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Statistics-Chapter1-Presenting data-Part1


Understanding data

Presenting data

Stem and Leaf plot

The stem and leaf plot is a simple plot which separates the data entries into leading digits or stems and trailing digits or leaves. The plot basically shows how the values distribute and cluster over the range of the observations in the data set.

In R there is a function called stem which is very suitable to display this data. It takes in 4 arguments out of which the first 2 are the most important. The first argument is an ordered vector of a variable and the second argument controls the height of the plot.

library(xlsx)
## Warning: package 'xlsx' was built under R version 3.2.3
## Loading required package: rJava
## Warning: package 'rJava' was built under R version 3.2.3
## Loading required package: xlsxjars
## Warning: package 'xlsxjars' was built under R version 3.2.3
mf_return <- read.xlsx(file = "./data/Mutual Funds.xls", sheetName = "Data")
head(mf_return)
##                                  Fund       Type Objective Assets Fees
## 1 ABN Amro Montag & Calswell Growth I large cap     Growth 1184.9   No
## 2             AIM Aggressive Growth A small cap     Growth 2639.2  Yes
## 3                   Aim Basic Value A large cap      Value 1863.9  Yes
## 4              Aim Small Cap Growth A small cap     Growth  602.2  Yes
## 5                        AIM Summit I large cap     Growth 1912.5  Yes
## 6                         AIM Value B large cap     Growth 9232.2   No
##   Expense.ratio Return.2001 X3.Yr.Return X5.Yr.Return Turnover    Risk
## 1          0.77       -13.1         -0.2         11.7      low average
## 2          1.04       -26.0          3.4          5.4      low    high
## 3          1.32         0.1         16.7         16.7  average     low
## 4          1.13       -13.8         17.7         18.5      low    high
## 5          0.72       -33.8         -5.2          7.3  average    high
## 6          1.77       -13.6         -2.1          8.8  average average
##   Best.Quarter Worst.Quarter
## 1         27.1         -15.5
## 2         31.4         -24.7
## 3         21.1         -17.3
## 4         38.1         -24.4
## 5         37.1         -30.3
## 6         27.1         -16.0
# Order the 5 Year returns of the 259 Mutual funds
mf_order_5yr_return <- mf_return$X5.Yr.Return[order(mf_return$X5.Yr.Return)]

