# 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] 2.7519513 2.5232595 3.5011759 2.3035515 3.5939936 1.7810003 2.4764538
## [8] 1.7156249 3.9795201 2.1853658 3.5714412 4.5382105 0.1631699 1.5732996
## [15] 0.8471329 0.9493139 1.1829094 1.5580658 3.7611175 1.1151472
data[1:300] # display the first 300 elements
## [1] 2.75195132 2.52325951 3.50117586 2.30355146 3.59399356 1.78100034
## [7] 2.47645381 1.71562487 3.97952007 2.18536582 3.57144122 4.53821047
## [13] 0.16316991 1.57329956 0.84713285 0.94931388 1.18290938 1.55806580
## [19] 3.76111752 1.11514721 0.85080909 1.59382404 5.09807384 0.90371130
## [25] 2.30220968 -0.27987137 2.18374689 0.54947223 1.13989482 3.16994626
## [31] 0.11301514 2.99246480 0.49726755 3.33924390 -0.09443747 4.74326685
## [37] 1.19019878 3.25811935 3.44699338 1.32231014 0.93647501 -0.08530590
## [43] 6.00942316 4.90895456 3.07523180 1.15222308 2.61778838 0.29745381
## [49] -0.26553477 5.73004883 1.92058700 0.68899487 0.00815470 1.94550258
## [55] 2.49944547 0.94665848 0.30496732 2.94731173 0.63680815 1.58274698
## [61] 3.54039488 1.33882035 1.37810173 -1.86628245 0.08376554 3.76063093
## [67] -0.72678245 2.23978110 2.79306574 1.45951892 0.48897308 1.56277251
## [73] 3.76091247 3.04761241 1.70790161 1.81735950 0.44205090 0.72721364
## [79] 0.21132646 -0.04292910 5.00402067 3.64949540 4.32259732 0.76411807
## [85] 0.55372634 2.15562325 0.72804262 0.22046264 1.12574296 2.67637555
## [91] 1.65098994 2.38646599 3.57537459 2.68336475 6.22927254 4.08401136
## [97] 1.32981572 2.77465829 2.81373931 4.31290477 1.05723085 5.25243380
## [103] 0.52332137 1.42024011 2.42186470 2.36217035 2.88447315 -0.92215038
## [109] 4.57331223 1.70074164 3.41793918 2.59764480 1.36471500 2.52327492
## [115] 0.67264959 2.31134372 0.32716368 1.81957598 2.42297962 2.64839166
## [121] 0.23093270 1.13233266 1.73317171 2.42023004 1.10078217 3.39522195
## [127] 1.27661382 3.89763166 4.18639007 0.28510842 2.79089818 2.28071202
## [133] 0.02599559 3.53635708 2.35130588 3.60507522 1.53650341 2.87174540
## [139] 1.60917209 2.26552661 1.23390568 4.41965211 -0.75976082 2.76779945
## [145] 0.46631012 2.78171237 4.64026682 2.55348744 0.38258995 -1.10883519
## [151] 0.88669985 2.03242690 0.48453992 -1.76370889 0.85359109 0.19381765
## [157] -0.45261459 2.63501046 -0.20932873 2.04744973 5.94943856 2.94490461
## [163] 0.19238329 5.24925693 3.90133988 1.84862971 2.82477069 3.83963952
## [169] 2.23795811 1.24993488 2.99714102 0.21789456 2.73428974 5.31872660
## [175] 3.03695634 -1.05209495 1.27395543 -0.35722121 2.65354121 1.90586242
## [181] 2.87064456 -0.79146103 4.59532962 0.24109017 1.66833315 1.14220631
## [187] 0.62946378 1.70215017 1.48383977 2.08980815 4.44808394 1.22944605
## [193] 3.27829181 -1.67651326 -1.43565002 4.43959172 1.86763713 0.40667335
## [199] 3.08648904 0.15759502 1.52921261 2.67963551 3.37702986 2.89795788
## [205] 1.49511597 0.45447916 2.96123626 -0.40149371 1.98387861 4.15999468
## [211] 0.33155848 0.03025465 1.53059717 2.59430959 3.58765649 1.15308964
## [217] 3.26316300 3.70573071 4.51558310 3.48237513 1.11813666 2.50355868
## [223] 1.26216368 2.85708475 5.34305917 2.36052326 0.48639449 4.44464890
## [229] 0.68454618 1.52801618 0.49282623 2.09123284 2.68869948 2.36523120
## [235] 4.60545393 0.69578129 4.19599078 1.67812401 3.43476216 1.46644219
## [241] 0.28817075 3.74686532 3.19449093 -0.77646471 4.99945474 0.50837837
## [247] 4.04845795 4.45933796 3.14973737 3.49173492 4.22201873 1.94748176
## [253] 2.43794217 0.79367852 3.41197085 4.85987712 1.40403128 -0.36110536
## [259] 2.50991607 3.93662857 1.73866247 1.12221266 1.57354058 -0.17628975
## [265] 3.28506836 2.85325244 2.00640600 3.02738796 1.17086406 2.36540405
## [271] 3.83520135 0.80792303 1.60216593 1.47124423 2.55324221 2.80866617
## [277] 3.06600559 3.90294675 3.72083972 0.99213767 2.53377182 2.00059521
## [283] 4.51962945 2.40901022 2.97579982 1.81782728 1.48407890 2.69425871
## [289] 1.13219012 2.41600874 2.83249458 1.72203877 2.76427004 0.15057107
## [295] 1.12650242 4.43999106 0.70684689 1.68308755 2.58773410 4.50247987
# 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)
# 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] -4.10132903 -3.98631556 -3.87130209 -3.75628863 -3.64127516 -3.52626169
