Source files: [https://github.com/djlofland/DATA606_F2019/tree/master/Lab2]

Some define Statistics as the field that focuses on turning information into knowledge. The first step in that process is to summarize and describe the raw information - the data. In this lab, you will gain insight into public health by generating simple graphical and numerical summaries of a data set collected by the Centers for Disease Control and Prevention (CDC). As this is a large data set, along the way you’ll also learn the indispensable skills of data processing and subsetting.

Getting started

The Behavioral Risk Factor Surveillance System (BRFSS) is an annual telephone survey of 350,000 people in the United States. As its name implies, the BRFSS is designed to identify risk factors in the adult population and report emerging health trends. For example, respondents are asked about their diet and weekly physical activity, their HIV/AIDS status, possible tobacco use, and even their level of healthcare coverage. The BRFSS Web site (http://www.cdc.gov/brfss) contains a complete description of the survey, including the research questions that motivate the study and many interesting results derived from the data.

We will focus on a random sample of 20,000 people from the BRFSS survey conducted in 2000. While there are over 200 variables in this data set, we will work with a small subset.

We begin by loading the data set of 20,000 observations into the R workspace. After launching RStudio, enter the following command.

## ── Attaching packages ──────────────────────────────────────────────────────────────────────────────── tidyverse 1.2.1 ──
## ✔ ggplot2 3.2.1     ✔ purrr   0.3.2
## ✔ tibble  2.1.3     ✔ dplyr   0.8.3
## ✔ tidyr   0.8.3     ✔ stringr 1.4.0
## ✔ readr   1.3.1     ✔ forcats 0.4.0
## ── Conflicts ─────────────────────────────────────────────────────────────────────────────────── tidyverse_conflicts() ──
## ✖ dplyr::filter() masks stats::filter()
## ✖ dplyr::lag()    masks stats::lag()

The data set cdc that shows up in your workspace is a data matrix, with each row representing a case and each column representing a variable. R calls this data format a data frame, which is a term that will be used throughout the labs.

To view the names of the variables, type the command

## [1] "genhlth"  "exerany"  "hlthplan" "smoke100" "height"   "weight"  
## [7] "wtdesire" "age"      "gender"

This returns the names genhlth, exerany, hlthplan, smoke100, height, weight, wtdesire, age, and gender. Each one of these variables corresponds to a question that was asked in the survey. For example, for genhlth, respondents were asked to evaluate their general health, responding either excellent, very good, good, fair or poor. The exerany variable indicates whether the respondent exercised in the past month (1) or did not (0). Likewise, hlthplan indicates whether the respondent had some form of health coverage (1) or did not (0). The smoke100 variable indicates whether the respondent had smoked at least 100 cigarettes in her lifetime. The other variables record the respondent’s height in inches, weight in pounds as well as their desired weight, wtdesire, age in years, and gender.

  1. How many cases are there in this data set? How many variables? For each variable, identify its data type (e.g. categorical, discrete).
## 'data.frame':    20000 obs. of  9 variables:
##  $ genhlth : Factor w/ 5 levels "excellent","very good",..: 3 3 3 3 2 2 2 2 3 3 ...
##  $ exerany : num  0 0 1 1 0 1 1 0 0 1 ...
##  $ hlthplan: num  1 1 1 1 1 1 1 1 1 1 ...
##  $ smoke100: num  0 1 1 0 0 0 0 0 1 0 ...
##  $ height  : num  70 64 60 66 61 64 71 67 65 70 ...
##  $ weight  : int  175 125 105 132 150 114 194 170 150 180 ...
##  $ wtdesire: int  175 115 105 124 130 114 185 160 130 170 ...
##  $ age     : int  77 33 49 42 55 55 31 45 27 44 ...
##  $ gender  : Factor w/ 2 levels "m","f": 1 2 2 2 2 2 1 1 2 1 ...
## [1] good      very good excellent fair      poor     
## Levels: excellent very good good fair poor
## [1] m f
## Levels: m f

There are 20000 observations and 9 variables in the cdc dataset.

