May 3rd, 2019

## Where are the cancers?

Highest kidney cancer rates in US (1980–1989) were in rural areas

## Highest rates of kidney cancer are in rural areas

Why? Maybe…

• the health care there is worse than in major cities
• High stress due to poverty
• are exposed to more harmful chemicals
• Drink Alcohol
• Smoke Tobacco

## Lowest rates of kidney cancer are in rural areas

Why? Maybe…

• No air and water pollution.
• No stress
• Eat healthy foods

Something is wrong, of course. The rural lifestyle cannot explain both very high and very low incidence of kidney cancer.

## Remember some concepts

Random variable
number that depends on the outcome of a random process
Average of population
a fixed number, that we do not know and want to know
Average of a sample
a random variable, that we get from an experiment

## Many things are averages

• concentration
• proportion
• density
• GC content
• frequency of events

What is the relationship between sample average and population average?

Can we learn the population average from the sample average?

## Hemocytometer

Each square is a sample. Volume is fixed. The cell count is an average of cell counts of some squares.

We want population cell density

We have a sample of cell densities

## Different samples have different average

N <- 4800
pop_LD <- sample(rep(0:9,N/10))
pop_MD <- sample(rep(0:29,N/30))
pop_HD <- sample(rep(0:79,N/80))
pop_HD
   [1] 64 77 47 76 43 17  0 47  6 14 28 46 69 18 43 17
[17]  8  6 13 44 58 21  8  3 31 36 75 74 57 12 14  0
[33] 72 21 23 56 37  6 79 14  0 27 22 34 22 53 76 65
[49]  7 72 25 32 59 61  6 77 61 58 66 19 59 67 49 40
[65] 36 35 15 69 47 72 63 38 70 36 77 53 56 66 21 46
[81] 64 24 76 42 66 41 55 49 69 71  1  1 38 68 33 71
[97]  1 16 77 57 50 62 15 62  4 19 28 57 73  7 62  3
[113] 52  9 20 43 47 54 77 58  4 33 27 12 44 60 23 35
[129] 21 68 16 49 65 63 65 59 46 29 42  6 34 18 70 68
[145] 54 32 73 57 23 18 61 20 39 54 77 54 76 71 72 20
[161] 47 53 73 31  2 12 34  7 18 76 53 50 24  4 36 16
[177] 37 40 18 27 47 31  7 18 36 38 63  9 19  1 51 67
[193] 29 54 27 56 75 64 37 21 73 56 31 12 36 76 75 63
[209] 73 63 66 50 29 33 65 23 63 53 61 20 23 55 30  0
[225] 10  8 20  1 61 70 53 45  3 41 49 48 59 68 55 77
[241] 74 72  2 35 65 51 71  7 54 62 67 31 72 28 20 64
[257] 55  4 56 16 67 14 13 57 60 32 36 39 74 75 76 24
[273] 16 76 69 21 66 14 60 54 19 48 36 66 47 32  3 64
[289] 25 14 59  6 57 40 39 79 72  1 47 64 18 42 31 38
[305] 13 42 16 75 68 32  7 61 23 18 74 39 73  7  0 53
[321] 69 40 50 75 70 77 28 71 55 42 15 72 78 14 26 47
[337] 49 55 77 65 77 14 73 21 60 76 53 16 38 35 75 55
[353] 46 16 51 73 65 27  6 39 66 39  7 54  4 76 23 79
[369]  7 48  3 16 21 24 61 54 75 41 29  1 78 13  0 48
[385] 22 70 33 31 18  2 71  6 18 42 60 54 38 19 34 43
[401]  6 67 67  5 72 45  9  4  2 15 51 69 27 15 66 54
[417] 75 62  2 54 75 42 66 71 55  4 77 20 76 50 13 47
[433] 46 53  9 34 26 59 34 20 22 46 23 65  0 39 42 36
[449] 11 36 10 34 37 26 66 30 55 48 38  8 64 34 28 43