## [7] -3.41124822 -3.29623476 -3.18122129 -3.06620782 -2.95119435 -2.83618089
## [13] -2.72116742 -2.60615395 -2.49114048 -2.37612702 -2.26111355 -2.14610008
## [19] -2.03108661 -1.91607315 -1.80105968 -1.68604621 -1.57103274 -1.45601928
## [25] -1.34100581 -1.22599234 -1.11097887 -0.99596541 -0.88095194 -0.76593847
## [31] -0.65092500 -0.53591154 -0.42089807 -0.30588460 -0.19087113 -0.07585767
## [37] 0.03915580 0.15416927 0.26918273 0.38419620 0.49920967 0.61422314
## [43] 0.72923660 0.84425007 0.95926354 1.07427701 1.18929047 1.30430394
## [49] 1.41931741 1.53433088 1.64934434 1.76435781 1.87937128 1.99438475
## [55] 2.10939821 2.22441168 2.33942515 2.45443862 2.56945208 2.68446555
## [61] 2.79947902 2.91449249 3.02950595 3.14451942 3.25953289 3.37454636
## [67] 3.48955982 3.60457329 3.71958676 3.83460023 3.94961369 4.06462716
## [73] 4.17964063 4.29465410 4.40966756 4.52468103 4.63969450 4.75470797
## [79] 4.86972143 4.98473490 5.09974837 5.21476184 5.32977530 5.44478877
## [85] 5.55980224 5.67481571 5.78982917 5.90484264 6.01985611 6.13486958
## [91] 6.24988304 6.36489651 6.47990998 6.59492345 6.70993691 6.82495038
## [97] 6.93996385 7.05497732 7.16999078 7.28500425
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%
## -4.1013290 0.9674866 1.9464922 3.0022499 7.2850043
## 0% 25% 50% 75% 100%
## -2.2594856 0.9644725 1.9874866 3.0121048 6.4532431
# 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.
## -4.1013 0.9675 1.9465 1.9859 3.0023 7.2850
## Min. 1st Qu. Median Mean 3rd Qu. Max.
## -2.2595 0.9645 1.9875 1.9425 3.0121 6.4532
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%
## -0.4040498
## 5%
## -0.711858
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%
## 4.465054
## 95%
## 4.347003
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%
## -0.4040498
## 5%
## -0.711858
# 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 FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [37] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
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## [793] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [805] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
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## [829] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [841] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [853] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [865] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE TRUE 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 TRUE FALSE FALSE FALSE FALSE FALSE 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 FALSE FALSE FALSE FALSE
## [949] FALSE FALSE FALSE FALSE FALSE TRUE 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 TRUE
## [985] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE
## [997] FALSE FALSE FALSE FALSE
# count all true values using the sum command
sum(Low5Percent[TRUE]) # there are 50 of them
## [1] 50
## [1] 50
# filter those 50 values that are smaller than the cutoff
data[Low5Percent==TRUE]
## [1] -1.8662825 -0.7267824 -0.9221504 -0.7597608 -1.1088352 -1.7637089
## [7] -0.4526146 -1.0520950 -0.7914610 -1.6765133 -1.4356500 -0.7764647
## [13] -0.8998951 -0.9855219 -0.4534577 -1.2748261 -2.3524156 -1.2225345
## [19] -0.5424577 -1.1272103 -1.6120844 -1.2181001 -0.5243288 -1.4580809
## [25] -1.0491111 -0.6266367 -0.6592201 -0.7983205 -2.1656424 -2.0683561
## [31] -0.7429001 -0.6703316 -1.3430252 -0.9129997 -0.9965473 -1.0620155
## [37] -1.7387775 -0.4652635 -0.7643273 -0.9596343 -1.7531541 -0.5785948
## [43] -0.6996214 -2.2214518 -0.7220290 -0.7016613 -1.6636330 -1.3927933
## [49] -0.5748690 -4.1013290
# How to get the top 5%
(Cutoff <- quantile(data,prob = 0.95))
## 95%
## 4.465054
## 95%
## 4.347003
(Top5Percent <- (data >= Cutoff))
