A COLLECTION OF CONNECTIONS

Same math.
Different stories.

Seven datasets to sharpen your intuition. Every example is simulated for teaching, not evidence about the real world.

01 / STRONG POSITIVE

Study hours × exam score

More study time accompanies higher scores in this simulated sample. Prior knowledge, teaching and many other factors still matter.

r = 0.870
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Study hoursExam score
0.50053.937
1.00054.887
1.50041.846
2.00061.372
2.50058.998
3.00051.609
3.50062.600
4.00059.058
4.50067.537
5.00062.951
5.50059.187
6.00078.801
6.50066.571
7.00065.511
7.50079.566
8.00072.640
8.50078.581
9.00074.809
9.50078.475
10.00091.062
Exam score versus Study hoursFocus or hover a data point to inspect its coordinates. Potential outliers use diamond markers. -0.635.92.351.25.366.58.281.711.197Study hoursExam score
02 / WEAK POSITIVE

Height × weight

A weak positive association with substantial individual variation. These illustrative measurements are synthetic.

r = 0.300
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Height (cm)Weight (kg)
152.00062.957
154.00062.261
156.00041.262
158.00067.521
160.00062.001
162.00049.204
164.00063.079
166.00055.863
168.00066.093
170.00057.364
172.00049.827
174.00076.214
176.00056.391
178.00052.779
180.00071.098
182.00058.974
184.00065.520
186.00057.971
188.00061.216
190.00077.405
Weight (kg) versus Height (cm)Focus or hover a data point to inspect its coordinates. Potential outliers use diamond markers. 147.436.9159.248.117159.3182.870.5194.681.7Height (cm)Weight (kg)
03 / STRONG NEGATIVE

Exercise × resting heart rate

Higher exercise values accompany lower resting heart rates in this simulated dataset. An observational relationship is not a causal conclusion.

r = -0.860
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Exercise (hours/week)Resting heart rate (bpm)
0.33381.473
0.66779.569
1.00066.806
1.33379.322
1.66774.838
2.00066.462
2.33372.353
2.66766.963
3.00070.904
3.33364.704
3.66759.142
4.00071.726
4.33359.592
4.66756.129
5.00064.397
5.33356.381
5.66758.352
6.00052.783
6.33352.988
6.66760.117
Resting heart rate (bpm) versus Exercise (hours/week)Focus or hover a data point to inspect its coordinates. Potential outliers use diamond markers. -0.449.31.558.23.567.15.5767.484.9Exercise (hours/week)Resting heart rate (bpm)
04 / TEMPERATURE & SALES

Temperature × ice cream sales

A simulated seasonal relationship. Temperature can also explain a correlation between ice cream sales and swimming activity.

r = 0.760
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Temperature (°C)Ice cream sales
13.000174.690
14.000177.562
15.00069.778
16.000219.567
17.000196.147
18.000133.068
19.000215.357
20.000182.697
21.000245.120
22.000204.209
23.000169.799
24.000320.284
25.000218.911
26.000205.891
27.000312.406
28.000252.989
29.000295.336
30.000260.859
31.000285.214
32.000380.116
Ice cream sales versus Temperature (°C)Focus or hover a data point to inspect its coordinates. Potential outliers use diamond markers. 10.732.516.6128.722.5224.928.4321.234.3417.4Temperature (°C)Ice cream sales
05 / NO CORRELATION

Unrelated variables

These synthetic coordinates are constructed to have zero sample linear correlation. Random samples will not usually have exactly zero r.

r = -0.000
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Variable XVariable Y
1.00015.030
2.00014.229
3.000-0.761
4.00017.276
5.00013.104
6.0003.846
7.00013.228
8.0007.871
9.00014.706
10.0008.291
11.0002.709
12.00020.835
13.0006.668
14.0003.829
15.00016.317
16.0007.529
17.00011.789
18.0006.199
19.0008.153
20.00019.152
Variable Y versus Variable XFocus or hover a data point to inspect its coordinates. Potential outliers use diamond markers. -1.3-3.44.63.310.51016.416.722.323.4Variable XVariable Y
06 / NONLINEAR RELATIONSHIP

A relationship hiding in plain sight

The U-shape is perfectly predictable, yet Pearson r is zero. Always look at the scatter plot before interpreting a number.

r = 0.000
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XX²
-10.000100.000
-9.00081.000
-8.00064.000
-7.00049.000
-6.00036.000
-5.00025.000
-4.00016.000
-3.0009.000
-2.0004.000
-1.0001.000
0.0000.000
1.0001.000
2.0004.000
3.0009.000
4.00016.000
5.00025.000
6.00036.000
7.00049.000
8.00064.000
9.00081.000
10.000100.000
X² versus XFocus or hover a data point to inspect its coordinates. Potential outliers use diamond markers. -12.4-12-6.2190506.28112.4112XX²
07 / OUTLIER EXAMPLE

One point. A different story.

An extreme point can substantially change Pearson r. A flag is an invitation to investigate, never permission to discard an observation.

r = -0.433
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XY
1.0005.618
2.0005.937
3.000-2.256
4.0009.366
5.0007.662
6.0002.908
7.0009.337
8.0006.923
9.00011.823
10.0008.774
11.0006.226
12.00017.901
13.00010.201
14.0009.297
15.00017.590
16.00013.117
17.00016.474
18.00013.920
19.00015.892
20.00023.291
48.000-30.000
Y versus XFocus or hover a data point to inspect its coordinates. Potential outliers use diamond markers. -4.6-36.49.9-19.924.5-3.439.113.253.629.7XY