Univariate statistics Continued: Histograms
ggplot(mtcars) +
geom_histogram(aes(mpg) , binwidth = 5)
Illya Mowerman
February 20, 2018
What makes cars more efficient. Efficiency measured in MPG
Variables:
[, 1] mpg Miles/(US) gallon [, 2] cyl Number of cylinders [, 3] disp Displacement (cu.in.) [, 4] hp Gross horsepower [, 5] drat Rear axle ratio [, 6] wt Weight (1000 lbs) [, 7] qsec 1/4 mile time [, 8] vs V/S [, 9] am Transmission (0 = automatic, 1 = manual) [,10] gear Number of forward gears [,11] carb Number of carburetors
summary(mtcars)## mpg cyl disp hp
## Min. :10.40 Min. :4.000 Min. : 71.1 Min. : 52.0
## 1st Qu.:15.43 1st Qu.:4.000 1st Qu.:120.8 1st Qu.: 96.5
## Median :19.20 Median :6.000 Median :196.3 Median :123.0
## Mean :20.09 Mean :6.188 Mean :230.7 Mean :146.7
## 3rd Qu.:22.80 3rd Qu.:8.000 3rd Qu.:326.0 3rd Qu.:180.0
## Max. :33.90 Max. :8.000 Max. :472.0 Max. :335.0
## drat wt qsec vs
## Min. :2.760 Min. :1.513 Min. :14.50 Min. :0.0000
## 1st Qu.:3.080 1st Qu.:2.581 1st Qu.:16.89 1st Qu.:0.0000
## Median :3.695 Median :3.325 Median :17.71 Median :0.0000
## Mean :3.597 Mean :3.217 Mean :17.85 Mean :0.4375
## 3rd Qu.:3.920 3rd Qu.:3.610 3rd Qu.:18.90 3rd Qu.:1.0000
## Max. :4.930 Max. :5.424 Max. :22.90 Max. :1.0000
## am gear carb
## Min. :0.0000 Min. :3.000 Min. :1.000
## 1st Qu.:0.0000 1st Qu.:3.000 1st Qu.:2.000
## Median :0.0000 Median :4.000 Median :2.000
## Mean :0.4062 Mean :3.688 Mean :2.812
## 3rd Qu.:1.0000 3rd Qu.:4.000 3rd Qu.:4.000
## Max. :1.0000 Max. :5.000 Max. :8.000ggplot(mtcars) +
geom_histogram(aes(mpg) , binwidth = 5)
We created a new categorical varible for MPG. This new variable is called efficient, where 1 is efficient and 0 is not. The cutoff for efficient is 22.8, which is the this quartile.
mtcars2 <- mtcars %>%
mutate(efficient = ifelse(mpg < 22.8 , 0 , 1))