## [1] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [13] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [25] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [37] FALSE FALSE FALSE FALSE FALSE FALSE TRUE TRUE FALSE FALSE FALSE FALSE
## [49] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [61] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [73] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [85] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE
## [97] FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [109] TRUE 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 FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [145] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [157] FALSE FALSE FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE
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## [289] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
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## [361] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [373] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [385] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [397] FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE 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 TRUE 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 FALSE FALSE FALSE FALSE FALSE FALSE
## [469] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [481] FALSE FALSE TRUE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE
## [493] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [505] FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [517] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [529] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [541] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [553] FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE
## [565] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [577] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [589] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [601] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [613] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [625] FALSE FALSE FALSE FALSE FALSE TRUE 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 FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [661] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [673] TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE 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 FALSE 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 FALSE TRUE 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 FALSE FALSE FALSE
## [817] TRUE 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 FALSE FALSE
## [853] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [865] FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE TRUE
## [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 TRUE FALSE FALSE FALSE FALSE
## [913] FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE
## [925] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [937] FALSE FALSE FALSE FALSE FALSE FALSE FALSE 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 FALSE
## [973] FALSE FALSE FALSE FALSE FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE
## [985] FALSE FALSE TRUE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE FALSE
## [997] FALSE FALSE FALSE FALSE
sum(Top5Percent[TRUE]) # counts all the true values
## [1] 50
## [1] 50
data[Top5Percent==TRUE]
## [1] 4.538210 5.098074 4.743267 6.009423 4.908955 5.730049 5.004021 6.229273
## [9] 5.252434 4.573312 4.640267 5.949439 5.249257 5.318727 4.595330 4.515583
## [17] 5.343059 4.605454 4.999455 4.859877 4.519629 4.502480 4.822705 7.285004
## [25] 4.563243 4.586380 5.441591 4.777574 5.692598 5.274925 5.227739 4.595354
## [33] 5.452934 4.629991 4.568528 5.951936 5.466546 4.489514 4.533919 4.546299
## [41] 4.594370 4.661387 5.170625 4.497297 5.016685 5.229844 5.097235 5.940167
## [49] 4.997812 4.806916