[465] 17 61 37  7 11 47 26 48 43 30 33  3  1 68 72 15
[481] 72 28 66 10 34 58 56 27 47 17 45 17 38 51 40 49
[497] 19 67 73 35 18 73 68 28 11 29 53 22 65 45 26 33
[513] 61  2  7 14 60  8 15 48 56 59 35 44 52 59 44 24
[529] 77 33 59 41 52  5 23 66 61 15 64 65 52 62 13  4
[545] 53 73 46 77 43  9 43 32 35 76 10 29 44 29 27 54
[561] 30 62 48 46 63 72  0 10 58 42 12 73 52 47 38  6
[577] 77 44 23 27 60 65  6 41 62 58 52  6 66 11 78 25
[593] 17 48 64 43 23 42 37 48 40 57 22 33  6 38 71 25
[609]  1 46 20  7 42 70 21 15 38 54 79 72 54  3 78 60
[625] 52  8 50 48 70  4 43 59 38 38 78  7 12 29 45 12
[641] 29 19 18 15 38 52 50 13 38 24 47 49 34 72 73 67
[657]  8  6 12 59 76 76 10 21 70 58 14 46 54 14 16 79
[673] 60 64 45  2 32 46 42 74 68 72 74 26 79 43  8 14
[689]  4 54 52 27 29 33 48 68 77 64 46 55 33 68 57 25
[705] 72 12 58 60 44 68  7 79  5 62 55 49 46 29 57 18
[721]  5 42 33 68 25  0 35 29 47 51 11 14 12 20 50 77
[737] 19 30 64 52 58 59 30 65  4 44 60 70  6 12 74 31
[753] 27 52  1 55 27 14 32 23 71 10 73 33  1 67 59 15
[769] 62 40  7 18 48 69 22 55 75 52 32 62 44 78 70  0
[785] 62 13 77  9 66 78 24  8 74 68 65 27 77 48 12  2
[801] 68 72 56 72 12 23 42 22 70 78 26 54 57 71 15  1
[817]  3  1  9 59 51 14 12 61 35 49 16 51 26 78 10 52
[833] 30  6 40 21 31 30 23 19 53  3 18 28 36  8 78 16
[849] 63 75 51 32 76  3 15 29 61 79 33 20 63 28 56 22
[865] 35 24 64 18 20 63 13 39 15 31 33 26 43  2 55 63
[881] 24 41 20 25  6  0 53 67 71 14  3 30 22 19 68 24
[897] 37 14 12 18 74 39 51 24 42 49 64 53 20 32 62 58
[913] 53 50 48 63 22 64 69 77 52 62  7 45 13 12 68 36
[929] 54 20 26 45 21 25 78 56 19 50 69  9 37 42  8 16
[945] 62 75 62 20  9 71 20 45  9 42 31 58 35 69 21 34
[961] 62 52  6 37 65 10 16  2 68 72 24 47 26 72 63 75
[977] 49 15 40 13 29 41 79 50 50 20 79 24 25  2 49 60
[993] 73 18 31  7 51 45 20 69 44 40 51 64 64 28 44  5
[1009] 19 19 78 54 45 52 29 36 61 51 48 32 77  6 68 77
[1025] 74 42 21 67 65 29 62 70 56 61 20 16 66 58 71 69
[1041] 48 60 17 35 13 14 18 19  8 25 49 47 12 18  1 62
[1057] 26 17 76 56 45 48 76 48 66 55 39 53 31 26 43 61
[1073]  2 52 59 16  5 59 31 17  2 10 72 72 61  3 62 54
[1089] 38 53 35  3 56 77  9 71 28 18 79 12 12 31 11 74
[1105] 33 71 28  4 37 48 41 28 79 32  9 68 14 36 49 71
[1121] 30 55 63  9 51 25 44  2  9 27 60  5 71  3 18  8
[1137] 75 69 51 44 16 12 69  8 23 21  8 53 20 53 25  1
[1153] 14 10 73 42 11 32 54 75 12 41 32 44 47 15 27 30
[1169] 10 44 31 39 30 35 61 57 26 34 14 24 52  9 53  7
[1185] 13 66 49 14 43 34 40 71 60 19 49 77 37 60 58 67
[1201] 22 15  2 25 34 77 55 30 49 63 61 13 21 29 53 14
[1217]  1 64 64 59 33 57 59 44 35 19 57  4 33 14 34 43
[1233] 58 36 20 29 78 26 13 39  0  7 70 79 43 77 59 23
[1249] 17 21 52 58 73 79 15 66 37 56  6 45 56 40  7 79
[1265] 77  8  9  6 38 68 71 45  0  2  8 46 57 63 32 46
[1281] 49 33 39 33 40 30  3 18 23  0 60 19 31 64 62 54
[1297] 29 44 15 64 52 72  3  0 13 43 73 50 25  6 57 70
[1313] 42 16 77  9 56 52 67 75 55 73 77 65 47  9 51 11
[1329] 68 52 69 77 67 78  0 74 53 49 31 10 23 47 69 71
