# Introduction to descriptr

## Introduction

Descriptive statistics are used to summarize data. It enables us to present the data in a more meaningful way and to discern any patterns existing in the data. They can be useful for two purposes:

• provide basic information about variables in a data set
• highlight potential relationships between variables

This document introduces you to a basic set of functions that describe data. There is a second vignette which provides details about functions which help visualize statistical distributions.

## Data

We have modified the mtcars data to create a new data set mtcarz. The only difference between the two data sets is related to the variable types.

str(mtcarz)
## '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 : Factor w/ 3 levels "4","6","8": 2 2 1 2 3 2 3 1 1 2 ...
##  $disp: num 160 160 108 258 360 ... ##$ 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  : Factor w/ 2 levels "0","1": 1 1 2 2 1 2 1 2 2 2 ...
##  $am : Factor w/ 2 levels "0","1": 2 2 2 1 1 1 1 1 1 1 ... ##$ gear: Factor w/ 3 levels "3","4","5": 2 2 2 1 1 1 1 2 2 2 ...
##  $carb: Factor w/ 6 levels "1","2","3","4",..: 4 4 1 1 2 1 4 2 2 4 ... ## Data Screening The ds_screener() function will screen a data set and return the following: - Column/Variable Names - Data Type - Levels (in case of categorical data) - Number of missing observations - % of missing observations ds_screener(mtcarz) ## ----------------------------------------------------------------------- ## | Column Name | Data Type | Levels | Missing | Missing (%) | ## ----------------------------------------------------------------------- ## | mpg | numeric | NA | 0 | 0 | ## | cyl | factor | 4 6 8 | 0 | 0 | ## | disp | numeric | NA | 0 | 0 | ## | hp | numeric | NA | 0 | 0 | ## | drat | numeric | NA | 0 | 0 | ## | wt | numeric | NA | 0 | 0 | ## | qsec | numeric | NA | 0 | 0 | ## | vs | factor | 0 1 | 0 | 0 | ## | am | factor | 0 1 | 0 | 0 | ## | gear | factor | 3 4 5 | 0 | 0 | ## | carb | factor |1 2 3 4 6 8| 0 | 0 | ## ----------------------------------------------------------------------- ## ## Overall Missing Values 0 ## Percentage of Missing Values 0 % ## Rows with Missing Values 0 ## Columns With Missing Values 0 ## Summary Statistics The ds_summary_stats function returns a comprehensive set of statistics for continuous data. ds_summary_stats(mtcarz, mpg) ## Univariate Analysis ## ## N 32.00 Variance 36.32 ## Missing 0.00 Std Deviation 6.03 ## Mean 20.09 Range 23.50 ## Median 19.20 Interquartile Range 7.38 ## Mode 10.40 Uncorrected SS 14042.31 ## Trimmed Mean 19.95 Corrected SS 1126.05 ## Skewness 0.67 Coeff Variation 30.00 ## Kurtosis -0.02 Std Error Mean 1.07 ## ## Quantiles ## ## Quantile Value ## ## Max 33.90 ## 99% 33.44 ## 95% 31.30 ## 90% 30.09 ## Q3 22.80 ## Median 19.20 ## Q1 15.43 ## 10% 14.34 ## 5% 12.00 ## 1% 10.40 ## Min 10.40 ## ## Extreme Values ## ## Low High ## ## Obs