Summarizing Numerical Associations

STAT 20: Introduction to Probability and Statistics

Concept Questions

Estimate the correlation

What is the (Pearson) correlation coefficient between these two variables?

This plot has a very strong negative linear association; the correlation coefficient is -.98.

The goal of this exercise is threefold: 1. Start calibrating the ability to associate the correlation with the structure in a scatter plot. 2. Make it clear that this is still just an informal eye-balling procedure (therefore multiple answers can be reasonable, but some are unreasonable). 3. Set some of them up for a challenge on the next one; some will conflate correlation with association.

If you’d like to train their eye a bit more, you can open up https://www.guessthecorrelation.com/ in a browser and play the game a few times.

01:00

Which four plots exhibit the strongest association?

01:00

Student will likely identify B, D, E, and F. F only appears to be highly associated because of the scaling of the y-axis. This is a good opportunity to review definition of association. The following slides demonstrate the effects of that scaling.

Pearson Correlation and Scale

df_f

Pearson Correlation and Scale

df_f

        x            y
1   -8.00 -1.307007369
2   -7.95 -0.466886973
3   -7.90 -1.317638373
4   -7.85  1.008059015
5   -7.80  1.153714017
6   -7.75  0.136000314
7   -7.70  1.081268517
8   -7.65  0.027234440
9   -7.60 -0.755489564
10  -7.55  0.643581028
11  -7.50 -2.221816425
12  -7.45 -0.352711462
13  -7.40  0.788325702
14  -7.35  0.348209977
15  -7.30  0.888132692
16  -7.25  1.494640228
17  -7.20  0.539665177
18  -7.15  1.534245783
19  -7.10 -0.636288267
20  -7.05  1.400750746
21  -7.00  0.857266821
22  -6.95  0.689240640
23  -6.90 -0.578689332
24  -6.85 -0.247484190
25  -6.80 -1.616711187
26  -6.75 -0.211921880
27  -6.70 -1.389650022
28  -6.65 -0.667030427
29  -6.60  0.265923179
30  -6.55  0.368512913
31  -6.50  0.159370185
32  -6.45 -0.803029578
33  -6.40 -1.181853575
34  -6.35  0.799141786
35  -6.30 -0.119741600
36  -6.25  1.055099362
37  -6.20  0.260610303
38  -6.15  1.215058556
39  -6.10 -2.093776227
40  -6.05 -0.340475843
41  -6.00  1.098181964
42  -5.95 -1.022920292
43  -5.90  0.997308309
44  -5.85  0.442282185
45  -5.80  2.024841893
46  -5.75 -0.206236931
47  -5.70 -0.366921396
48  -5.65 -0.540982671
49  -5.60  0.326545855
50  -5.55  1.185875818
51  -5.50  0.163270582
52  -5.45 -0.252746010
53  -5.40 -0.538783647
54  -5.35  0.239830058
55  -5.30  0.227238979
56  -5.25  0.146973095
57  -5.20  1.206565164
58  -5.15  0.274237725
59  -5.10  0.291223416
60  -5.05 -0.048435420
61  -5.00 -0.160503335
62  -4.95  0.982662625
63  -4.90 -0.009812706
64  -4.85 -1.236449694
65  -4.80 -0.220086594
66  -4.75  0.916450245
67  -4.70  0.312158589
68  -4.65 -0.271222289
69  -4.60 -0.362834150
70  -4.55  0.690127669
71  -4.50  0.954579372
72  -4.45 -1.002334741
73  -4.40 -0.827510811
74  -4.35 -0.481050964
75  -4.30  0.213284850
76  -4.25  0.694018655
77  -4.20  0.318705891
78  -4.15  1.359291786
79  -4.10  0.299499214
80  -4.05 -0.136853535
81  -4.00 -0.907524278
82  -3.95  0.580039124
83  -3.90 -0.451761432
84  -3.85 -0.637651429
85  -3.80 -0.377287558
86  -3.75 -0.108975033
87  -3.70  0.266174878
88  -3.65  1.950537958
89  -3.60  2.481797153
90  -3.55  0.024020307
91  -3.50 -0.271974465
92  -3.45 -0.324420253
93  -3.40  0.914990966
94  -3.35 -0.177070403
95  -3.30  1.668125591
96  -3.25 -0.196626395
97  -3.20  0.909735944
98  -3.15  0.379802619
99  -3.10 -0.717674077
100 -3.05 -0.137978296
101 -3.00 -0.032596143
102 -2.95 -1.151078285
103 -2.90 -0.035147120
104 -2.85  1.289653594
105 -2.80  1.301664628
106 -2.75 -1.314453018
107 -2.70  0.279983464
108 -2.65 -0.134496134
109 -2.60 -0.993968666
110 -2.55 -1.125493956
111 -2.50 -0.585075814
112 -2.45 -0.360440229
113 -2.40 -0.128946258
114 -2.35 -1.763949074
115 -2.30  0.249921212
116 -2.25  1.356315758
117 -2.20 -0.221241869
118 -2.15 -0.255945298
119 -2.10  0.764032808
120 -2.05 -0.429023148
121 -2.00 -0.506416626

Pearson Correlation and Scale

df_f |>
    ggplot(aes(x = x,
               y = y)) +
    geom_point() +
    ylim(-9, 9) +
    theme_bw()

Pearson Correlation and Scale

df_f |>
    ggplot(aes(x = x,
               y = y)) +
    geom_point() +
    ylim(-9, 9) +
    theme_bw()

Pearson Correlation and Scale

df_f |>
    ggplot(aes(x = x,
               y = y)) +
    geom_point() +
    theme_bw()

Pearson Correlation and Scale

df_f |>
    ggplot(aes(x = x,
               y = y)) +
    geom_point() +
    theme_bw()

Pearson Correlation and Scale

df_f |>
    summarize(pearson = cor(x, y))

Pearson Correlation and Scale

df_f |>
    summarize(pearson = cor(x, y))

      pearson
1 -0.03106456

Describing Associations

When considering the structure in a scatter plot, pay attention to:

1. The strength of the association
   • How much does the conditional distribution of y change when you move along the x?

2. The shape of the association
   • Linear? Quadratic? Cubic or beyond?

3. The direction of the (linear) association
   • Positive or negative?

Guess: Which state has the highest poverty rate? The lowest?

This can be asked without poll-everywhere, without group discussion, just asking students to shout out which states they think will have highest and lowest.

As they answer, ask where they’re thinking those states points are; how they’re using this plot to think about the highest povery rate and the lowest poverty rate. Point out that these two questions don’t even take advantage of the scatter plot, which allows you to see the relationship between two variables. Each question just deals with the marginal distribution of poverty or graduation rate, which could just as well be two separate histograms, as shown on the next slide.

This is an equally-useful set of plots to answer the previous question.

If we want to ask a question that requires a scatter plot (and a linear model) to answer, we need to ask a conditional question.

Which state has the highest graduation rate given its poverty rate?

01:00

This assesses whether or not students have an informal sense of what a residual is.

Worksheet: Summarizing Numerical Associations

20:00

Break

05:00

Lab Part 2: Flights

25:00