[1345] 46 34 74 67 68 58 41 46 21 30 37 46 60 38 37 45
[1361] 18 23 73 79 44 12 54  7 77 47 19 39 72 22 11 44
[1377] 13 46 77  9 71 79 14 74 23 46 67 46 35 67  9 40
[1393] 56 11  6 30  1 53 68 70 57 44 61 36 59 43 11 23
[1409] 60 21  1 58 59  3 78 11 14  2 62 35 70 37 64 65
[1425] 22 34 65 16  8  2 39 15 57 68 79 36 38 72 52 63
[1441] 13 16 20  7 70 68 32 12  8 12 47 54 61 40 10 41
[1457] 21 67 38 28 70 24 55 44 56 69 45 59 28 73 35 31
[1473]  6 57 38 22 19 44 16 79 67 48 45 18  7 44 78 72
[1489] 32 44 35 13 54 66 72 72  4 70 15 67 69 40 37 45
[1505] 43 76  0 44 28  6 19 30 19 45 30 51 68  4 22 35
[1521] 70 47  6 55 43 50 49 75  1 23 24 19 32 42 49 59
[1537] 59 75 59 46 49  2 29 21 42 28 50 10 36 50 21 70
[1553] 67 20 51 78 11 65  7 10 66  7 19 36 30 18 77 24
[1569] 44 59 25 47 12 44 40 54 63 47 68 71 79 38 35 63
[1585]  4 17 49 45 65 78 15 67 41 13 57  3 21 29  0 41
[1601] 29 48  4 71  5 31 77 36 30 41 35 31 74 31 42 73
[1617] 12 44 15 17 44  5 63  4 60  1 54 65 33 77 64 12
[1633]  3 16  4 52 69 25 57 17  9 51 48 70 55 37 14 15
[1649] 14 17 59 25 49 57  8 33 41 61 52 43 22  0 55 76
[1665] 71 26 49 40 53 41 25 61 22 28 26 27 65 67 41 15
[1681] 78 59 37 29 76 41 48 71 47 44 47 24 77 34 71  7
[1697] 28 55 43 12 26 10 66 50 38 53 58 65 40 34  0  8
[1713] 59 11 23 61 57 17 15 20 50  0 63  6 38  8 60 37
[1729] 28 35 59 35  8 10 24  5 66 58  3 13 61  2 21 30
[1745] 64 74 78 67  0 35 21  8 16 20 31  5  1 58 53  8
[1761] 29 51  4  2 16 58 47 61 34 73 11 52 40 28 49 21
[1777] 65 75 54  4 13 50 25 26 49 18 55 56 14 39 12 16
[1793] 34  9 57 20 68  9 40 28 56 65 38  4 76  0  0 56
[1809] 22 51  5 68  1 11 36 75 51 79 64  7  4 37 69 43
[1825]  4 39 44  9 54 75 35 64 24 34 59 60  7  3 79 48
[1841] 29 12  0 32 65 59 62 55 73 65  7 70 10 25 32 13
[1857] 36  3 47 44 62 23  1 13 51 42 25 79 19 71 24 11
[1873] 65 37 67 13 46 69 32 79  9 19 22 40 62 48 32 74
[1889] 22 23 53 40  0 51  2 73 69 18 50 57 45 55 26 52
[1905] 20  2 62 79 24 22 14 51 79 28  8 74  8  5 64  1
[1921] 60 56 73 70  7 58 44 62 22 40 52 18 68 24  2 64
[1937] 43 20 72 20 71 70 25 37 46 42 65 54  1  7 43 56
[1953] 14 43 38 64 29 34 74 55 37 39 42 38 61 23 18 60
[1969] 29  4 56 27 47 35 45 38  0  4 21 60 74 10 59 42
[1985] 17 51 49 39 58 74 39 77  0 65 79 16 36 79 25  3
[2001] 35  5 52 47 70 13 31 65 69 20 71 25 49 27  1 25
[2017] 57 29 41 56 16 63 56  9 24 41 17 43  6 26 48 23
[2033] 19 72 53 66 44 59 32 57  3 11 72 39 29  3 52 36
[2049] 19 39 63 11 41 41 66 26 73  8 10 24 40  8 60 76
[2065] 35 12 19 75 42 44 61 52  1 58 29 45 52 37 62 24
[2081] 41 39 49 74 33 31 32 26 55 12  6 69 51 54 10  0
[2097] 15 32 70 27 39 55 73 40 46 11  1 27 36 69 33 11
[2113] 74 69 45 34 70 51 62 52 37 22 51  3 56 79 38 45
[2129]  1 11 75 26 31  1 33 69 69 45 16 23 61 67 60 45
[2145] 49 52 42 76 16 68 56 31  5 23 59 41 77 40  4 36
[2161] 22 11 77 75 23 66 30 66 67 56 39  3 34 51 55  2
[2177] 36 21  2  0 73 55 52 70 47 40 68 30 68 48 28 26
[2193] 59 59 25 67 60 38 15 35 63  2 38 12 57 35 61 67