Value Obs Value ## 15 10.4 20 33.9 ## 16 10.4 18 32.4 ## 24 13.3 19 30.4 ## 7 14.3 28 30.4 ## 17 14.7 26 27.3 ## Cross Tabulation The ds_cross_table() function creates two way tables of categorical variables. ds_cross_table(mtcarz, cyl, gear) ## Cell Contents ## |---------------| ## | Frequency | ## | Percent | ## | Row Pct | ## | Col Pct | ## |---------------| ## ## Total Observations: 32 ## ## ---------------------------------------------------------------------------- ## | | gear | ## ---------------------------------------------------------------------------- ## | cyl | 3 | 4 | 5 | Row Total | ## ---------------------------------------------------------------------------- ## | 4 | 1 | 8 | 2 | 11 | ## | | 0.031 | 0.25 | 0.062 | | ## | | 0.09 | 0.73 | 0.18 | 0.34 | ## | | 0.07 | 0.67 | 0.4 | | ## ---------------------------------------------------------------------------- ## | 6 | 2 | 4 | 1 | 7 | ## | | 0.062 | 0.125 | 0.031 | | ## | | 0.29 | 0.57 | 0.14 | 0.22 | ## | | 0.13 | 0.33 | 0.2 | | ## ---------------------------------------------------------------------------- ## | 8 | 12 | 0 | 2 | 14 | ## | | 0.375 | 0 | 0.062 | | ## | | 0.86 | 0 | 0.14 | 0.44 | ## | | 0.8 | 0 | 0.4 | | ## ---------------------------------------------------------------------------- ## | Column Total | 15 | 12 | 5 | 32 | ## | | 0.468 | 0.375 | 0.155 | | ## ---------------------------------------------------------------------------- ds_twoway_table() will return a tibble. ds_twoway_table(mtcarz, cyl, gear) ## Joining, by = c("cyl", "gear", "count") ## # A tibble: 8 x 6 ## cyl gear count percent row_percent col_percent ## <fct> <fct> <int> <dbl> <dbl> <dbl> ## 1 4 3 1 0.0312 0.0909 0.0667 ## 2 4 4 8 0.250 0.727 0.667 ## 3 4 5 2 0.0625 0.182 0.400 ## 4 6 3 2 0.0625 0.286 0.133 ## 5 6 4 4 0.125 0.571 0.333 ## 6 6 5 1 0.0312 0.143 0.200 ## 7 8 3 12 0.375 0.857 0.800 ## 8 8 5 2 0.0625 0.143 0.400 A plot method has been defined which will generate: ### Grouped Bar Plots k <- ds_cross_table(mtcarz, cyl, gear) plot(k) ### Stacked Bar Plots k <- ds_cross_table(mtcarz, cyl, gear) plot(k, stacked = TRUE) ### Proportional Bar Plots k <- ds_cross_table(mtcarz, cyl, gear) plot(k, proportional = TRUE) ## Frequency Table (Categorical Data) The ds_freq_table() function creates frequency tables for categorical variables. ds_freq_table(mtcarz, cyl) ## Variable: cyl ## ----------------------------------------------------------------------- ## Levels Frequency Cum Frequency Percent Cum Percent ## ----------------------------------------------------------------------- ## 4 11 11 34.38 34.38 ## ----------------------------------------------------------------------- ## 6 7 18 21.88 56.25 ## ----------------------------------------------------------------------- ## 8 14 32 43.75 100 ## ----------------------------------------------------------------------- ## Total 32 - 100.00 - ## ----------------------------------------------------------------------- ### Bar Plot A barplot method has been defined. k <- ds_freq_table(mtcarz, cyl) plot(k) ## Frequency Table (Continuous Data) The ds_freq_cont function creates frequency tables for continuous variables. The default number of intervals is 5. ds_freq_cont(mtcarz, mpg, 4) ## Variable: mpg ## |---------------------------------------------------------------------------| ## | Bins | Frequency | Cum Frequency | Percent | Cum Percent | ## |---------------------------------------------------------------------------| ## | 10.4 - 16.3 | 10 | 10 | 31.25 | 31.25 | ## |---------------------------------------------------------------------------| ## | 16.3 - 22.1 | 13 | 23 | 40.62 | 71.88 | ## |---------------------------------------------------------------------------| ## | 22.1 - 28 | 5 | 28 | 15.62 | 87.5 | ## |---------------------------------------------------------------------------| ## | 28 - 33.9 | 4 | 32 | 12.5 | 100 | ## |---------------------------------------------------------------------------| ## | Total | 32 | - | 100.00 | - | ## |---------------------------------------------------------------------------| ### Histogram A hist method has been defined. k <- ds_freq_cont(mtcarz, mpg, 4) plot(k) ## Group Summary The ds_group_summary() function returns descriptive statistics of a continuous variable for the different levels of a categorical variable. k <- ds_group_summary(mtcarz, cyl, mpg) k ## mpg by cyl ## ----------------------------------------------------------------------------------------- ## | Statistic/Levels| 4| 6| 8| ## ----------------------------------------------------------------------------------------- ## | Obs| 11| 7| 14| ## | Minimum| 21.4| 17.8| 10.4| ## | Maximum| 33.9| 21.4| 19.2| ## | Mean| 26.66| 19.74| 15.1| ## | Median| 26| 19.7| 15.2| ## | Mode| 22.8| 21| 10.4| ## | Std. Deviation| 4.51| 1.45| 2.56| ## | Variance| 20.34| 2.11| 6.55| ## | Skewness| 0.35| -0.26| -0.46| ## | Kurtosis| -1.43| -1.83| 0.33| ## | Uncorrected SS| 8023.83| 2741.14| 3277.34| ## | Corrected SS| 203.39| 12.68| 85.2| ## | Coeff Variation| 16.91| 7.36| 16.95| ## | Std. Error Mean| 1.36| 0.55| 0.68| ## | Range| 12.5| 3.6| 8.8| ## | Interquartile Range| 7.6| 2.35| 1.85| ## ----------------------------------------------------------------------------------------- ds_group_summary() returns a tibble which can be used for further analysis. k$tidy_stats
## # A tibble: 3 x 15
##   cyl   length   min   max  mean median  mode    sd variance skewness
##   <fct>  <int> <dbl> <dbl> <dbl>  <dbl> <dbl> <dbl>    <dbl>    <dbl>
## 1 4         11  21.4  33.9  26.7   26.0  22.8  4.51    20.3     0.348
## 2 6          7  17.8  21.4  19.7   19.7  21.0  1.45     2.11   -0.259
## 3 8         14  10.4  19.2  15.1   15.2  10.4  2.56     6.55   -0.456
## # ... with 5 more variables: kurtosis <dbl>, coeff_var <dbl>,
## #   std_error <dbl>, range <dbl>, iqr <dbl>

