Track 9 · Correlation and Regression · Lesson 1 of 15

Covariance and correlation

Calculate covariance and Pearson's correlation, test whether a correlation differs from zero, and recognise what r cannot see.

35 minAdvanced8 graded exercisesFree

A running club wants to tell members how strongly weekly training volume is related to race times, and a facilities manager wants to know how closely a building's energy use tracks the outdoor temperature. One number — the correlation coefficient — summarises a straight-line relationship, but only if you know when it can be trusted.

Every relationship between two numeric variables starts with a scatter plot: one point per observation, the explanatory variable on the horizontal axis and the response on the vertical axis. Describe what you see in three words before computing anything:

Direction — do larger $x$ values go with larger $y$ values (positive) or smaller ones (negative)?
Form — do the points follow a straight line, a curve, or clusters?
Strength — how tightly do the points hug that form?

Weekly TV spend and units sold for a juice brand (156 weeks, synthetic). The column of points at zero spend are weeks between TV campaigns.

Show the data as a table

TV spend (thousand SAR)	Units sold (packs)
0	4367
0	5380
75.3	7627
61.3	7919
62.6	6688
58.6	7884
66	7739
0	5879
0	4796
0	6707
0	5775
60	7577
67.7	8206
72.8	8885
72.3	7832
73	9029
65	8250
68.4	8799
69	8145
0	5831
0	6570
0	5668
0	6953
0	6784
49.4	8785
60.9	9182
51.2	9505
64.1	8240
0	7408
0	7306
0	6773
0	6541
55.3	8455
54.5	8957
53.7	9229
53.1	8703
55.1	7460
51.9	8039
0	5905
0	5477
0	6114
80.8	8945
75.1	9202
93.5	9112
85.5	8624
86.6	8919
0	6021
0	5725
0	5676
0	5329
83.5	8201
62.9	7164
75.5	8819
79.1	8230
0	5154
0	5265
0	6139
82.9	8482
75.9	8031
89.7	9162
88.7	8621
79.1	9362
88.7	10276
93.9	8955
78	8466
0	6361
0	6240
0	6718
0	6151
0	5667
106.3	9681
106.7	9296
96.4	9560
123.9	10197
100.6	10482
0	7032
0	7468
0	6456
0	6870
0	7602
0	7245
37.9	8807
42.1	7896
35.5	9124
36.4	8515
40.5	8897
41.6	9238
38.6	7039
0	5860
0	6382
0	5542
0	5216
0	5785
0	6580
109.6	9390
117	8750
105.1	9130
122.9	10112
114.5	9322
123	9794
0	6321
0	7010
0	5697
0	5463
0	5649
0	6896
0	5482
81.7	8272
92.3	8680
85.8	8411
91.3	8105
87.7	9232
0	5353
0	7431
0	6588
0	6596
0	6173
0	6563
89.4	9363
94.9	9345
92.4	8873
94.2	9133
92.5	9302
91.3	10174
0	7128
0	6167
0	7733
0	6950
102.2	9416
111.7	10983
106.2	11032
123.8	9927
95	9762
118.4	11045
122.8	11248
0	7315
0	7662
0	7385
0	7136
36.8	8404
36.7	7713
34.9	8159
33.8	8526
35.3	8267
39.1	7926
0	6386
0	6829
0	6636
0	6738
0	6582
0	6256
27.9	7878
29.8	7685
28.4	8746
28.7	6855
0	6151

Covariance measures whether the two variables move together. For each observation, multiply how far $x$ is from its mean by how far $y$ is from its mean. When both are above (or both below) their means the product is positive; when one is above and the other below it is negative. The covariance averages these products.

s_{xy} = \Cov(x, y) = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})(y_i - \bar{y}) = \frac{S_{xy}}{n-1}

Sample covariance

$x_i, y_i$: the $i$ -th pair of observations
$\bar{x}, \bar{y}$: the two sample means
$S_{xy}$: the sum of cross-products $\sum (x_i - \bar{x})(y_i - \bar{y})$
$n$: the number of pairs; dividing by $n - 1$ matches the sample variance

The covariance has awkward units (kilometres × minutes, SAR × packs) and its size depends on the scale of measurement, so it cannot tell you whether a relationship is strong. Dividing by both standard deviations removes the units and gives Pearson's correlation coefficient $r$ .

r = \frac{s_{xy}}{s_x\, s_y} = \frac{S_{xy}}{\sqrt{S_{xx}\, S_{yy}}}

Pearson's correlation coefficient

$s_x, s_y$: the sample standard deviations of $x$ and $y$
$S_{xx}, S_{yy}$: the sums of squared deviations $\sum (x_i - \bar{x})^2$ and $\sum (y_i - \bar{y})^2$

Worked example

Covariance and r by hand

Five pairs: $x$ = 1, 2, 3, 4, 5 and $y$ = 2, 4, 5, 4, 5. Find the covariance and the correlation.