[2209] 62 30 55 66  8 18 41 63  5 26 52 25 67 46  2 17
[2225] 34 69 43 78 39 78  3 21 72 29 65 62  2 27 61 15
[2241] 23 22 25 29 30 78 60 49 63  6 31 18 39 73 12 17
[2257] 77 21 21 72 11 50 11 45 15 78 22 57 53  2  2 37
[2273] 32 41 74 70 44 34 12 78 34 13 46  3  5 55 43 62
[2289] 52 13 32 40 65 57 76 78  1 35 17  1 27 43 52 15
[2305] 54 67 46 42 20 30 14 12 58 42 14 76 64 35 25 51
[2321] 52 27 36 60 79 36 14 10 45 11 43 44 69 17  8 40
[2337] 73  4 37  1 17 75  0 41 58 74 42  4 53 39  6 18
[2353] 78 46 11 46  1 24 11 71 23  2 29 57 78 74  3 70
[2369] 22 58 18 36 54 61 49 76 42 67 53 23 28 16 25 78
[2385] 71 39 44 43 57 13 35  9 52 51 54 21 23 33 65 39
[2401] 21 31 15 58 26 79 58 57 48 57  9 11 19 38  9 58
[2417] 46 18 53  2 61 76 34 27 56  5 48 63  7 20 38  0
[2433] 28  5  3 59 43 46 68 60 64 35  1 77 33 76 37 42
[2449] 23 63 30 44  7 10  4 59 74 15 40 75 69 78 43 31
[2465] 30 72 37 44 69  9 37 77 61 54 43  1 29  5 43 30
[2481]  0 65 79  1  9 16 66 56 33 38 73 64 46 53 48 20
[2497] 22 79 23  5 44 12  7 49 46 78 28 10 78  3 13 79
[2513] 55 62 12 14 29 64 48 43 61 50 42 36 72 58 50 74
[2529] 67 29 78 12  2 16  9 37 74 69 56 15 54 12 51 14
[2545] 39 63  7 26 46 59 49 40 25 75 14 70 36 48 77 49
[2561] 33 53 79 15 14 35 24  7 71 17 60 40 19 13 19 65
[2577] 22  5 38 73 79  3  7 10 57 39 36 41  3 58 25 22
[2593] 51 39 19 17 73 75  2 11 54 57 76 61 56  5 71 72
[2609] 34 36 57 76 28 57 49 64 68 78 23 17 27 38 75 64
[2625] 60  2 13 28 29 73 19 18 13  6 42 25 43 40  6  2
[2641] 23 41 26 21 45 57 72 17 67 62 40 41 42 46  2 43
[2657] 66 63 46 54 73 58 43 20  6 47 26 70 28 19 29 10
[2673] 24 26  6 51 63 11 34 24 79 31 36  5 60 63 20 74
[2689] 18 20 35 42 41 63 32 61  3 25  7 40 14  4 16 11
[2705] 10 12 51 57 57 68 19 25 71  8 60 34 24 63 79 17
[2721] 33 18 16 53  3 37  5 70 10 15 31 46 45  9 38 78
[2737] 38 79 38 28 10 26 69 55 69 25 39 41 76 69 28 54
[2753] 31 71 59 79  9 43 51 63 57 64 43  8 50 23  4 69
[2769] 14 50 41 13  5 43 66 41 18 78 69 69  0 43 50 64
[2785] 17 56 44 12 40 60 68 67 49 73 53 39 64  3 36  3
[2801] 14  1 67 22 27 50 69 67 21 34 12 27 66 47 68 20
[2817] 67 11 25 43 33 28 30 33 40 56  6 79 22 12  4 27
[2833] 62 41  2 28 75 19  9  5  5  5  0 78 16 35 62 12
[2849] 41 77 76 34 14 71 24  5 66 49 41 16 75 55 20 50
[2865] 33 73 28 43 53 75 15 50 27 39 45 63 22 50 70  5
[2881] 26 71 56 67 21 16 31 18  9 18 61  3 61 53 36 55
[2897] 18 33 20  7 65 15  0 55 76 58 26 12 49 76 48 41
[2913] 47 69 44 67 56 74 64 56 61 57 39 39 58 60 57 33
[2929] 79 51  5  2  8 30 22 38 40 76 49 37 10 25 78 73
[2945]  9 36 38  7 72 27 58 44 29 73 27 58 53 60 73 61
[2961] 30 50 62 34 10 17  9 55 64 64 76 40 48 41 76 25
[2977] 39  6 56 56  6 33 39 43 65 74 70 73  4 17 14 76
[2993] 50  7  4  3 65 10  6 12  0 14 45 63 55 39 70 59
[3009] 12 22 69 44 30 54 32 75 31 33 33 25 27 22  4 21
[3025]  1 19  0 40  4 17 38 75 28 13 29 59 10  5  2 16
[3041] 23 22  8 57 14 78  9 39 68 40 51 14 32 16 48 72
[3057] 50 77 11 42 15 42  5 51 16 26 56 48 48 62 42 62