### Box Plot

A boxplot() method has been defined.

k <- ds_group_summary(mtcarz, cyl, mpg)
plot(k)

## Multiple Variable Statistics

The ds_multi_stats() function generates summary/descriptive statistics for variables in a data frame/tibble.

ds_multi_stats(mtcarz, mpg, disp, hp)
## # A tibble: 3 x 16
##   vars    min   max  mean t_mean median  mode range variance  stdev  skew
##   <chr> <dbl> <dbl> <dbl>  <dbl>  <dbl> <dbl> <dbl>    <dbl>  <dbl> <dbl>
## 1 disp   71.1 472   231    228    196   276   401    15361   124    0.420
## 2 hp     52.0 335   147    144    123   110   283     4701    68.6  0.799
## 3 mpg    10.4  33.9  20.1   20.0   19.2  10.4  23.5     36.3   6.03 0.672
## # ... with 5 more variables: kurtosis <dbl>, coeff_var <dbl>, q1 <dbl>,
## #   q3 <dbl>, iqrange <dbl>

## Multiple One Way Tables

The ds_oway_tables() function creates multiple one way tables by creating a frequency table for each categorical variable in a data frame.

ds_oway_tables(mtcarz)
##                              Variable: cyl
## -----------------------------------------------------------------------
## Levels     Frequency    Cum Frequency       Percent        Cum Percent
## -----------------------------------------------------------------------
##    4          11             11              34.38            34.38
## -----------------------------------------------------------------------
##    6           7             18              21.88            56.25
## -----------------------------------------------------------------------
##    8          14             32              43.75             100
## -----------------------------------------------------------------------
##  Total        32              -             100.00              -
## -----------------------------------------------------------------------
##
##                              Variable: vs
## -----------------------------------------------------------------------
## Levels     Frequency    Cum Frequency       Percent        Cum Percent
## -----------------------------------------------------------------------
##    0          18             18              56.25            56.25
## -----------------------------------------------------------------------
##    1          14             32              43.75             100
## -----------------------------------------------------------------------
##  Total        32              -             100.00              -
## -----------------------------------------------------------------------
##
##                              Variable: am
## -----------------------------------------------------------------------
## Levels     Frequency    Cum Frequency       Percent        Cum Percent
## -----------------------------------------------------------------------
##    0          19             19              59.38            59.38
## -----------------------------------------------------------------------
##    1          13             32              40.62             100
## -----------------------------------------------------------------------
##  Total        32              -             100.00              -
## -----------------------------------------------------------------------
##
##                             Variable: gear
## -----------------------------------------------------------------------
## Levels     Frequency    Cum Frequency       Percent        Cum Percent
## -----------------------------------------------------------------------
##    3          15             15              46.88            46.88
## -----------------------------------------------------------------------
##    4          12             27              37.5             84.38
## -----------------------------------------------------------------------
##    5           5             32              15.62             100
## -----------------------------------------------------------------------
##  Total        32              -             100.00              -
## -----------------------------------------------------------------------
##
##                             Variable: carb
## -----------------------------------------------------------------------
## Levels     Frequency    Cum Frequency       Percent        Cum Percent
## -----------------------------------------------------------------------
##    1           7              7              21.88            21.88
## -----------------------------------------------------------------------
##    2          10             17              31.25            53.12
## -----------------------------------------------------------------------
##    3           3             20              9.38             62.5
## -----------------------------------------------------------------------
##    4          10             30              31.25            93.75
## -----------------------------------------------------------------------
##    6           1             31              3.12             96.88
## -----------------------------------------------------------------------
##    8           1             32              3.12              100
## -----------------------------------------------------------------------
##  Total        32              -             100.00              -
## -----------------------------------------------------------------------

## Multiple Two Way Tables

The ds_tway_tables() function creates multiple two way tables by creating a cross table for each unique pair of categorical variables in a data frame.

