# Mindanao State University
# General Santos City
# Introduction to R base commands
# Submitted by: Danzel A. Ocenar
# Math Department
# March 24, 2023
# Processing of continuous data
# Using random number generators
# Exer1: generate data with mean 2 and standard deviation 1.5 using rnorm() command
data <- rnorm(1000,2,1.5) # 1 thousand values
length(data) # count number of elements
## [1] 1000
data[1:20] # display first 20 elements
## [1] 0.9660600 2.3336249 4.9699421 2.2435277 2.2046704 -0.6662515
## [7] 0.5517051 0.9808605 0.8703542 1.2574640 2.3856223 3.6672273
## [13] 0.7294806 3.8231475 2.1191374 3.7317091 2.0748177 3.3159550
## [19] -0.6304804 1.5521240
data[1:300] # display the first 300 elements
## [1] 0.966060025 2.333624855 4.969942063 2.243527699 2.204670422
## [6] -0.666251501 0.551705077 0.980860453 0.870354174 1.257463969
## [11] 2.385622253 3.667227287 0.729480570 3.823147493 2.119137416
## [16] 3.731709142 2.074817675 3.315955040 -0.630480353 1.552123950
## [21] 1.667583205 0.260411526 0.786156049 4.670953313 0.587367659
## [26] 2.535524806 3.189836998 -0.137552212 2.526475109 -0.597970675
## [31] 1.680668427 2.678998486 1.037714773 -0.316731125 0.372598160
## [36] 4.572019399 -0.282490775 3.793485965 2.199625752 1.534764529
## [41] 2.161428760 0.605471704 4.269597270 3.951453925 4.190556890
## [46] 1.386908611 1.232692371 1.062829032 3.783576009 1.926946141
## [51] 3.432200573 0.236415446 1.038380492 1.598381349 1.170117424
## [56] 1.258560762 -0.750785105 2.234140144 2.195912022 2.297645345
## [61] 2.322320463 2.128675120 -0.192927199 3.440224082 0.582426379
## [66] 1.096039708 0.949077821 2.335753902 1.648240127 -1.408825062
## [71] 0.495130502 5.668204643 2.386767368 3.168006311 0.260520657
## [76] -0.229292902 4.725145452 -2.126789324 2.786514673 1.890230125
## [81] 3.666617648 1.913259513 3.024439138 2.008636060 0.009716954
## [86] 3.266153834 2.886604893 0.334993563 2.640970361 1.916113620
## [91] 1.977773363 1.250005463 1.280131827 2.434171750 2.630885745
## [96] -0.968517049 0.246018501 2.122885350 2.242586683 3.371277601
## [101] 1.943039520 0.955963449 0.210834647 3.482257860 -1.446316134
## [106] 1.979755081 4.446742099 0.762660519 -0.650690567 2.550255174
## [111] 1.745877280 4.436342462 0.676233535 1.733144936 3.933386460
## [116] 1.496821415 1.540352398 0.871628014 2.036951093 0.983485936
## [121] -0.264125926 2.287831114 3.673377898 4.130642846 1.704241108
## [126] 0.835304076 0.279147793 0.995056055 3.705538316 0.673698958
## [131] 0.393661099 2.881378633 1.914883551 5.244359150 0.846570943
## [136] 3.249374135 2.756341675 3.560975035 1.238383157 2.017807415
## [141] 2.567061919 1.522895737 1.957019864 0.978735682 1.478677051
## [146] 0.937627953 4.197046454 2.671358332 