[3073]  3 62 42 79 16 56 64  2 26 29 26 39 61 71 17 36
[3089]  9 32  1 51 75 78 21 48 14 76 58 55 75 67 45 41
[3105]  1 14 76 39 60 45 30 73 50 33 78 73 64 46 33  7
[3121] 47 37 30 75 41 49 42 28 18  3 24 31  8  3 17 55
[3137] 20  1 77  3  6 26 75 16 22 32 28 51 27 17 18 67
[3153] 28 77 67 66 78 41 19 71  6 21 72 52 37  9 47 70
[3169]  2 28 60  7 29 40 78 55 41  2 24 11 11 54 43 62
[3185] 22 48  8 52  0  0 66 61 59 59 34 41 32  4  4 75
[3201] 78 30 54 61 27  2 25 62 49 60  5 72  4 77 67 58
[3217] 28  2 62 28 11 74  4  3 47 34  3 48 64 39 71 31
[3233] 79 24 75  4 50  5 60 21 35 27 10 46 24 59 19 57
[3249] 48 75 77  3 73 41 79 30 37 55 73 73 45 27 10 65
[3265] 31 53 63 16 36 15 35 71  0  5  7 63 22  8 46 45
[3281] 33 26 63 25 26 78 49 25 19 75 39 66 40 33 20 28
[3297] 21 38 76 24 75 65 11 34 63 24 58 25 70 47 57 33
[3313] 25 75 25 23 26 62 20 68 51 62 76 28  2  8 72 38
[3329] 62 32 29 10 57  9 19 34 38 39 26 21 13 28 46 45
[3345] 78 79 36 13 77 17 13 47 45  0 37 12 34 42 27 46
[3361] 74 64 44 16 26 60 23 32  8 68  5 50 29 27 52 53
[3377]  5 33 72 76 48 49  0 43 79 23 65 64 72 40 13 18
[3393] 50 40 30 59 41 68 72 38  4 17 33 43  6 72  5  1
[3409] 58 32 59 30 59 47  9 27 50 58 35 22 31 19 37  9
[3425] 45 29 39 51 33 23 12 36  1 60 39 10 17 59  0 25
[3441] 54 10 19 75  2 29 34 52 50 29 51 58 56 48 44 15
[3457] 73 20 45 43 41  5 53 34 19 47 37 47 76 15 34 71
[3473] 64 19  4 50 30 21 55 55 41 74 15 54 33 68  4 74
[3489] 51  0 62 14  3 63 50 42 49 66 22 57 12  8  5 20
[3505] 25 78 32 25 28 58 18  7 73  6 62 58 79 23 68 54
[3521] 35 48 27 23 50  1  0 52 32 55 17 54 43 41 35 64
[3537] 47 40 65 30 18  0 29 54 65 20 27  7 68 65  0 58
[3553] 24 44 75 23 71  7 21 71 10 21 56 66 66 55 66 51
[3569] 52 47  8 24 50 56 68 12  2  1 13 32 30 61 64  5
[3585] 41  3 35 32 23 20 70 51  4  9 62 15 74  8 55 65
[3601] 39 12 54 42  1 16 42 22 66 22 48  3 38  4 23 22
[3617] 76 15 33 27 30 75 12 71 67 67 27 27  9 28 35 70
[3633] 37 76 27 18 54 63 74 51  1 66 77 50 66 61 75 52
[3649] 70 46 63 20 46 72 76 29 75 72 68 21 56 19 28 21
[3665] 36 10 13  9 50 47 21 47 78 50 24 39  7 11 41 60
[3681]  7 49 21 65 16 11 17 13 64 72 74 12 45 74 47 27
[3697] 24 66  3 40 20 26  1  5 45 15 34 20 17 21 40 52
[3713] 71 20 74 13  4  7 14 78 74 42 24 37  8 48 33 24
[3729] 41 51 61 56 76  0 49 10 36 21 65 26 16 76 69 59
[3745] 21  1 45 19  6 69 32 70 40 78 61 37 46 60 15 42
[3761] 21 10 42 32 56 17 74  2 28  1 21  1 38 64 27 78
[3777] 52 31 59 15 67 74 31 62 65 34 24  3 24 42 37 34
[3793] 57 41  2 56 70 34 16 71 68 53 48 32  0 24 33 13
[3809] 35 11 56 24 37 40 31 66 43 47 45 36 19 37 31 67
[3825] 46  5 46 56 46 58 33 11 65 18 69  4 76  7 71 36
[3841] 54 71 14 17  7 77 36  2 23 55 79 22 68 65 13 61
[3857] 20 61 68 61 33 42  8 26 15 53 11 36  5 29 17 49
[3873] 42 27 17 45 39 44 43 36 58 72 53  2 18 10 11 54
[3889] 50 38 34 19 58 53 78 27 68 73 35  1 65 12 24 44
[3905] 24 13 17 30  5 15 50 64 37 50 64 33 10 11 30 45
[3921] 35 38  4 10 35 78 63 25 21 76 26 27 56  3 53 65