ds_tway_tables(mtcarz)
##     Cell Contents
##  |---------------|
##  |     Frequency |
##  |       Percent |
##  |       Row Pct |
##  |       Col Pct |
##  |---------------|
##
##  Total Observations:  32
##
##                          cyl vs vs
## -------------------------------------------------------------
## |              |                     vs                     |
## -------------------------------------------------------------
## |          cyl |            0 |            1 |    Row Total |
## -------------------------------------------------------------
## |            4 |            1 |           10 |           11 |
## |              |        0.031 |        0.312 |              |
## |              |         0.09 |         0.91 |         0.34 |
## |              |         0.06 |         0.71 |              |
## -------------------------------------------------------------
## |            6 |            3 |            4 |            7 |
## |              |        0.094 |        0.125 |              |
## |              |         0.43 |         0.57 |         0.22 |
## |              |         0.17 |         0.29 |              |
## -------------------------------------------------------------
## |            8 |           14 |            0 |           14 |
## |              |        0.438 |            0 |              |
## |              |            1 |            0 |         0.44 |
## |              |         0.78 |            0 |              |
## -------------------------------------------------------------
## | Column Total |           18 |           14 |           32 |
## |              |        0.563 |        0.437 |              |
## -------------------------------------------------------------
##
##
##                          cyl vs am
## -------------------------------------------------------------
## |              |                     am                     |
## -------------------------------------------------------------
## |          cyl |            0 |            1 |    Row Total |
## -------------------------------------------------------------
## |            4 |            3 |            8 |           11 |
## |              |        0.094 |         0.25 |              |
## |              |         0.27 |         0.73 |         0.34 |
## |              |         0.16 |         0.62 |              |
## -------------------------------------------------------------
## |            6 |            4 |            3 |            7 |
## |              |        0.125 |        0.094 |              |
## |              |         0.57 |         0.43 |         0.22 |
## |              |         0.21 |         0.23 |              |
## -------------------------------------------------------------
## |            8 |           12 |            2 |           14 |
## |              |        0.375 |        0.062 |              |
## |              |         0.86 |         0.14 |         0.44 |
## |              |         0.63 |         0.15 |              |
## -------------------------------------------------------------
## | Column Total |           19 |           13 |           32 |
## |              |        0.594 |        0.406 |              |
## -------------------------------------------------------------
##
##
##                                 cyl vs gear
## ----------------------------------------------------------------------------
## |              |                           gear                            |
## ----------------------------------------------------------------------------
## |          cyl |            3 |            4 |            5 |    Row Total |
## ----------------------------------------------------------------------------
## |            4 |            1 |            8 |            2 |           11 |
## |              |        0.031 |         0.25 |        0.062 |              |
## |              |         0.09 |         0.73 |         0.18 |         0.34 |
## |              |         0.07 |         0.67 |          0.4 |              |
## ----------------------------------------------------------------------------
## |            6 |            2 |            4 |            1 |            7 |
## |              |        0.062 |        0.125 |        0.031 |              |
## |              |         0.29 |         0.57 |         0.14 |         0.22 |
## |              |         0.13 |         0.33 |          0.2 |              |
## ----------------------------------------------------------------------------
## |            8 |           12 |            0 |            2 |           14 |
## |              |        0.375 |            0 |        0.062 |              |
## |              |         0.86 |            0 |         0.14 |         0.44 |
## |              |          0.8 |            0 |          0.4 |              |
## ----------------------------------------------------------------------------
## | Column Total |           15 |           12 |            5 |           32 |
## |              |        0.468 |        0.375 |        0.155 |              |
## ----------------------------------------------------------------------------
##
##
##                                                       cyl vs carb
## -------------------------------------------------------------------------------------------------------------------------
## |              |                                                  carb                                                  |
## -------------------------------------------------------------------------------------------------------------------------
## |          cyl |            1 |            2 |            3 |            4 |            6 |            8 |    Row Total |
## -------------------------------------------------------------------------------------------------------------------------
## |            4 |            5 |            6 |            0 |            0 |            0 |            0 |           11 |
## |              |        0.156 |        0.188 |            0 |            0 |            0 |            0 |              |
## |              |         0.45 |         0.55 |            0 |            0 |            0 |            0 |         0.34 |
## |              |         0.71 |          0.6 |            0 |            0 |            0 |            0 |              |
## -------------------------------------------------------------------------------------------------------------------------
## |            6 |            2 |            0 |            0 |            4 |            1 |            0 |            7 |
## |              |        0.062 |            0 |            0 |        0.125 |        0.031 |            0 |              |
## |              |         0.29 |            0 |            0 |         0.57 |         0.14 |            0 |         0.22 |
## |              |         0.29 |            0 |            0 |          0.4 |            1 |            0 |              |
## -------------------------------------------------------------------------------------------------------------------------
## |            8 |            0 |            4 |            3 |            6 |            0 |            1 |           14 |
## |              |            0 |        0.125 |        0.094 |        0.188 |            0 |        0.031 |              |
## |              |            0 |         0.29 |         0.21 |         0.43 |            0 |         0.07 |         0.44 |
## |              |            0 |          0.4 |            1 |          0.6 |            0 |            1 |              |
## -------------------------------------------------------------------------------------------------------------------------
## | Column Total |            7 |           10 |            3 |           10 |            1 |            1 |           32 |
## |              |        0.218 |        0.313 |        0.094 |        0.313 |        0.031 |        0.031 |              |
## -------------------------------------------------------------------------------------------------------------------------
##
##
##                           vs vs am
## -------------------------------------------------------------
## |              |                     am                     |
## -------------------------------------------------------------
## |           vs |            0 |            1 |    Row Total |
## -------------------------------------------------------------
## |            0 |           12 |            6 |           18 |
## |              |        0.375 |        0.188 |              |
## |              |         0.67 |         0.33 |         0.56 |
## |              |         0.63 |         0.46 |              |
## -------------------------------------------------------------
## |            1 |            7 |            7 |           14 |
## |              |        0.219 |        0.219 |              |
## |              |          0.5 |          0.5 |         0.44 |
## |              |         0.37 |         0.54 |              |
## -------------------------------------------------------------
## | Column Total |           19 |           13 |           32 |
## |              |        0.594 |        0.407 |              |
## -------------------------------------------------------------
##
##
##                                 vs vs gear
## ----------------------------------------------------------------------------
## |              |                           gear                            |
## ----------------------------------------------------------------------------
## |           vs |            3 |            4 |            5 |    Row Total |
## ----------------------------------------------------------------------------
## |            0 |           12 |            2 |            4 |           18 |
## |              |        0.375 |        0.062 |        0.125 |              |
## |              |         0.67 |         0.11 |         0.22 |         0.56 |
## |              |          0.8 |         0.17 |          0.8 |              |
## ----------------------------------------------------------------------------
## |            1 |            3 |           10 |            1 |           14 |
## |              |        0.094 |        0.312 |        0.031 |              |
## |              |         0.21 |         0.71 |         0.07 |         0.44 |
## |              |          0.2 |         0.83 |          0.2 |              |
## ----------------------------------------------------------------------------
## | Column Total |           15 |           12 |            5 |           32 |
## |              |        0.469 |        0.374 |        0.156 |              |
## ----------------------------------------------------------------------------
##
##
##                                                        vs vs carb
## -------------------------------------------------------------------------------------------------------------------------
## |              |                                                  carb                                                  |
## -------------------------------------------------------------------------------------------------------------------------
## |           vs |            1 |            2 |            3 |            4 |            6 |            8 |    Row Total |
## -------------------------------------------------------------------------------------------------------------------------
## |            0 |            0 |            5 |            3 |            8 |            1 |            1 |           18 |
## |              |            0 |        0.156 |        0.094 |         0.25 |        0.031 |        0.031 |              |
## |              |            0 |         0.28 |         0.17 |         0.44 |         0.06 |         0.06 |         0.56 |
## |              |            0 |          0.5 |            1 |          0.8 |            1 |            1 |              |
## -------------------------------------------------------------------------------------------------------------------------
## |            1 |            7 |            5 |            0 |            2 |            0 |            0 |           14 |
## |              |        0.219 |        0.156 |            0 |        0.062 |            0 |            0 |              |
## |              |          0.5 |         0.36 |            0 |         0.14 |            0 |            0 |         0.44 |
## |              |            1 |          0.5 |            0 |          0.2 |            0 |            0 |              |
## -------------------------------------------------------------------------------------------------------------------------
## | Column Total |            7 |           10 |            3 |           10 |            1 |            1 |           32 |
## |              |        0.219 |        0.312 |        0.094 |        0.312 |        0.031 |        0.031 |              |
## -------------------------------------------------------------------------------------------------------------------------
##
##
##                                 am vs gear
## ----------------------------------------------------------------------------