Means. $\bar{x} = 3$ , $\bar{y} = 4$ .
Deviations. $x - \bar{x}$ : −2, −1, 0, 1, 2. $y - \bar{y}$ : −2, 0, 1, 0, 1.
Cross-products $(x - \bar{x})(y - \bar{y})$ : 4, 0, 0, 0, 2, so $S_{xy} = 6$ . Also $S_{xx} = 4 + 1 + 0 + 1 + 4 = 10$ and $S_{yy} = 4 + 0 + 1 + 0 + 1 = 6$ .
Covariance. $s_{xy} = 6/(5 - 1) = 1.50$ .
Correlation. $r = 6/\sqrt{10 \times 6} = 6/\sqrt{60} = 0.7746$ .

Covariance 1.50; $r = 0.775$ — a fairly strong positive linear association.

Property	What it means in practice
$-1 \le r \le 1$	$r = \pm 1$ only when every point lies exactly on a straight line
No units	Changing kilometres to miles or minutes to hours leaves $r$ unchanged (covariance changes)
Symmetric	The correlation of $x$ with $y$ equals the correlation of $y$ with $x$
Linear only	$r$ measures how closely points follow a straight line; a strong curve can give $r$ near 0
Not resistant	One extreme point can create, destroy or reverse a correlation

Properties of

r

Generate data with a chosen correlation, then switch the pattern to curved or add an outlier.

Daily energy use against outdoor temperature for one office building (synthetic). Pearson's r is only 0.31 because the pattern is a curve, not a line.