Variable Datatype
genhlth Factor (Categorial, ordinal)
exerany Numeric (Float, logical)
hlthplan Numeric (Float, logical)
smoke100 Numeric (Float, logical)
height Numeric (Float, Discrete)
weight Integer (Discrete)
wtdesire Integer (Discrete)
age Integer (Discrete)
gender Factor (Categorial, nominal)

We can have a look at the first few entries (rows) of our data with the command

##     genhlth exerany hlthplan smoke100 height weight wtdesire age gender
## 1      good       0        1        0     70    175      175  77      m
## 2      good       0        1        1     64    125      115  33      f
## 3      good       1        1        1     60    105      105  49      f
## 4      good       1        1        0     66    132      124  42      f
## 5 very good       0        1        0     61    150      130  55      f
## 6 very good       1        1        0     64    114      114  55      f

and similarly we can look at the last few by typing

##         genhlth exerany hlthplan smoke100 height weight wtdesire age
## 19995      good       0        1        1     69    224      224  73
## 19996      good       1        1        0     66    215      140  23
## 19997 excellent       0        1        0     73    200      185  35
## 19998      poor       0        1        0     65    216      150  57
## 19999      good       1        1        0     67    165      165  81
## 20000      good       1        1        1     69    170      165  83
##       gender
## 19995      m
## 19996      f
## 19997      m
## 19998      f
## 19999      f
## 20000      m

You could also look at all of the data frame at once by typing its name into the console, but that might be unwise here. We know cdc has 20,000 rows, so viewing the entire data set would mean flooding your screen. It’s better to take small peeks at the data with head, tail or the subsetting techniques that you’ll learn in a moment.

Summaries and tables

The BRFSS questionnaire is a massive trove of information. A good first step in any analysis is to distill all of that information into a few summary statistics and graphics. As a simple example, the function summary returns a numerical summary: minimum, first quartile, median, mean, second quartile, and maximum. For weight this is

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##    68.0   140.0   165.0   169.7   190.0   500.0

R also functions like a very fancy calculator. If you wanted to compute the interquartile range for the respondents’ weight, you would look at the output from the summary command above and then enter

## [1] 50
## [1] 50

R also has built-in functions to compute summary statistics one by one. For instance, to calculate the mean, median, and variance of weight, type

## [1] 169.683
## [1] 1606.484
## [1] 165
## [1] 40.08097

While it makes sense to describe a quantitative variable like weight in terms of these statistics, what about categorical data? We would instead consider the sample frequency or relative frequency distribution. The function table does this for you by counting the number of times each kind of response was given. For example, to see the number of people who have smoked 100 cigarettes in their lifetime, type

## 
##     0     1 
## 10559  9441

or instead look at the relative frequency distribution by typing

## 
##       0       1 
## 0.52795 0.47205

Notice how R automatically divides all entries in the table by 20,000 in the command above. This is similar to something we observed in the Introduction to R; when we multiplied or divided a vector with a number, R applied that action across entries in the vectors. As we see above, this also works for tables. Next, we make a bar plot of the entries in the table by putting the table inside the barplot command.

Notice what we’ve done here! We’ve computed the table of cdc$smoke100 and then immediately applied the graphical function, barplot. This is an important idea: R commands can be nested. You could also break this into two steps by typing the following:

Here, we’ve made a new object, a table, called smoke (the contents of which we can see by typing smoke into the console) and then used it in as the input for barplot. The special symbol <- performs an assignment, taking the output of one line of code and saving it into an object in your workspace. This is another important idea that we’ll return to later.

  1. Create a numerical summary for height and age, and compute the interquartile range for each. Compute the relative frequency distribution for gender and exerany. How many males are in the sample? What proportion of the sample reports being in excellent health?
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   48.00   64.00   67.00   67.18   70.00   93.00
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##   18.00   31.00   43.00   45.07   57.00   99.00
## [1] 6
## [1] 26
## 
##     m     f 
##  9569 10431
## 
##     0     1 
##  5086 14914
## 
## excellent very good      good      fair      poor 
##   0.23285   0.34860   0.28375   0.10095   0.03385

There are 9569 males in the sample and 23.29% report to be in excellent health.

The table command can be used to tabulate any number of variables that you provide. For example, to examine which participants have smoked across each gender, we could use the following.

##    
##        0    1
##   m 4547 5022
##   f 6012 4419

Here, we see column labels of 0 and 1. Recall that 1 indicates a respondent has smoked at least 100 cigarettes. The rows refer to gender. To create a mosaic plot of this table, we would enter the following command.