mf_order_5yr_return
##   [1] -6.1 -2.8 -1.2 -0.7  0.5  1.8  1.9  2.5  2.8  2.8  3.3  3.5  3.8  3.8
##  [15]  4.0  4.2  4.3  4.4  4.5  4.6  4.8  5.0  5.1  5.2  5.4  5.4  5.5  5.5
##  [29]  5.8  5.9  6.0  6.2  6.3  6.4  6.5  6.5  6.6  6.6  6.6  6.8  6.8  7.0
##  [43]  7.0  7.1  7.1  7.2  7.2  7.3  7.5  7.5  7.6  7.6  7.6  7.6  7.8  7.8
##  [57]  7.8  7.9  7.9  8.1  8.1  8.1  8.2  8.2  8.2  8.2  8.3  8.3  8.4  8.4
##  [71]  8.4  8.4  8.4  8.5  8.5  8.5  8.5  8.6  8.8  8.8  8.8  8.8  8.8  9.0
##  [85]  9.0  9.0  9.1  9.1  9.1  9.2  9.3  9.3  9.3  9.4  9.4  9.4  9.4  9.5
##  [99]  9.5  9.5  9.5  9.5  9.6  9.6  9.6  9.7  9.8  9.8  9.9  9.9  9.9  9.9
## [113]  9.9 10.0 10.0 10.0 10.1 10.1 10.1 10.1 10.2 10.2 10.3 10.3 10.4 10.4
## [127] 10.5 10.5 10.5 10.5 10.5 10.5 10.6 10.6 10.7 10.7 10.7 10.8 10.9 10.9
## [141] 10.9 10.9 11.0 11.0 11.0 11.1 11.1 11.1 11.1 11.2 11.2 11.2 11.3 11.3
## [155] 11.3 11.3 11.4 11.5 11.5 11.5 11.6 11.7 11.7 11.7 11.8 11.9 11.9 11.9
## [169] 12.0 12.1 12.1 12.1 12.2 12.2 12.2 12.2 12.2 12.3 12.3 12.3 12.4 12.4
## [183] 12.5 12.5 12.5 12.5 12.5 12.6 12.7 12.9 12.9 12.9 12.9 12.9 13.0 13.0
## [197] 13.1 13.2 13.2 13.3 13.3 13.4 13.4 13.5 13.7 13.7 13.9 14.0 14.1 14.2
## [211] 14.7 14.8 14.8 14.9 14.9 15.0 15.0 15.1 15.2 15.2 15.4 15.4 15.6 15.6
## [225] 15.7 15.7 15.8 15.8 15.8 16.0 16.3 16.5 16.7 16.9 17.0 17.0 17.0 17.1
## [239] 17.6 17.7 17.8 18.1 18.1 18.2 18.2 18.5 18.5 18.5 18.6 18.7 18.9 19.7
## [253] 21.4 21.4 21.4 22.0 22.9 25.0 26.3
stem(mf_order_5yr_return, scale= 2)
## 
##   The decimal point is at the |
## 
##   -6 | 1
##   -5 | 
##   -4 | 
##   -3 | 
##   -2 | 8
##   -1 | 2
##   -0 | 7
##    0 | 5
##    1 | 89
##    2 | 588
##    3 | 3588
##    4 | 0234568
##    5 | 012445589
##    6 | 02345566688
##    7 | 001122355666688899
##    8 | 111222233444445555688888
##    9 | 000111233344445555566678899999
##   10 | 00011112233445555556677789999
##   11 | 00011112223333455567778999
##   12 | 01112222233344555556799999
##   13 | 0012233445779
##   14 | 01278899
##   15 | 00122446677888
##   16 | 03579
##   17 | 0001678
##   18 | 1122555679
##   19 | 7
##   20 | 
##   21 | 444
##   22 | 09
##   23 | 
##   24 | 
##   25 | 0
##   26 | 3

Interpretation

There are 4 mutual funds delivering -ve returns and 7 delivering returns > 20%. Most of the funds have returns between 8% to 12% returns.

What is the stem and leaf graph good for

  • It separates each data into groups: Separation
  • It shows asymmetry in the data: Assymetry
  • It visually shows irregularities: Irregularities
  • It shows where most of the data is centered: Centering