Show the data as a table

Outdoor temperature (°C)	Energy use (kWh)
9.4	2711.1
12.3	2658.9
13.3	2560.5
10.2	2670.3
10.8	2549.4
7.6	1320.4
5.6	1446.6
9.3	2721.3
8.9	2689.9
8.2	2664
9.3	2533.5
11.1	2555.6
7.8	1263
7.9	1312.8
7.6	2809.5
9.5	2656.7
9.3	2797.6
10	2673.8
10.2	2776.5
6.6	1319.4
6.6	1528
6.4	2897.5
9.5	2680.7
11.7	2471.4
11.2	2509.6
10.8	1338
9.9	1309.9
7.3	2723.8
6.5	2791.4
6.3	2728.4
7.6	2878.2
9.1	2609.5
8.4	1474.9
8.6	1413
12.4	2383.9
12.5	2308.3
12.7	2454.6
13.1	2439.8
11.7	2531.7
11.5	1234.6
13.4	1162
12.8	2273.3
13.2	2337.3
14.2	2386.4
14	2469.5
12.7	1427.1
11.9	1271.3
11.5	2543.8
12.1	2548.1
13.9	2465
12	2561.1
14.1	2303
13.4	1054
10.7	1159.2
11.2	2546.3
12	2333.9
12.5	2397.7
11.9	2440
11.3	2395.2
11.3	1156.4
12.8	1203.2
15.6	2299.3
13.6	2448.1
14.1	2440.2
14.3	2410.8
13.4	2394.3
14.1	1156.6
14.2	1098.7
11.1	2512.2
10.8	2624.7
10.7	2503.1
10.6	2469
13.2	2324.4
13.5	1146.2
15.8	1098.5
16.2	2332.2
17.9	2401.6
16.2	2337.9
14.8	2469.3
16.9	2528.5
17.2	1145.8
17.8	1255.2
20.2	2311.1
18.8	2302
17.7	2293
17	2226.6
14	2353.5
14.4	1133.8
13.8	1159.2
13.6	2616.7
13.3	2545.1
15.3	2298
16.4	2227.9
15.3	2453.6
15.3	1182
17.9	1259.9
17	2528.1
16.8	2450
17.5	2498.5
18.3	2375.7
18	2364
20.1	1253
20.1	1182.4
21.7	2372.2
19.1	2400.8
21.1	2393.6
21.6	2167.9
24	2408.8
27.1	1290
24.8	1196.6
23.9	2455.4
24.2	2329.4
21	2202.8
20.7	2359.3
20.9	2377
22	1215.2
22	1078.6
22.3	2160.3
22.1	2251.2
22.7	2263.9
22.6	2148.4
22.7	2399.8
22.3	1220.4
25.3	1120.1
22.6	2382.6
24.8	2662.1
27.4	2849.4
28.2	2923.6
26.9	2731
29.1	1485.6
28.8	1511.2
28.1	2901.3
29.6	3032.4
29.9	3094.4
27.1	2854.8
26.7	2834.6
26.7	1353.5
24.8	1344.5
26.8	2752.8
28.9	2961.8
30.5	3162.3
29.8	3064.1
28.9	1570.8
26.7	1468.5
26.7	2666.4
28	2913.2
31.3	3214.6
32.7	3378.3
31.7	3256.5
32.6	1609.1
31.7	1347.5
30.6	2903.5
28.5	2941.1
28.3	2856.8
30.8	3205.7
32.3	3263.8
33	1562
33.6	1557.2
32.4	3117.8
27.5	2712.4
28.9	2992.2
27.8	2988.1
30	3155.9
31.1	1561.6
32.2	1510.6
28.3	2791.1
29.9	2848.2
31	3054.4
28.8	2848.6
29.6	2977.3
32.1	1569.9
34.6	1652.4
31.1	3175.8
28.6	2942.7
30.2	3168.3
33	3323.9
32.1	3261.9
32.5	1712.1
30.6	1579.2
30.1	3124.9
34	3397.9
31.7	3329.6
35.1	3578.5
33.5	3492.7
32.3	1588.5
33.8	1585.4
30	3036.2
30.5	3056.5
33.8	3319.4
34.6	3390.2
34	3276.4
37.1	1675.2
36.4	1680.8
34.7	3392.2
33.8	3429.9
32.7	3395
33.1	3217.6
34.2	3566.9
34.2	1713.5
33.6	1534
31.2	2978.3
33	3267.7
32.9	3358
35.2	3531.8
35.1	3686.2
35	1677.4
33.5	1662.7
35	3660.8
35.9	3747.1
37.2	3768.8
38	3752.5
34.8	3564.5
33.2	1609.8
32.3	1602.4
33.9	3413.6
30.4	3169.3
28.5	2982.8
28.9	3056.2
29.4	3091.5
30.6	1545.3
31.6	1531.1
31.5	3322.2
30.6	3118.2
30.5	3089
31.6	3163.2
33.6	3358.9
33.9	1577.5
34.9	1610.9
32.7	3175.1
33.3	3296.6
32.5	3250.5
32.3	3488.4
32.8	3383
31.9	1676.6
31.1	1434.7
35.4	3636.3
33.5	3500.1
30.8	3159.2
30	2956.7
29	2990.6
31.3	1408.8
29.7	1325.3
31	3047.3
32.1	3202.5
30.4	3130.7
31.9	3184.5
31.1	2988.8
29.2	1280.1
28.3	1317.2
30.5	2961.7
29.3	2956.5
32.6	3317.4
28.8	2838.7
25	1294
24	1287.8
23.9	2569.6
25.7	2710.5
25.7	2692.8
26.8	2718.1
29.8	3032.9
27.9	1409
28.1	1433.8
24.9	2491.8
25.5	2475.5
25.8	2646.3
27.6	2869.3
26.1	2753
25.4	1271.9
25.6	1270.3
24.8	2447.4
24.1	2384
22.1	2225.1
23.6	2414.3
24.9	2556.7
24.8	1303.6
26.3	1448.9
25.6	2689.4
23.8	2537
24.2	2512
24.2	2543.6
23.7	2319
24.9	1119.1
23.3	1122.6
21.5	2287.9
23.3	2319.8
21.3	2101.5
25.5	2631.1
25.1	2567.2
20.3	1064.6
20.4	1015.5
20	2217.2
20.7	2318
18	2229.6
18.5	2422.9
14.3	2556.3
17.5	973
15.4	1132.1
15.7	2353.1
17.3	2260.3
16.9	2303.9
19.6	2451
18.7	2504
19.2	1174.5
18.9	1218.9
16.7	2356.5
16.6	2399.9
16.7	2327.5
13.1	2470.5
13.9	2244.6
17.3	1041.2
14	1080
14.2	2227.7
15.6	2190.2
13.9	2435.8
12.9	2606.5
10.7	1416.9
10.4	1332.9
11.3	2544.6
12.6	2544.6
13	2551
13.1	2582.6
15.8	2432.2
13.8	1298.7
12.3	1196.1
10.9	2514
8.5	2761.9
10.3	2630.1
9.3	2718
11.8	2563.6
13.9	1158.9
15.8	1100.6
13.8	2491.6
14.8	2460
14.1	2445.1
12.4	2488.4
12.6	2529
12.7	1251.4
13.1	1273
12	2395.6
10.4	2409.1
10.3	2567.2
10.2	2638.7
14.1	2418.5
13.1	1094.6
11.8	1267.8
12.3	2551.4
11	2543.3
9.3	2675.8
8.4	2706.7
8.9	2634.8
8.7	1384.5
8.3	1417.1
7.7	2804.7
6.8	2688.1
7.3	2726.9
8.4	2748
8.2	2677.3
9	1293.3
9.9	1367.7
10.3	2675

Is the correlation real or sampling noise? The population correlation is written $\rho$ (rho). To test $H_0: \rho = 0$ , convert $r$ into a $t$ statistic with $n - 2$ degrees of freedom. A confidence interval for $\rho$ uses Fisher's $z$ -transformation (the lab reports it).

t = r\sqrt{\frac{n-2}{1-r^2}}, \qquad df = n - 2

Test of H0: ρ = 0

$r$: the sample correlation
$n$: the number of pairs

Worked example

Training volume and race time

In the running club data (500 finishers), is weekly training distance correlated with finish time?