1.140201018 0.849052713
## [151] 0.475673128 0.630921262 2.810083888 0.795463896 1.864760752
## [156] 3.327044961 1.955878717 1.422230600 2.223370426 -0.194163923
## [161] 1.940463710 4.155448153 0.998717041 3.170495987 1.402528174
## [166] 1.597172907 4.961266487 2.439374427 2.780289757 2.018994251
## [171] 2.238528217 2.814415897 0.248379946 2.584051803 1.309024425
## [176] 2.776398031 2.918304358 2.484648240 3.033290964 1.678153241
## [181] 4.971714148 2.442030952 2.722214994 4.158800705 3.446200910
## [186] -0.911690293 0.073324256 6.109577205 0.130874993 3.255019955
## [191] 2.838659582 3.850714443 1.336037713 4.014514978 2.422574209
## [196] 0.649273943 4.545010119 2.837063009 3.055217101 -0.496845705
## [201] -0.012503507 3.015139276 2.074651113 0.427258146 1.241955821
## [206] 2.421724965 0.903455155 1.543491391 2.151260901 0.922635426
## [211] -1.242388305 4.335545085 -1.514765268 1.310358645 0.938473584
## [216] 1.653004174 -1.254561622 1.051784375 3.338639294 2.036046010
## [221] 2.381591436 -1.770573957 1.591851501 2.676307789 1.367850688
## [226] 0.338359258 2.071010951 4.112063673 3.413787646 3.049937929
## [231] 2.626732031 1.497519634 1.312605207 2.754580415 0.838691117
## [236] -1.148710356 1.505033440 4.463016349 2.099347173 3.979760190
## [241] 1.962513976 2.124567220 2.964195559 2.657262140 -1.051091328
## [246] 2.351906069 2.758155676 2.670590552 4.692108436 -0.168620801
## [251] -0.249427340 1.736153033 1.536954214 0.992068507 4.480827969
## [256] 0.914441441 4.051306642 -0.540921406 -0.504263022 2.943754527
## [261] 2.693479570 1.465611849 1.583014751 1.153396272 1.460622066
## [266] 1.882194459 2.303362346 1.328372149 -1.468093341 4.140818748
## [271] 2.293068782 1.575372199 1.414180024 2.953850762 2.476653636
## [276] 4.137490657 1.828934373 1.277780907 1.725825859 3.122312846
## [281] 0.314828271 0.576739965 3.634549111 4.171997053 3.232358268
## [286] 0.009232337 1.927379062 1.485916556 3.561794060 4.544491255
## [291] 0.922933257 1.742552671 1.993800355 3.561602980 -0.321958321
## [296] 4.583017781 3.496225958 1.996230713 1.894106967 1.828085129
# Exer2: Draw histogram with one main title and different thickness
maintitle <- "Histogram and Density Plot"
hist(data, breaks=20,col="lightblue",main = maintitle)
hist(data, breaks=40,col="lightblue",main = maintitle)
hist(data, breaks=100,col="gray",main = maintitle)
hist(data, breaks=300,col="gray",main = maintitle)
# Exer3: Draw histogram with one main title and sub title
subtitle <- "This is my second title"
maintitle <- paste0("Histogram and Density Plot \n",subtitle)
hist(data, breaks=20,col="lightblue",main = maintitle)
hist(data, breaks=40,col="lightblue",main = maintitle)
hist(data, breaks=100,col="gray",main = maintitle)