head(mtcars2)## mpg cyl disp hp drat wt qsec vs am gear carb efficient
## 1 21.0 6 160 110 3.90 2.620 16.46 0 1 4 4 0
## 2 21.0 6 160 110 3.90 2.875 17.02 0 1 4 4 0
## 3 22.8 4 108 93 3.85 2.320 18.61 1 1 4 1 1
## 4 21.4 6 258 110 3.08 3.215 19.44 1 0 3 1 0
## 5 18.7 8 360 175 3.15 3.440 17.02 0 0 3 2 0
## 6 18.1 6 225 105 2.76 3.460 20.22 1 0 3 1 0mean(mtcars2$efficient)## [1] 0.28125First a correlation
cor.test(mtcars$mpg , mtcars$cyl)##
## Pearson's product-moment correlation
##
## data: mtcars$mpg and mtcars$cyl
## t = -8.9197, df = 30, p-value = 6.113e-10
## alternative hypothesis: true correlation is not equal to 0
## 95 percent confidence interval:
## -0.9257694 -0.7163171
## sample estimates:
## cor
## -0.852162At alpha = 0.05, the correlation is significant. The correlation is strong and negative.
As vehicles have more cylinders, their mpg decrease
Below is a t-test between efficiency and cylinders
t.test(mtcars2$cyl ~ mtcars2$efficient , alternative = 'greater')##
## Welch Two Sample t-test
##
## data: mtcars2$cyl by mtcars2$efficient
## t = 10.969, df = 22, p-value = 1.094e-10
## alternative hypothesis: true difference in means is greater than 0
## 95 percent confidence interval:
## 2.567024 Inf
## sample estimates:
## mean in group 0 mean in group 1
## 7.043478 4.000000The mean number of cylinders for efficient cars is significantly less at the alpha = .05 level.
Vehicles with more than 4 cylinders are innegicient in MPG
We created a table of counts between efficiency and cylinders
CrossTable(mtcars2$cyl , mtcars2$efficient , chisq = T)## Warning in chisq.test(t, correct = FALSE, ...): Chi-squared approximation
## may be incorrect##
##
## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 32
##
##
## | mtcars2$efficient
## mtcars2$cyl | 0 | 1 | Row Total |
## -------------|-----------|-----------|-----------|
## 4 | 2 | 9 | 11 |
## | 4.412 | 11.276 | |
## | 0.182 | 0.818 | 0.344 |
## | 0.087 | 1.000 | |
## | 0.062 | 0.281 | |
## -------------|-----------|-----------|-----------|
## 6 | 7 | 0 | 7 |
## | 0.770 | 1.969 | |
## | 1.000 | 0.000 | 0.219 |
## | 0.304 | 0.000 | |
## | 0.219 | 0.000 | |
## -------------|-----------|-----------|-----------|
## 8 | 14 | 0 | 14 |
## | 1.541 | 3.938 | |
## | 1.000 | 0.000 | 0.438 |
## | 0.609 | 0.000 | |
## | 0.438 | 0.000 | |
## -------------|-----------|-----------|-----------|
## Column Total | 23 | 9 | 32 |
## | 0.719 | 0.281 | |
## -------------|-----------|-----------|-----------|
##
##
## Statistics for All Table Factors
##
##
## Pearson's Chi-squared test
## ------------------------------------------------------------
## Chi^2 = 23.90514 d.f. = 2 p = 6.442659e-06
##
##
## CrossTable(mtcars2$am , mtcars2$efficient , chisq = T)## Warning in chisq.test(t, correct = TRUE, ...): Chi-squared approximation
## may be incorrect
## Warning in chisq.test(t, correct = TRUE, ...): Chi-squared approximation
## may be incorrect##
##
## Cell Contents
## |-------------------------|
## | N |
## | Chi-square contribution |
## | N / Row Total |
## | N / Col Total |
## | N / Table Total |
## |-------------------------|
##
##
## Total Observations in Table: 32
##
##
## | mtcars2$efficient
## mtcars2$am | 0 | 1 | Row Total |
## -------------|-----------|-----------|-----------|
## 0 | 17 | 2 | 19 |
## | 0.819 | 2.092 | |
## | 0.895 | 0.105 | 0.594 |
## | 0.739 | 0.222 | |
## | 0.531 | 0.062 | |
## -------------|-----------|-----------|-----------|
## 1 | 6 | 7 | 13 |
## | 1.197 | 3.058 | |
## | 0.462 | 0.538 | 0.406 |
## | 0.261 | 0.778 | |
## | 0.188 | 0.219 | |
## -------------|-----------|-----------|-----------|
## Column Total | 23 | 9 | 32 |
## | 0.719 | 0.281 | |
## -------------|-----------|-----------|-----------|
##
##
## Statistics for All Table Factors
##
##
## Pearson's Chi-squared test
## ------------------------------------------------------------
## Chi^2 = 7.165562 d.f. = 1 p = 0.007431642
##
## Pearson's Chi-squared test with Yates' continuity correction
## ------------------------------------------------------------
## Chi^2 = 5.182812 d.f. = 1 p = 0.02281138
##
## Efficiency has a dependerncy on the transmition type, at the alpha = 0.05 level.
The relative risk of being efficient when a vehicle has an automatic transmition is: 3.5 (0.778/0.222)
The likelihood of a veicle being automatics when it is efficient is 77.8%
If a vehicle is automatic, the likelihood of it being efficient is 53.8%. This opposed to manual transmition cars that only have a 10.5% likelihood of being efficient
Through the exploratory data analysis presented in this study it is observed that automatic vehicles with less cylinders are more efficient in terms of MPG.
In addition, radios in vehicles have no impact on MPG.