We could have accomplished this in two steps by saving the table in one line and applying mosaicplot in the next (see the table/barplot example above).

  1. What does the mosaic plot reveal about smoking habits and gender?

A higher portions of males report having smoked compared with females.

Interlude: How R thinks about data

We mentioned that R stores data in data frames, which you might think of as a type of spreadsheet. Each row is a different observation (a different respondent) and each column is a different variable (the first is genhlth, the second exerany and so on). We can see the size of the data frame next to the object name in the workspace or we can type

## [1] 20000     9

which will return the number of rows and columns. Now, if we want to access a subset of the full data frame, we can use row-and-column notation. For example, to see the sixth variable of the 567th respondent, use the format

## [1] 160

which means we want the element of our data set that is in the 567th row (meaning the 567th person or observation) and the 6th column (in this case, weight). We know that weight is the 6th variable because it is the 6th entry in the list of variable names

## [1] "genhlth"  "exerany"  "hlthplan" "smoke100" "height"   "weight"  
## [7] "wtdesire" "age"      "gender"

To see the weights for the first 10 respondents we can type

##  [1] 175 125 105 132 150 114 194 170 150 180

In this expression, we have asked just for rows in the range 1 through 10. R uses the : to create a range of values, so 1:10 expands to 1, 2, 3, 4, 5, 6, 7, 8, 9, 10. You can see this by entering

##  [1]  1  2  3  4  5  6  7  8  9 10

Finally, if we want all of the data for the first 10 respondents, type

##      genhlth exerany hlthplan smoke100 height weight wtdesire age gender
## 1       good       0        1        0     70    175      175  77      m
## 2       good       0        1        1     64    125      115  33      f
## 3       good       1        1        1     60    105      105  49      f
## 4       good       1        1        0     66    132      124  42      f
## 5  very good       0        1        0     61    150      130  55      f
## 6  very good       1        1        0     64    114      114  55      f
## 7  very good       1        1        0     71    194      185  31      m
## 8  very good       0        1        0     67    170      160  45      m
## 9       good       0        1        1     65    150      130  27      f
## 10      good       1        1        0     70    180      170  44      m

By leaving out an index or a range (we didn’t type anything between the comma and the square bracket), we get all the columns. When starting out in R, this is a bit counterintuitive. As a rule, we omit the column number to see all columns in a data frame. Similarly, if we leave out an index or range for the rows, we would access all the observations, not just the 567th, or rows 1 through 10. Try the following to see the weights for all 20,000 respondents fly by on your screen