Estimate. $r = -0.640$ : runners who train more tend to finish faster (lower times).
Test. $t = -0.6396\sqrt{498/(1 - (-0.6396)^2)} = -18.57$ on 498 df, two-sided p < 0.0001.
Interval. 95% CI for $\rho$ : (−0.689, −0.585).
Strength. $r^2 = 0.409$ : about 41% of the variation in finish times is associated with a straight-line relationship to training volume.

A moderately strong negative linear association, far too strong to be sampling noise. It does not show that training causes faster times — fitter runners may simply choose to train more (see Correlation is not causation).

Quick check

A dataset records distance in kilometres and time in minutes, with $r = -0.64$ . The analyst converts distance to miles and time to hours. What happens to $r$ ?

Exercises

The lesson is complete when every exercise is answered correctly. Hints are free; viewing a worked solution before solving means the exercise won't count towards mastery.

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Exercise 1 · Guided, difficulty 1 of 3

Covariance and r from seven pairs

Not started

Seven students recorded their weekly study hours ( $x$ : 10, 11, 2, 3, 6, 8, 13) and their quiz scores ( $y$ , out of 100: 94, 102, 57, 55, 76, 66, 106). Calculate the sample covariance and Pearson's correlation $r$ . Give the covariance to 2 decimal places and $r$ to 3.

Study hours x (n = 7)

10, 11, 2, 3, 6, 8, 13

Quiz score y (n = 7)

94, 102, 57, 55, 76, 66, 106

Exercise 2 · Practice, difficulty 1 of 3

Write the formula for r

Not started

Write an expression for Pearson's correlation $r$ in terms of the sum of cross-products Sxy and the sums of squares Sxx and Syy.

Exercise 3 · Practice, difficulty 1 of 3

Match the scatter to r

Not started

Match each description of a scatter plot (50 points each) to the most plausible Pearson correlation.

Exercise 4 · Practice, difficulty 2 of 3

Is the correlation significant?

Not started

A survey of $n = 40$ employees found a correlation of $r = 0.35$ between weekly overtime hours and a wellbeing score. Test $H_0: \rho = 0$ against a two-sided alternative: give the $t$ statistic (2 decimal places) and the p-value (4 decimal places).

Exercise 5 · Application, difficulty 2 of 3

Training and finish time

Not started

In the running_club dataset, test the correlation between training_km_week and finish_min (Pearson, two-sided). Report $r$ , the $t$ statistic and the 95% confidence interval for $\rho$ . Use the lab's Test tab (correlation).

Data: running_club · Open in the lab to explore and calculate.

Exercise 6 · Challenge, difficulty 2 of 3

One unusual point

Not started

Nine machines' ages ( $x$ , years) and annual maintenance costs ( $y$ , thousand SAR) are shown. A tenth machine, aged 22 years with a cost of 20.9 thousand SAR (it was rebuilt last year), is then added.

Calculate $r$ for the nine machines and for all ten. Give both to 3 decimal places.

Age x (years) (n = 9)

5, 7, 10, 4, 6, 8, 1, 3, 2

Cost y (k SAR) (n = 9)

29, 34.8, 40.9, 29.6, 32.7, 33.7, 26.3, 27.4, 23.9

Exercise 7 · Interpretation, difficulty 2 of 3

A weak r for a strong relationship

Not started

For the 360 days with a meter reading in energy_use, the correlation between outdoor_temp_c and energy_kwh is $r = 0.31$ . Plot the data in the lab (Visualise → scatter). Which conclusion is justified?

Data: energy_use · Open in the lab to explore and calculate.

Exercise 8 · Interpretation, difficulty 2 of 3

Report the correlation

Not started

For the running club, $r = -0.64$ between weekly training distance and 10 km finish time (n = 500, 95% CI −0.69 to −0.58). Write two or three sentences for the club newsletter describing the relationship: its direction, its strength, and what it does not show.

What you can now do

Describe a scatter plot by direction, form and strength
Calculate the sample covariance and Pearson's r, and explain why r has no units
Test H0: ρ = 0 and report a confidence interval for a correlation
Recognise when r misleads: curved relationships and influential outliers