Question

The following bivariate data set contains an outlier.

x	y
48.9	43.4
33.2	483.2
63.3	-182.8
44.5	-22.7
40.6	-84.6
71.1	75.3
45.9	-126.9
16.1	343.7
44.3	77.1
53	-294
30.5	-94.2
53.9	2.1
6.9	491.9
65.5	-13
263.7	2849.6

What is the correlation coefficient with the outlier?
r_w =

What is the correlation coefficient without the outlier?
r_wo =

Would inclusion of the outlier change the evidence for or against a significant linear correlation?

• Yes. Including the outlier changes the evidence regarding a linear correlation.
• No. Including the outlier does not change the evidence regarding a linear correlation.

Answer 1

correlation coefficient with the outlier-

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
48.9	43.4	97.22	37303.06	1904.36
33.2	483.2	653.31	60841.16	-6304.63
63.3	-182.8	20.61	175846.04	-1903.80
44.5	-22.7	203.35	67205.38	3696.76
40.6	-84.6	329.79	103130.90	5831.90
71.1	75.3	152.28	25998.34	-1989.70
45.9	-126.9	165.38	132088.63	4673.84
16.1	343.7	1819.88	11483.27	-4571.45
44.3	77.1	209.09	25421.11	2305.50
53	-294	33.18	281472.69	3055.91
30.5	-94.2	798.63	109388.95	9346.71
53.9	2.1	23.62	54962.11	1139.38
6.9	491.9	2689.46	65208.73	-13242.97
65.5	-13	45.43	62270.21	-1681.90
263.7	2849.6	42000.40	6828082.56	535520.52

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	881.4	3548.1	49241.62	8040703.14	537780.4
mean	58.760	236.540	SSxx	SSyy	SSxy

correlation coefficient , r = Sxy/√(Sx.Sy)   =537780.4/√(49241.62*8040703.14) = 0.8547

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correlation coefficient without the outlier-

X	Y	(x-x̅)²	(y-ȳ)²	(x-x̅)(y-ȳ)
48.9	43.4	22.83	42.16	-31.03
33.2	483.2	119.28	187755.08	-4732.33
63.3	-182.8	367.82	54145.97	-4462.72
44.5	-22.7	0.14	5269.72	-27.48
40.6	-84.6	12.40	18088.33	473.61
71.1	75.3	727.84	645.52	685.45
45.9	-126.9	3.16	31255.71	-314.44
16.1	343.7	785.20	86322.64	-8232.90
44.3	77.1	0.03	740.23	4.86
53	-294	78.83	118262.30	-3053.28
30.5	-94.2	185.54	20762.75	1962.75
53.9	2.1	95.62	2284.16	-467.35
6.9	491.9	1385.43	195370.31	-16452.14
65.5	-13	457.04	3955.51	-1344.56

	ΣX	ΣY	Σ(x-x̅)²	Σ(y-ȳ)²	Σ(x-x̅)(y-ȳ)
total sum	617.7	698.5	4241.184	724900.39	-35991.5
mean	44.121	49.893	SSxx	SSyy	SSxy

correlation coefficient , r = Sxy/√(Sx.Sy)   = -35991.5/√(4241.184*724900.39) =    -0.6491

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Yes. Including the outlier changes the evidence regarding a linear correlation