[3937] 68 73 54 55 35 36 75 11 46 66 19  4  6 61 64 25
[3953] 32 30 43 70  6 37  5 59 37  5 73 77 23 73 66 25
[3969] 26 19 18 58  0  9 53 37 10 57 61 44 31 11 47 36
[3985] 41  6 13 11 57 28  6 49 10 72 59  8 64 74 19 14
[4001] 65 19 52 45 61 53 43 47 74 63 42 44 55  8 31 38
[4017] 71 45 29 43 34 56 31 59  9 10 57 53 71 62 79 63
[4033] 10 58 61 26 53 45 17 17 22 44 14 67 74 55 18 18
[4049]  3 42 60 51 23 27 37 77 15 37 69  8 34 14 48 63
[4065] 11 48 22 11 46 64 66 52 52 11  0 63 17 44 22 68
[4081] 55 25 68 15 17 55 16 60 20 23  3 24 26 10 20 16
[4097] 60 72 20 22 31 63 53 37 37 62 16  8 16 69 30 63
[4113]  1 46 71 53 10 71 18 28 28 31 29 11 15 62 70 76
[4129]  5 34  5 36 55 73 48 30 75 51  5 49  3 10  5 21
[4145] 51  6 74  4 75 45 53  6 40 48 59 33 65 50 22 15
[4161] 11 44 35 70 70 15 66 24 46 13 70 21 37 31 59 67
[4177] 33  0 75 52 61 23 51 43  9 55 66 17 63 74 19 22
[4193] 27 11  9 52 18 13 53 26 26 51 56 39  9 41 50 30
[4209] 67 48 31 16 49 11 11  2 28 62 53 30 49 79 32 31
[4225] 54 19 51 41 14  9  4 29 71 11 12 34 69  4 12 31
[4241] 18 52 73 13 23 47  2 76 34 32 23 68 22 41 36 52
[4257] 62 37  9 49 34 33  8 70 18 67 25 43 26  3 22 39
[4273] 61 47 61 38 69  9 17 61 36 74 39 21 70 48 69 58
[4289] 54 30 48 67  2  5 34 43 30 29  9 58 32 10 11  9
[4305] 52 47 17 45 27 32 53 16 40 35 54 72  1 25 51  6
[4321] 32 66 13 71 36 36 47 31 66 20 40 28 27 68 18 75
[4337] 15 28 63 46 37 69 54 29 13 75 22 79 29 69 36 50
[4353] 77 32 25 32 57 57 37 62  6 76 47 72  5 66 45 70
[4369] 48 24 78 56 32 73 70 49 20 24 13 35 19 38 39 56
[4385] 70 33 67 38 73 63 69  4 62  7  4  9 32 76 30 11
[4401]  7 75 23 79  7 56 16 64 47 59 40 60 65  0 17 63
[4417] 13 69 61 44 42 50 15 73 22 48 50 40 13 70 74 31
[4433] 48 44 39 46 79 78 27 26 28  8 28  5 39 10 79 49
[4449] 66 34 74  6 24 30 41 24 70 61 27 34 65 45 58 57
[4465] 35 47 57 35 29 55  1 11 24 71 33 79 45 29 62 55
[4481] 78 38 27 20 63 19 32 33 22 52 17 74  6 46 44 55
[4497] 72 58 23 40 38 60  0 65 14 18 68 21  0  4 19 69
[4513] 57 67 46  6 31  2 10 29  6 70 24 67 35  0 56 66
[4529]  8 41 24 63 74 74 51  7 51 31 66 58 28 75 23 60
[4545] 75 49  7  8 68  8 70 37 72 30 53 15 48 30 79 53
[4561] 20  8  1 26 60 44 30  0 50 40 16 35 39 57 77 41
[4577] 58 37 66 34 50 27 76  8 65 47  7  9 14  6 48 13
[4593]  8  6 29  6 72 26 60  1 38 16 74  8 50  7 33 17
[4609] 72 15 32 66 45 72 50 63 12 17 47 46 14 35 77  4
[4625] 76  4 14  8 68 23 65 25 45  3 18 13 26  0 69 22
[4641] 72 31 42 35 40 24 60 38 77  3 59 60  4 54 71 64
[4657] 76 62 21 38 78 26  6 59 77 10 25 74 31 67 77 42
[4673] 50 10 17 54 79 20 30 63 46 38  5 28 53 32 30  1
[4689] 54 76 45 37 36 32 13 13 58 74 46 60 57  9 40 60
[4705] 50 26  0 58 14 31  8 69  5 34 74 78 35 44 71 19
[4721] 74 27 17 71 49 34 73 68 49 16 69 31 51 57 45 69
[4737]  2 52 76 71  5 52 13 77 69 22 42 79 42 27 36 16
[4753] 12 70 25 33 66 17 75 67 67 66 32  3 79 49 70 19
[4769] 62  5  2 60 10 56 78  7 25 56 30 66  2  1 45 32
[4785] 36 26 63 58 39 28 15 73 10 64 78 64 17 60 16 67