# Question: What causes the subtitle to be in the second line?
# Exer4: Draw histogram with main title and sub title
subtitle <- "This is my second title"
maintitle <- paste0("Histogram and Density Plot \n",subtitle)
hist(data, breaks=20,col="lightblue",main = maintitle)
# # Add density curve. We define the range of the density curve
x = seq(from=min(data), to=max(data), length.out=100)
norm_dist = dnorm(x, mean=mean(data), sd=sd(data)) * (max(data)-min(data))/ 20*length(data)
lines(x, norm_dist, col='violet',lwd=4)
# Exer5: Draw histogram with main title and sub title
subtitle <- "This is my second title"
maintitle <- paste0("Histogram and Density Plot \n",subtitle)
hist(data, breaks=20,col="lightblue",main = maintitle)
# Add density curve and the location of the mean value
(x = seq(from=min(data), to=max(data), length.out=100))
## [1] -2.52629899 -2.43231975 -2.33834051 -2.24436127 -2.15038204 -2.05640280
## [7] -1.96242356 -1.86844432 -1.77446508 -1.68048584 -1.58650660 -1.49252736
## [13] -1.39854813 -1.30456889 -1.21058965 -1.11661041 -1.02263117 -0.92865193
## [19] -0.83467269 -0.74069345 -0.64671421 -0.55273498 -0.45875574 -0.36477650
## [25] -0.27079726 -0.17681802 -0.08283878 0.01114046 0.10511970 0.19909893
## [31] 0.29307817 0.38705741 0.48103665 0.57501589 0.66899513 0.76297437
## [37] 0.85695361 0.95093284 1.04491208 1.13889132 1.23287056 1.32684980
## [43] 1.42082904 1.51480828 1.60878752 1.70276675 1.79674599 1.89072523
## [49] 1.98470447 2.07868371 2.17266295 2.26664219 2.36062143 2.45460067
## [55] 2.54857990 2.64255914 2.73653838 2.83051762 2.92449686 3.01847610
## [61] 3.11245534 3.20643458 3.30041381 3.39439305 3.48837229 3.58235153
## [67] 3.67633077 3.77031001 3.86428925 3.95826849 4.05224772 4.14622696
## [73] 4.24020620 4.33418544 4.42816468 4.52214392 4.61612316 4.71010240
## [79] 4.80408163 4.89806087 4.99204011 5.08601935 5.17999859 5.27397783
## [85] 5.36795707 5.46193631 5.55591554 5.64989478 5.74387402 5.83785326
## [91] 5.93183250 6.02581174 6.11979098 6.21377022 6.30774946 6.40172869
## [97] 6.49570793 6.58968717 6.68366641 6.77764565
norm_dist = dnorm(x, mean=mean(data), sd=sd(data)) * (max(data)-min(data))/ 20*length(data)
lines(x, norm_dist, col='violet',lwd=4)
abline(v = mean(data), col="black", lwd=3, lty=2)
# Add legend to top right position
legend("topright",
legend = c("Histogram", "density curve", "Average"),
col = c("lightblue", "violet", "black"),
lty = 1)
# Compute Quartile values Q1,Q2 and Q3
# These quantities divide the data into 4 equal parts
hist(data, breaks=20,col="lightblue",main = maintitle)
Quantiles = quantile(data)
Quantiles
## 0% 25% 50% 75% 100%
## -2.5262990 0.9782641 2.0062251 2.9077209 6.7776456
# locate the quartiles Q1, Q2, Q3
abline(v = Quantiles[2], col="black", lwd=3, lty=3)
abline(v = Quantiles[3], col="red", lwd=3, lty=3)
abline(v = Quantiles[4], col="blue", lwd=3, lty=3)
# Exer6: Locate the lowest 5% and the highest 5% of the distribution data
summary(data)
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -2.5263 0.9783 2.0062 1.9689 2.9077 6.7776
subtitle <- "This is my second title"
maintitle <- paste0("Histogram and Density Plot \n",subtitle)
hist(data, breaks=20,col="lightblue",main = maintitle)
# Add density curve (define the range of the density curve)
x = seq(from=min(data), to=max(data), length.out=100)
norm_dist = dnorm(x, mean=mean(data), sd=sd(data)) * (max(data)-min(data))/ 20*length(data)
lines(x, norm_dist, col='violet',lwd=4)
abline(v = mean(data), col="black", lwd=3, lty=2)
# Add legend
legend("topright", # Add legend to plot
legend = c("Histogram", "density curve", "Average"),
col = c("lightblue", "violet", "black"),
lty = 1)
# Locate lowest 5% and highest 5% of the data
quantile(data,prob = 0.05)
## 5%
## -0.5274741
abline(v = quantile(data,prob = 0.05), col="red", lwd=3, lty=3)
# Area to the left of the red line is lowest 5% of the data
quantile(data,prob = 0.95)
## 95%
## 4.436439
abline(v = quantile(data,prob = 0.95), col="blue", lwd=3, lty=3)
# Area to the right of the blue line is top 5% of the data
# Area in between the red line and the blue line is 90% of the data
# How to get the values (lowest 5%)
(Cutoff <- quantile(data,prob = 0.05))
## 5%
## -0.5274741
# mark those values that is lower than -.42 as true
# and higher than -.42 as false
(Low5Percent <- (data < Cutoff))
## [1] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [13] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [25] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
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## [841] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [853] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
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## [877] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [889] FALSE FALSE TRUE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
## [901] FALSE TRUE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
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## [925] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [937] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [949] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [961] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [973] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [985] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [997] FALSE FALSE FALSE FALSE
# count all true values using the sum command
sum(Low5Percent[TRUE]) # there are 50 of them
