Before we do an analysis, we need data! Fortunately R provides a large number of sample data sets. Also most packages either use one of the these or supply their own to help users learn the package. You can list all the data sets you have available using the data() function. You will see all the data sets listed on the top left pane. We will start with one called mtcars.

Before doing any analysis it is a good idea to explore your data. This can help catch any problems or potential difficulties, and also help guide your analysis. There are some helpful functions to use as well.


View(mtcars)  # Displays the data frame in the top left pane

> head(mtcars)
                   mpg cyl disp  hp drat    wt  qsec vs am gear carb
Mazda RX4         21.0   6  160 110 3.90 2.620 16.46  0  1    4    4
Mazda RX4 Wag     21.0   6  160 110 3.90 2.875 17.02  0  1    4    4
Datsun 710        22.8   4  108  93 3.85 2.320 18.61  1  1    4    1
Hornet 4 Drive    21.4   6  258 110 3.08 3.215 19.44  1  0    3    1
Hornet Sportabout 18.7   8  360 175 3.15 3.440 17.02  0  0    3    2
Valiant           18.1   6  225 105 2.76 3.460 20.22  1  0    3    1

> tail(mtcars)
                mpg cyl  disp  hp drat    wt qsec vs am gear carb
Porsche 914-2  26.0   4 120.3  91 4.43 2.140 16.7  0  1    5    2
Lotus Europa   30.4   4  95.1 113 3.77 1.513 16.9  1  1    5    2
Ford Pantera L 15.8   8 351.0 264 4.22 3.170 14.5  0  1    5    4
Ferrari Dino   19.7   6 145.0 175 3.62 2.770 15.5  0  1    5    6
Maserati Bora  15.0   8 301.0 335 3.54 3.570 14.6  0  1    5    8
Volvo 142E     21.4   4 121.0 109 4.11 2.780 18.6  1  1    4    2

> typeof(mtcars)
[1] "list"

> class(mtcars)
[1] "data.frame"

> names(mtcars)
 [1] "mpg"  "cyl"  "disp" "hp"   "drat" "wt"   "qsec" "vs"   "am"   "gear" "carb"

# Displays the structure of the object
> str(mtcars)
'data.frame':	32 obs. of  11 variables:
 $ mpg : num  21 21 22.8 21.4 18.7 18.1 14.3 24.4 22.8 19.2 ...
 $ cyl : num  6 6 4 6 8 6 8 4 4 6 ...
 $ disp: num  160 160 108 258 360 ...
 $ hp  : num  110 110 93 110 175 105 245 62 95 123 ...
 $ drat: num  3.9 3.9 3.85 3.08 3.15 2.76 3.21 3.69 3.92 3.92 ...
 $ wt  : num  2.62 2.88 2.32 3.21 3.44 ...
 $ qsec: num  16.5 17 18.6 19.4 17 ...
 $ vs  : num  0 0 1 1 0 1 0 1 1 1 ...
 $ am  : num  1 1 1 0 0 0 0 0 0 0 ...
 $ gear: num  4 4 4 3 3 3 3 4 4 4 ...
 $ carb: num  4 4 1 1 2 1 4 2 2 4 ...
 
> help(mtcars)  # Displays the documentation for the mtcars data set.

We might be curious if there is a difference in mileage between transmission types. First lets look at some descriptive statistics.


> summary(mtcars$mpg)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
  10.40   15.43   19.20   20.09   22.80   33.90

If we use a different package called psych we can get some additional information. Packages are bundles of functions that can be imported and used. We’ll cover them more in another unit, but they are one of the best things about R!

> install.packages("psych")  # You probably don't need to install it
> library(psych)
> describe(mtcars$mpg)
   vars  n  mean   sd median trimmed  mad  min  max range skew kurtosis   se
X1    1 32 20.09 6.03   19.2    19.7 5.41 10.4 33.9  23.5 0.61    -0.37 1.07


# This function does the same thing but by group.
> describeBy(mtcars$mpg, group = mtcars$am)

 Descriptive statistics by group 
group: 0
   vars  n  mean   sd median trimmed  mad  min  max range skew kurtosis   se
X1    1 19 17.15 3.83   17.3   17.12 3.11 10.4 24.4    14 0.01     -0.8 0.88
------------------------------------------------------------------ 
group: 1
   vars  n  mean   sd median trimmed  mad min  max range skew kurtosis   se
X1    1 13 24.39 6.17   22.8   24.38 6.67  15 33.9  18.9 0.05    -1.46 1.71

It does look like automatic transmission cars get lower gas mileage. Is this a meaningful difference? For instance, if you take two coins and toss them 100 times, we expect them both to be about 50% heads, and 50% tails. If one lands 40/60 the other 52/48 is there a difference in how they flip? We know there is some randomness, so we need a way to handle that. For this we use statitistical tests, in this case one called a T test. Statistical tests like this help us to decide if we should believe the groups to be different or the same.


> trans_test <- t.test(mtcars$mpg~mtcars$am)
> trans_test

	Welch Two Sample t-test

data:  mtcars$mpg by mtcars$am
t = -3.7671, df = 18.332, p-value = 0.001374
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -11.280194  -3.209684
sample estimates:
mean in group 0 mean in group 1 
       17.14737        24.39231

The output may seem confusing at first so lets do some explanation. The first line just tells us the name of the test, followed by the data we used. Then we see t = -3.7671, df = 18.332, p-value = 0.001374. The t value is a way of measuring the size of the difference betweeen the two groups that is independent of the scale. df stands for degrees of freedom, its a way of describing the sample size.

t and df are used to compute the p-value which is actually the most meaningful value. The p-value is an estimate of the likelihood that you would get a difference between the groups as larger or larger by chance. That may seem confusing, but if we think about our coins again, we know we are unlikely to get 50 heads and 50 tails if we toss it 100 times. The further we are away from that, however, the more likely we are to believe the coin is loaded. In the case of our cars, we have a sample of cars, some with standard transmission, some with automatic. We wouldn’t expect them to be exactly the same even if the type of transmission had no effect because cars vary in so many different ways. However, at some point we might, the difference in mileage would be large enough for us to believe transmission does have an effect on gas mileage. In this case the p-value is 0.001374 which means we would expect a difference this large by chance only 0.1374% of the time. That seems pretty unlikely, and by convention any p-value of 0.05 (1 in 20 times) or less we consider to be signficant. So yes, statistically cars with standard transmissions get better gas mileage than cars with automatic transmissions.

It is tempting to just say we have found the answer and leave it at that. Always buy a standard transmission if you want to save money on gas. However, there is no substitute for good thinking.

First consider is this a meaningful difference? This is different from statistical significance. What you should ask in this case is “Is the difference between transmission types a meaningful difference in mileage”. A difference of 0.1 mpg might not be, but we see a difference of about 7mpg, which is meaningful to most people.

There may also be other reasons why we saw this difference. Is it possible automatic transmissions are found more often in larger cars, while standard transmissions are found in smaller sports cars?

Try it out.

See if you can test the hypothesis that smaller cars have standard transmissions.

  • Do descriptive statistics first, then a t-test.