## Representing the sample with a single value

\begin{aligned} \text{mean}(\mathbf x)&=\frac{1}{N}\sum_i x_i\\ &=\bar{\mathbf x}\end{aligned}

mean(x) is the average of x

$$\bar{\mathbf x}$$ is the value that results in the smallest squared error

mean(pop_LD)
[1] 4.5
mean(pop_MD)
[1] 14.5
mean(pop_HD)
[1] 39.5

## Variance: Quality of representation

$\text{var}(\mathbf x)=\frac{1}{N}\sum_i (x_i-\bar{\mathbf x})^2$

Variance is the mean squared error of the average

We explained this on the last semester

var(pop_LD)
[1] 8.25
var(pop_MD)
[1] 74.9
var(pop_HD)
[1] 533

## standard deviation

\begin{aligned} \text{sd}(\mathbf x)&=\sqrt{\text{var}(\mathbf x)}\\ &=\sqrt{\frac{1}{N}\sum_i (x_i-\bar{\mathbf x})^2}\end{aligned}

sd(pop_LD)
[1] 2.87
sd(pop_MD)
[1] 8.66
sd(pop_HD)
[1] 23.1

## Standard Deviation is the “width” of population

We care about sd(x) because it tells us how close is the mean to most of the population

Russian mathematician Chebyshev It can be proved that always $\Pr(\vert x_i-\bar{\mathbf x}\vert\geq k\cdot\text{sd}(\mathbf x))\leq 1/k^2$

In other words, the probability that “the distance between the mean $$\bar{\mathbf x}$$ and any element $$x_i$$ is bigger than $$k\cdot\text{sd}(\mathbf x)$$” is less than $$(1/k^2)$$

## This is Chebyshev’s inequality

It is always valid, for any probability distribution

(Later we will see better rules valid only sometimes)

It can also be written as $\Pr(\vert x_i-\bar{\mathbf x}\vert\leq k\cdot\text{sd}(\mathbf x))\geq 1-1/k^2$

The probability that “the distance between the mean $$\bar{\mathbf x}$$ and any element $$x_i$$ is less than $$k\cdot\text{sd}(\mathbf x)$$” is greater than $$1-1/k^2$$

## Some examples of Chebyshev’s inequality

Another way to understand the meaning of this theorem is $\Pr(\bar{\mathbf x} -k\cdot\text{sd}(\mathbf x)\leq x_i \leq \bar{\mathbf x} +k\cdot\text{sd}(\mathbf x))\geq 1-1/k^2$ Replacing $$k$$ for some values, we get \begin{aligned} \Pr(\bar{\mathbf x} -1\cdot\text{sd}(\mathbf x)\leq x_i \leq \bar{\mathbf x} +2\cdot\text{sd}(\mathbf x))&\geq 1-1/1^2=0\\ \Pr(\bar{\mathbf x} -2\cdot\text{sd}(\mathbf x)\leq x_i \leq \bar{\mathbf x} +2\cdot\text{sd}(\mathbf x))&\geq 1-1/2^2=0.75\\ \Pr(\bar{\mathbf x} -3\cdot\text{sd}(\mathbf x)\leq x_i \leq \bar{\mathbf x} +3\cdot\text{sd}(\mathbf x))&\geq 1-1/3^2=0.889 \end{aligned}

## Explanation from stats.libretexts.org

For any numerical data set

• at least 3/4 of the data lie within two standard deviations of the mean, that is, in the interval with endpoints $$\bar{\mathbf x}±2\cdot\text{sd}(\mathbf x)$$
• at least 8/9 of the data lie within three standard deviations of the mean, that is, in the interval with endpoints $$\bar{\mathbf x}±3\cdot\text{sd}(\mathbf x)$$
• at least $$1-1/k^2$$ of the data lie within $$k$$ standard deviations of the mean, that is, in the interval with endpoints $$\bar{\mathbf x}±k\cdot\text{sd}(\mathbf x),$$ where $$k$$ is any positive whole number greater than 1

## Example: the case of pop_HD

These values should be less than 1, 0.25 and 0.11

mean(abs(pop_HD-mean(pop_HD))> 1*sd(pop_HD))
[1] 0.425
mean(abs(pop_HD-mean(pop_HD))> 2*sd(pop_HD))
[1] 0
mean(abs(pop_HD-mean(pop_HD))> 3*sd(pop_HD))
[1] 0