##     [1] 175 125 105 132 150 114 194 170 150 180 186 168 185 170 170 185 156
##    [18] 185 200 125 200 160 160 165 105 190 190 160 115 185 166 180 182 185
##    [35] 220 117 160 190 125 160 124 143 118 210 200 145 175 130 112 141 179
##    [52] 135 140 170 150 172 125 168 120 180 186 144 250 160 164 155 140 125
##    [69] 140 190 140 134 217 215 190 215 112 136 142 210 180 120 225 200 148
##    [86] 143 135 115 193 140 265 166 145 138 137 175 230 140 180 100 118 142
##   [103] 110 114 175 180 140 193 211 220 210 160 132 163 132 207 145 190 200
##   [120] 170 280 125 120 224 115 185 160 250 190 205 200 160 144 185 140 200
##   [137] 280 180 250 172 215 138 185 220 136 190 150 130 172 107 187 125 110
##   [154] 205 140 128 180 165 204  97 130 180 195 187 142 225 180 180 180 128
##   [171] 130 215 180 152 151 143 175 210 235 190 220  92 185 150 129 135 140
##   [188] 110 183 212 165 171 205 170 170 155 222 200 150 210 130 200 155 135
##   [205] 200 134 128 115 155 112 116 135 135 180 140 179 149 135 140 165 126
##   [222] 180 107 165 120 210 145 200 125 142 135 185 152 225 207 155 212 122
##   [239] 160 125 170 135 148 160 150 165 130 135 190 125 175 200 212 178 132
##   [256] 255 175 150 170 156 170 185 198 186 123 168 162 185 113 220 160 165
##   [273] 270 220 175 206 210 175 330 200 230 175 180 180 170 100 108 107 205
##   [290] 124 163 130 125 250 295 185 105 168 225 185 200 142  90 145 100 138
##   [307] 140 145 130 220 300 142 180 130 185 113 140 175 225 153 180 240 195
##   [324] 150 110 115 160 147 132 195 250 190 170 175 170 120 155 155 170 127
##   [341] 185 150 200 155 160 165 125 195 140 283 180 180 173 165 150 175 185
##   [358] 290 115 128 140 130 155 130 115 190 190 115 160 200 125 170 150 160
##   [375] 120 163 160 184 160 270 275 165 122 123 145 112 208 103 180 182 195
##   [392] 133 230 180 205 185 200 200 175 130 120 169 182 200 155 105 101 240
##   [409] 178 200 164 145 140 130 280 220 150 145 190 135 160 187 220 140 180
##   [426] 215 146 170 190 125 180 190 155 210 130 195 160 150 102 185 190 145
##   [443] 210 250 180 130 196 130 110 160 155 140 210 185 180 238 220 185 119
##   [460] 240 180 174 218 202 180 230 145 180 160 135 190 125 132 240 230 200
##   [477] 135 200 150 140 201 195 210 165 157 118 185 170 186 150 161 253 180
##   [494] 170 171 175 125 180 182 149 133 180 180 185 140 174 135 120 135 198
##   [511] 285 110 160 150 210 185 144 128 150 179 100 195 160 230 100 135 150
##   [528] 110 220 140 185 250 143 130 195 165 150 182 185 125 115 168 185 170
##   [545] 175 178 170 160 140 175 110 200 155 208 125 150 235 220 190 200 141
##   [562] 175 180 138 170 132 160 172 123 120 180 190 195 140 192 200 135 185
##   [579] 242 170 180 165 230 165 170 155 130 164 176 130 140 140 165 150 130
##   [596] 195 165 130 230 150 235 185 215 135 180 115 180 280 170 180 155 155
##   [613] 200 185 108 101 328 155 230 148 201 140 150 170 162 145 155 165 230
##   [630] 198 132 185 200 220 111 196 154 162 135 155 150 155 198 190 170 180
##   [647] 133 200 170 130 200 145 235 135 215 145 170 150 128 180 134 120 168
##   [664] 180 126 113 160 250 230 146 120 116 121 175 205 185 248 140 170 245
##   [681] 120 145 275 121 185 170 115 152 125 138 167 200 208 185 245 220 179
##   [698] 140 238 135 200 145 110 128 132 165 145 136 190 135 180 166 165 200
##   [715] 190 140 145 100 167 105 140 135 150 125 130 185 170 198 155 160 275
##   [732] 130 173 150 198 144 120 135 140 130 185 150 163 130 165 190 165 150
##   [749] 200 150 250 210 180 215 170 190 165 110 170 148 175 200 180 124 114
##   [766] 135 165 130 380 170 170 170 141 130 255 190 210 150 145 175 172 175
##   [783] 108 160 200 147 120 145 145 130 172 280 