## [1] 50
# filter those 50 values that are smaller than the cutoff
data[Low5Percent==TRUE]
## [1] -0.6662515 -0.6304804 -0.5979707 -0.7507851 -1.4088251 -2.1267893
## [7] -0.9685170 -1.4463161 -0.6506906 -0.9116903 -1.2423883 -1.5147653
## [13] -1.2545616 -1.7705740 -1.1487104 -1.0510913 -0.5409214 -1.4680933
## [19] -1.5808149 -0.9858549 -1.3172021 -0.7309035 -1.4899810 -0.6127787
## [25] -0.5800145 -1.3353158 -0.9626062 -0.5799172 -1.2626621 -0.6719241
## [31] -0.8080242 -0.6835268 -1.3107704 -0.7714565 -0.8634210 -1.1238875
## [37] -0.7465499 -0.7990425 -0.5958343 -0.7079007 -1.5240626 -2.5262990
## [43] -1.0669064 -0.6284323 -1.2693399 -1.2048876 -0.8240495 -0.8888334
## [49] -0.5987788 -0.9263016
# How to get the top 5%
(Cutoff <- quantile(data,prob = 0.95))
## 95%
## 4.436439
(Top5Percent <- (data >= Cutoff))
## [1] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [49] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [73] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [97] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [109] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [121] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [133] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [169] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [181] TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
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## [205] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
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## [301] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE TRUE FALSE FALSE
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## [325] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [337] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [349] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [361] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [373] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [385] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [397] FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE FALSE
## [409] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [421] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [433] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [445] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [457] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [469] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [481] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [493] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [505] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [517] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [529] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [541] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [553] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [565] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [577] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [589] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [601] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [613] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [625] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [637] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [649] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [661] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE TRUE FALSE FALSE
## [673] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [685] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [697] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [709] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [721] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [733] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [745] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [757] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [769] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [781] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [793] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [805] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [817] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [829] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [841] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [853] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [865] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [877] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [889] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [901] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [913] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [925] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [937] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [949] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [961] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [973] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [985] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [997] FALSE FALSE FALSE FALSE
sum(Top5Percent[TRUE]) # counts all the true values
## [1] 50
data[Top5Percent==TRUE]
## [1] 4.969942 4.670953 4.572019 5.668205 4.725145 4.446742 5.244359 4.961266
## [9] 4.971714 6.109577 4.545010 4.463016 4.692108 4.480828 4.544491 4.583018
## [17] 5.527718 5.375826 5.683828 5.687822 5.302008 5.054866 4.905610 5.983110
## [25] 5.068425 4.955530 5.350370 4.601221 4.589088 4.800956 4.493155 4.954236
## [33] 5.011784 4.690234 5.199152 4.937980 5.209443 4.705893 4.518841 4.499070
## [41] 5.269730 4.855373 4.865170 4.709061 6.777646 4.529517 4.438283 4.868448
## [49] 4.806124 6.248886