## Simulating cell counting

one_sample <- function(m, population) {
return(sample(population, size=m))
}
one_sample(30, pop_HD)
 [1] 32 40 62 43 71 47 44 17 41 27 65 11 69 23 46 19  7
[18] 30 29 34 17 58 70 75 53 15 68 57 29 35

## Bad news! The sample average changes every time

Moreover, it is often different from the population average

## Average of smaller samples are more extreme

This explains why rural areas have the highest and lowest cancer rates

It is because the groups are smaller, so averages are taken from smaller groups

## Good news

When the sample size is big,

the sample average is closer to

the population average

## Log-log plot (for high density population)

plot(log(sd_sample_mean)~log(size))
model_HD <- lm(log(sd_sample_mean)~log(size))
lines(predict(model_HD)~log(size))

## Linear model

coef(model_HD)
(Intercept)   log(size)
3.242      -0.528 

$\log(\text{sd_sample_mean}) = 3.242 + -0.528\cdot\log(\text{size})$ $\text{sd_sample_mean} = \exp(3.242) \cdot\text{size}^{-0.528} =25.587\cdot\text{size}^{-0.528}$

## Models for all populations

$\text{sd_sample_mean} = A\cdot \text{size}^B$

A B std dev population
pop_LD 3.282 -0.5306 2.873
pop_MD 9.092 -0.5126 8.656
pop_HD 25.59 -0.528 23.09

Coefficient $$A$$ is the standard deviation of the population
Coefficient $$B$$ is -0.5

## What does this mean

If we know the population standard deviation, we can predict the sample standard deviation

$\text{sd(sample mean)}=\frac{\text{sd(population)}}{\sqrt{\text{sample size}}}$

## Law of Large Numbers

Using Chebyshev formula, we know that, with high probability $\vert \text{mean(sample)} -\text{mean(population)}\vert < k\cdot\frac{\text{sd(population)}}{\sqrt{\text{sample size}}}$

Therefore the population average is inside the interval $\text{mean(sample)} \pm k\cdot\frac{\text{sd(population)}}{\sqrt{\text{sample size}}}$ (probably)

## How to know population standard deviation?

Remember that we do not know neither the population mean nor the population variance

So we do not know the population standard deviation 😕

In most cases we can use the sample standard deviation

## Francis Galton, his cousin and his machine

English explorer, Inventor, Anthropologist
(1822–1911)

Cousin of Charles Darwin

He studied medicine and mathematics at Cambridge University.

He invented the phrase “nature versus nurture”

In his will, he donated funds for a professorship in genetics to University College London.

## Simulating the Galton Machine

We will simulate each ball one by one

one_ball <- function(M) {
return(sum(sample(c(-1,1), size=M, replace=TRUE)))
}

Here M is the number of “left-right” choices made by the ball

## Simulating the Galton Machine

Galton <- replicate(1000, one_ball(5))
barplot(table(Galton))

## Bigger M, wider variance

Galton <- replicate(10000, one_ball(50))
barplot(table(Galton))

## Mean and Variance

It is easy to see that the population mean is 0 for any M

If we think, we can show that the variance is M

Standard deviation will be sqrt(M)

## correcting variance M=5

Galton <- replicate(10000, one_ball(5))/sqrt(5)
barplot(table(Galton))

## correcting variance M=50

Galton <- replicate(10000, one_ball(50))/sqrt(50)
barplot(table(Galton))

## correcting variance M=500

Galton <- replicate(100000, one_ball(500))/sqrt(500)
barplot(table(Galton))

## correcting variance M=5000

Galton <- replicate(100000, one_ball(5000))/sqrt(5000)
barplot(table(Galton))

## The Normal distribution

This “bell shaped” curve is found in many experiments, specially when they involve the sum of many small contributions

• Measurement errors
• Height of a population
• Scores on University Admission exams

It is called Gaussian distribution, or also Normal distribution

## Simulating the Normal distribution

Instead of simulating the Galton machine several times, we can simulate the Normal distribution using the R function

rnorm(n, mean = 0, sd = 1)

The parameter n is mandatory. It is the sample size

You can also change the mean and the standard deviation of the simulation

## Exercise

In Class 8 (and Homework 8) we predicted the final outcome of the water formation for each value of r1_rate, and fixing the other values

rates_of_r1 <- 10^seq(from=-3, to=-1, length=50)
r2_rate=0.01, H_ini=2, O_ini=1, W_ini=0

Now we are not so sure about how much hydrogen we had at the beginning. Instead of H_ini=2 fixed, we will simulate taking values from

H_values <- rnorm(n=6, mean=2, sd=0.05)