140 200 157 210 165 110 165
##   [800] 135 162 155 150 177 175 130 113 165 145 170 150 182 140 135 135 115
##   [817] 230 165 248 202 190 200 200 165 210 165 220 240 160 140 140 135 120
##   [834] 185 248 140 110 135 160 340 115 160 115 135 180 170 115 160 240 134
##   [851] 170 140 140 260 226 133 220 135 210 180 400 180 220 205 180 152 105
##   [868] 160 105 199 200 145 115 235 125 200 160 180 250 165 168 160 185 190
##   [885] 170 135 260 190 120 157 250 270 178 133 150 137 136 120 180 150  95
##   [902] 140 153 175 165 140 250 124 197 158 103 234 240 270 195 117 190 148
##   [919] 176 120 190 145 198 180 140 157 100 170 150 145 145 160 165 148 170
##   [936] 135 140 155 185 123 255 135 125 156 162 170 180 150 140 175 200 123
##   [953] 154 125 110 150 300 120 198 190 185 165 140 148 160 195 170 125 195
##   [970] 168 160 160 180 144 150 170 150 175 120 205 115 172 235 190 125 110
##   [987] 113 230 114 195 130 145 200 168 230 198 185 203 174 155 210 295 260
##  [1004] 165 115 180 180 126 112 135 225 160 260 122 122 136 230 188 168 159
##  [1021] 155 165 180 187 170 155 165 140 170 185 135 180 215 130 137 145 165
##  [1038] 220 180 138 185 119 130 220 150 135 170 128 200 117 170 208 198 150
##  [1055] 140 180 165 100 180 125 130 140 110 158 145 175 138 218 212 160 178
##  [1072] 145 200 167 125 185 157 200 155 140 148 113 160 110 240 210 159 210
##  [1089] 200 164 230 200 180 150 190 145 192 215 154 197 151 188 260 160 115
##  [1106] 250 140 115 220 139 150 150 210 170 180 163 165 118 205 199 140 190
##  [1123] 150 170 105 150 190 180 115 120 235 171 175 265 170 180 177 170 150
##  [1140] 165 220 210 100 150 140 300 150 135 150 145 139 180 184 300 180 200
##  [1157] 150 178 175 220 190 110 134 170 180 160 176 160 125 170 130 197 200
##  [1174] 185 180 180 105 100 178 180 230 200 135 160 155 340 189 140 220 145
##  [1191] 137 191 180 199 140 215 163 138 120 150 200 147 175 155 190 270 160
##  [1208] 145 160  92 126 155 138 175 178 189 125 180 170 195 175 155 200 100
##  [1225] 159 123 200  95 240 165 280 158 172 170 135 205 138 225 181 185 180
##  [1242] 173 178 178 185 200 160 153 180 135 290 195 265 220 160 150 195 215
##  [1259] 135 121  85 220 117 148 190 150 110 150 180 165 192 180 150 170 160
##  [1276] 175 230 235 400 240 190 150 210 210 125 126 128 168 185 230 270 142
##  [1293] 170 180 197 160 165 225 250 195 225 240 175 175 136 130 130 144 170
##  [1310] 170 160 200 175 130 180 190 130 170  80 200 195 135 120 144 165 250
##  [1327] 165 110 210 190 178 138 150 160 156 180 175 170 138 168 220 209 180
##  [1344] 200 135 280 215 310 160 170 170 170 140 120 180 125 162 140 204 155
##  [1361] 130 165 190 127 100 154 210 210 228 180 140 135 149 160 212 120 240
##  [1378] 170 180 117 218 160 150 170 170 180 300 205 135 170 148 160 155 160
##  [1395] 135 175 127 142 135 126 163 150 190 135 270 138 145 140 204 185  98
##  [1412] 150 168 160 170 150 103 125 210 150 182 160 225 185 185 130 185 200
##  [1429] 150 113 200 215 160 195 120 189 225 160 155 230 115 125 128 102 132
##  [1446] 145 199 150 202 160 180 110 175 185 150 115 110 140 180 150 184 183
##  [1463] 178 185 170 225 139 215 200 130 175 165 180 170 135 175 108 150 203
##  [1480] 200 112 185 153 210 225 230 170 130 200 150 145 350 115 148 165 160
##  [1497] 220 185 192 225 270 195 120 175 130 170 110 110 157 146 187 160 160
##  [1514] 190 150 210 180 200 140 225 112 176 140 160 129 143 120 270 130 139
##  [1531] 168 150 130 125 120 170 145 215 145 215 165 160 109 135 140 145 204
##  [1548] 115 140 158 132 210 220 117 160  98 140 132 135 205 150 190 132 180
##  [1565] 170 195 175 118 160 175 200 192 125 140 220 170 143 274 150 200 195
##  [1582] 195 170 144 136 