, y_4)$ is a $4 \times 4$ ___ ___ _____ _____ ________ ________ ___ ________ ________ _______ ____ _____.
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