## |              |                           gear                            |
## ----------------------------------------------------------------------------
## |           am |            3 |            4 |            5 |    Row Total |
## ----------------------------------------------------------------------------
## |            0 |           15 |            4 |            0 |           19 |
## |              |        0.469 |        0.125 |            0 |              |
## |              |         0.79 |         0.21 |            0 |         0.59 |
## |              |            1 |         0.33 |            0 |              |
## ----------------------------------------------------------------------------
## |            1 |            0 |            8 |            5 |           13 |
## |              |            0 |         0.25 |        0.156 |              |
## |              |            0 |         0.62 |         0.38 |         0.41 |
## |              |            0 |         0.67 |            1 |              |
## ----------------------------------------------------------------------------
## | Column Total |           15 |           12 |            5 |           32 |
## |              |        0.469 |        0.375 |        0.156 |              |
## ----------------------------------------------------------------------------
##
##
##                                                        am vs carb
## -------------------------------------------------------------------------------------------------------------------------
## |              |                                                  carb                                                  |
## -------------------------------------------------------------------------------------------------------------------------
## |           am |            1 |            2 |            3 |            4 |            6 |            8 |    Row Total |
## -------------------------------------------------------------------------------------------------------------------------
## |            0 |            3 |            6 |            3 |            7 |            0 |            0 |           19 |
## |              |        0.094 |        0.188 |        0.094 |        0.219 |            0 |            0 |              |
## |              |         0.16 |         0.32 |         0.16 |         0.37 |            0 |            0 |          0.6 |
## |              |         0.43 |          0.6 |            1 |          0.7 |            0 |            0 |              |
## -------------------------------------------------------------------------------------------------------------------------
## |            1 |            4 |            4 |            0 |            3 |            1 |            1 |           13 |
## |              |        0.125 |        0.125 |            0 |        0.094 |        0.031 |        0.031 |              |
## |              |         0.31 |         0.31 |            0 |         0.23 |         0.08 |         0.08 |         0.41 |
## |              |         0.57 |          0.4 |            0 |          0.3 |            1 |            1 |              |
## -------------------------------------------------------------------------------------------------------------------------
## | Column Total |            7 |           10 |            3 |           10 |            1 |            1 |           32 |
## |              |        0.219 |        0.313 |        0.094 |        0.313 |        0.031 |        0.031 |              |
## -------------------------------------------------------------------------------------------------------------------------
##
##
##                                                       gear vs carb
## -------------------------------------------------------------------------------------------------------------------------
## |              |                                                  carb                                                  |
## -------------------------------------------------------------------------------------------------------------------------
## |         gear |            1 |            2 |            3 |            4 |            6 |            8 |    Row Total |
## -------------------------------------------------------------------------------------------------------------------------
## |            3 |            3 |            4 |            3 |            5 |            0 |            0 |           15 |
## |              |        0.094 |        0.125 |        0.094 |        0.156 |            0 |            0 |              |
## |              |          0.2 |         0.27 |          0.2 |         0.33 |            0 |            0 |         0.47 |
## |              |         0.43 |          0.4 |            1 |          0.5 |            0 |            0 |              |
## -------------------------------------------------------------------------------------------------------------------------
## |            4 |            4 |            4 |            0 |            4 |            0 |            0 |           12 |
## |              |        0.125 |        0.125 |            0 |        0.125 |            0 |            0 |              |
## |              |         0.33 |         0.33 |            0 |         0.33 |            0 |            0 |         0.38 |
## |              |         0.57 |          0.4 |            0 |          0.4 |            0 |            0 |              |
## -------------------------------------------------------------------------------------------------------------------------
## |            5 |            0 |            2 |            0 |            1 |            1 |            1 |            5 |
## |              |            0 |        0.062 |            0 |        0.031 |        0.031 |        0.031 |              |
## |              |            0 |          0.4 |            0 |          0.2 |          0.2 |          0.2 |         0.16 |
## |              |            0 |          0.2 |            0 |          0.1 |            1 |            1 |              |
## -------------------------------------------------------------------------------------------------------------------------
## | Column Total |            7 |           10 |            3 |           10 |            1 |            1 |           32 |
## |              |        0.219 |        0.312 |        0.094 |        0.312 |        0.031 |        0.031 |              |
## -------------------------------------------------------------------------------------------------------------------------