240 195 200 130 134 185 165 280 138 124 230 150 200
##  [1599] 164 200 140 120 120 145 192 130 205 186 198 185 125 170 147 168 150
##  [1616] 364 135 158 170 140 168 150 135 140 165 172 250 196 212 176 190 150
##  [1633] 185 184 145 185 190 236 185 115 150 160 160 208 132 120 190 165 180
##  [1650] 160 175 210 133 160 160 210 172 260 171 160 125 142 200 170 142 190
##  [1667] 210 174 140  98 140 185 215 170 115 120 190 225 128 170 285 108 145
##  [1684] 175 135 290 190 170 165 150 160 170 175 100 185 170 132 185 112 139
##  [1701] 157 185 125 137 200 141 130 194 260 130 157 186 180 174 155 240 162
##  [1718] 128 168 160 267 170 145 183 205 150 235 115 130 140 128 108 195 150
##  [1735] 165 220 128 170 235 155 169 150 225 180 243 128 138 150 170 172 130
##  [1752] 173 208 135 160 139 200 195 185 150  98 165 212 220 200 200 172 190
##  [1769] 171 174 101 145 160 128 260 132 180 150 253 130 220 210 160 140 180
##  [1786] 200 187 190 160 150 164 150 190 160 150 172 180 150 175 185 175 235
##  [1803] 200 135 195 140 240 235 170 178 141 136 275 200 200 153 150 240 362
##  [1820] 180 270 137 135 250 220 128 180 170 190 165 210 145 141 180 185 173
##  [1837] 175 194 230 157 145 134 110 130 149 250 174 210 145 165 130 192 145
##  [1854] 155 145 199 120 185 140 144 125 140 170 203 200 112 160 190 120 210
##  [1871] 175 167 145 161 180 130 115 150 210 155 215 155 150 190 150 149 160
##  [1888] 140 135 275 150 140 215 135 176 140 170 142 185 156 140 265 170 220
##  [1905] 225 135 185 135 130 145 185 185 145 135 137 220 140 192 150 130 170
##  [1922] 105 130 172 140 220 127 168 125 200 147 190 190 153 142 172 220 190
##  [1939] 120 150 158 170 155 124 230 140 185 135 192 170 176 150 188 150 200
##  [1956] 180 154 150 169 132 125 216 217 126 180 114 135 170 230 136 180 165
##  [1973] 210 199 190 145 145 182 140 195 185 155 160 185 217 135 150 170 203
##  [1990] 165 155 235 150 146 500 135 145 130 132 165 240 220 130 130 165 195
##  [2007] 225 190 155 140  90 124 170 213 105 225 160 125 240 192 200 177 146
##  [2024] 210 185 130 185 178 160 115 175 105 250 183 190 236 170 165 180 210
##  [2041] 185 165 160 115 107 165 196 142 126 245 210 165 180 130 135 155 140
##  [2058] 127 220 150 205 190 170 190 130 178 140 225 230 260 120 121 142 140
##  [2075] 135 165 200 173 132 178 200 228 185 230 155 110 165 150 138 185 149
##  [2092] 120 120 195 160 150 270 155 165 215 160 155 106 170  92 145 220 230
##  [2109] 190 160 150 170 140 210 155 184 200 128 125 185 188 190 210 175 131
##  [2126] 170 170 193 120 170 160 175 260 112 135 174 160 157 180 199 260 220
##  [2143] 120 120 190 165 160 135 125 212 130 150 150 220 130 125 125 147 200
##  [2160] 280 135 275 227 250 350 185 250 155 200 160 135 190 200 210 151 190
##  [2177] 182 205 140 160 127 150 190 227 160 205 140 110 173 220 160 180 115
##  [2194] 220 185 120 163 200 135 210 160 136 250 150 100 125 103 132 140 140
##  [2211] 175 135 175 180 130 140 145 173 135 104 120 120 185 190 161 215 228
##  [2228] 125 148 160 160 105 120 240 173 125 170 350 155 130 130 150 175 145
##  [2245] 128 150 134 124 225 190 285 145 140 140 160 150 250 150 160 150 175
##  [2262] 140 190 130 165 250 172 122 195 120 120 185 200 160 190 155 126 110
##  [2279] 165 206 180 160 140 230 220 130 195 200 177 120 200 162 140 155 198
##  [2296] 160 190 170 135 200 190 223 140 220 175 100 180 210 178 200 129 160
##  [2313] 195 180 200 125 113 155 140 210 110 175 115 200 330 121 215 150 150
##  [2330] 200 200 245 140 165 142 167 117 172 150 190 175 153 130 135 260 308
##  [2347] 145 140 125 160 158 142 130 167 195 150 150 225 140 190 165 200 150
##  [2364] 263 152 150 118 160 200 204 140 165 200 142 222 175 150 126 138 190
##  [2381] 160 120 127 185 110 149 155 152 170 198 205 180 180 125 200 230 128
##  [2398] 155 120 150 132 215 180 175 175 190 111 200 155 130 160 175 145 220
##  [2415] 175 230 135 260 178 162 140 165 190 127 208 105 200 215 235 225 200
##  [2432] 185 156 225 128 235 120 135 162 140 137 185 106 190 260 210 300 172
##  [2449] 230 160 250 145 195 130 205 165 130 110 300 128 265 172 155 132 206
##  [2466] 160 123 195 190 150 147 105 174 150 161 225 180 230 168 150 205 180
##  [2483] 160 175 210 160 174 140 150 188 155 225 135 128 210 145 195 160 220
##  [2500] 132 208 125 225 140 180 160 170 160 245 218 190 100 125 120 240 148
##  [2517] 150 250 190  90 240 220 176 154 185 245 180 118 100 155 145 210 110
##  [2534] 210 135 120 184 195 170 210 165 125 115 155 262 160 194 155 190 120
##  [2551] 230 147 175 150 185 179 115 116 125 157 127 153 160 145 180 130 140
##  [2568] 125 175 215 230 190 165 206 140 132 185 170 148 215 138 130 200 145
##  [2585] 110 216 350 170 195 185 220 190 150 140 185 180 178 172 200 135 167
##  [2602] 200 145 240 215 147 285 160 200 157 145 135 170 195 195 165 185 140
##  [2619] 210 250 150 180 190 140 170 180 170 185 230 120 235 150 120 140 165
##  [2636] 210 185 170 200 127 165 190 140 210 165 120 185 170 218 135 140 150
##  [2653] 130 217 210 187 212 165 400 139 170 117 240 140 215 240 140 130 150
##  [2670] 160 144 190 120 200 167 195 149 240 142 190 140 150 220 235 160 210
##  [2687] 240 195 178 165 195 193 260 185 190 130 185 320 200 130 149 185 180
##  [2704] 133 130 235 170 183 240 140 130 150 137 180 195 125 178 135 220 204
##  [2721] 170 200 145 200 160 175 190 150 138 129 165 135 160 200 225 225 126
##  [2738] 150 160 250 210 135 160 275 163 190 220 220 190 185 140 200 195 140
##  [2755] 170 165 218 140 199 170 142 195 197 109 133 168 175 154 215 155 170
##  [2772] 105 150 140 175 195 248 160 165 220 165 106 145 130 155 260 140 173
##  [2789] 210 125 185 194 180 118 190 140 210 138 107 155 134 250 180 157 170
##  [2806] 113 170 135 205 185 190 160 235 150 165 180 200 145 120 200 180 275
##  [2823] 180 150 155 180 165 175 160 170 160 140 195 180 150 130 190 160 120
##  [2840] 185 180 135 185 110 150 134 125 146 140 160 155 170 141 120 165 203
##  [2857] 170 122 195 195 200 185 195 170 140 140 165 185 175 185 190 175 106
##  [2874] 158 125 185 155 136 196 140 195 210 175 175 128 207 190 200 190 230
##  [2891] 165 126 205 130 160 250 150 177 150 135 125 126 265 150 130 116 190
##  [2908] 150 180 180 145 135 129 220 165 105 210 200 210 138 190 160 196 150
##  [2925] 160 160 212 267 180 125 165 185 190 170 138 171 130 165 220 160 139
##  [2942] 222 135 400 155 138 145 164 148 138 155 185 233 245 195 124 170 200
##  [2959] 250 212 115 165 170 180 180 210 245 185 110 186 165 142 212 180 192
##  [2976] 200 165 190 250 172 115 132 170 210 125 145 275 200 150 100 170 193
##  [2993] 185 165 255 155 130 145 175 160 230 145 186 210 165 185 225 165 152
##  [3010] 180 150 185 240 158 180 168 165 148 130 115 220 189 195 150 120 132
##  [3027] 118 155 131 215 215 137 160 135 260 125 160 123 137 250 170 115 110
##  [3044] 180 125 230 160 160 200 210 205 157 135 155 185 255 127 140 175 135
##  [3061] 240 135 175 220 149 155 125 130 200 150 151 165 150 182 189 160 128
##  [3078] 160 175 200 190 185 210 120 162 195 171 185 180 230 195 140 160 120
##  [3095] 185 128 130 170 137 132 170 150 121 180 102 124 170 170 310 162 122
##  [3112] 130 255 155 115 182 205 190 170 210 138 110 110 120 165 120 190 212
##  [3129] 165 168 110 140 166 175 139 220 154 118 220 155 174 165 185 150 143
##  [3146] 115 169 150 150 210 160 176 150 175 155 150 202 223 150 200 120 130
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