What does the stem and leaf graph not tell you

  1. Its not easy to visualize categorical plots. For example if one wants to visualize the 5 year returns based on mutual fund objectives
head(mf_return)
##                                  Fund       Type Objective Assets Fees
## 1 ABN Amro Montag & Calswell Growth I large cap     Growth 1184.9   No
## 2             AIM Aggressive Growth A small cap     Growth 2639.2  Yes
## 3                   Aim Basic Value A large cap      Value 1863.9  Yes
## 4              Aim Small Cap Growth A small cap     Growth  602.2  Yes
## 5                        AIM Summit I large cap     Growth 1912.5  Yes
## 6                         AIM Value B large cap     Growth 9232.2   No
##   Expense.ratio Return.2001 X3.Yr.Return X5.Yr.Return Turnover    Risk
## 1          0.77       -13.1         -0.2         11.7      low average
## 2          1.04       -26.0          3.4          5.4      low    high
## 3          1.32         0.1         16.7         16.7  average     low
## 4          1.13       -13.8         17.7         18.5      low    high
## 5          0.72       -33.8         -5.2          7.3  average    high
## 6          1.77       -13.6         -2.1          8.8  average average
##   Best.Quarter Worst.Quarter
## 1         27.1         -15.5
## 2         31.4         -24.7
## 3         21.1         -17.3
## 4         38.1         -24.4
## 5         37.1         -30.3
## 6         27.1         -16.0
#Get the levels of the Objectives: Growth and Value
levels(mf_return$Objective)
## [1] "Growth" "Value"
temp <- mf_return$X5.Yr.Return[mf_return$Objective %in% "Growth"]
(mf_return_5Yr_Growth <- temp[order(temp)])
##   [1] -6.1 -2.8 -1.2 -0.7  0.5  1.8  1.9  2.5  2.8  3.3  3.5  3.8  3.8  4.0
##  [15]  4.2  4.3  4.5  4.6  5.0  5.1  5.2  5.4  5.5  5.8  5.9  6.0  6.2  6.3
##  [29]  6.5  6.5  7.0  7.1  7.1  7.2  7.2  7.3  7.5  7.6  7.6  7.8  7.8  7.8
##  [43]  7.9  8.1  8.1  8.2  8.3  8.3  8.4  8.5  8.5  8.5  8.6  8.8  8.8  8.8
##  [57]  9.0  9.0  9.1  9.1  9.1  9.2  9.3  9.3  9.5  9.5  9.5  9.5  9.6  9.6
##  [71]  9.7  9.8  9.9  9.9  9.9  9.9 10.0 10.1 10.1 10.1 10.2 10.3 10.3 10.4
##  [85] 10.5 10.5 10.5 10.5 10.5 10.5 10.6 10.7 10.7 10.8 10.9 11.0 11.0 11.1
##  [99] 11.1 11.1 11.2 11.2 11.3 11.3 11.3 11.3 11.4 11.5 11.5 11.5 11.6 11.7
## [113] 11.7 11.9 11.9 12.2 12.2 12.3 12.3 12.4 12.5 12.7 12.9 12.9 12.9 13.0
## [127] 13.1 13.2 13.4 13.4 13.7 13.7 13.9 14.1 14.7 14.8 14.9 15.0 15.7 15.8
## [141] 15.8 16.0 16.9 17.0 17.0 17.6 17.8 18.1 18.1 18.2 18.5 18.5 18.7 18.9
## [155] 21.4 22.0 22.9 26.3
temp <- mf_return$X5.Yr.Return[mf_return$Objective %in% "Value"]
(mf_return_5Yr_Value <- temp[order(temp)])
##   [1]  2.8  4.4  4.8  5.4  5.5  6.4  6.6  6.6  6.6  6.8  6.8  7.0  7.5  7.6
##  [15]  7.6  7.9  8.1  8.2  8.2  8.2  8.4  8.4  8.4  8.4  8.5  8.8  8.8  9.0
##  [29]  9.3  9.4  9.4  9.4  9.4  9.5  9.6  9.8  9.9 10.0 10.0 10.1 10.2 10.4
##  [43] 10.6 10.7 10.9 10.9 10.9 11.0 11.1 11.2 11.7 11.8 11.9 12.0 12.1 12.1
##  [57] 12.1 12.2 12.2 12.2 12.3 12.4 12.5 12.5 12.5 12.5 12.6 12.9 12.9 13.0
##  [71] 13.2 13.3 13.3 13.5 14.0 14.2 14.8 14.9 15.0 15.1 15.2 15.2 15.4 15.4
##  [85] 15.6 15.6 15.7 15.8 16.3 16.5 16.7 17.0 17.1 17.7 18.2 18.5 18.6 19.7
##  [99] 21.4 21.4 25.0
stem(mf_return_5Yr_Growth, scale=2)
## 
##   The decimal point is at the |
## 
##   -6 | 1
##   -5 | 
##   -4 | 
##   -3 | 
##   -2 | 8
##   -1 | 2
##   -0 | 7
##    0 | 5
##    1 | 89
##    2 | 58
##    3 | 3588
##    4 | 02356
##    5 | 0124589
##    6 | 02355
##    7 | 0112235668889
##    8 | 1123345556888
##    9 | 00111233555566789999
##   10 | 0111233455555567789
##   11 | 00111223333455567799
##   12 | 2233457999
##   13 | 01244779
##   14 | 1789
##   15 | 0788
##   16 | 09
##   17 | 0068
##   18 | 1125579
##   19 | 
##   20 | 
##   21 | 4
##   22 | 09
##   23 | 
##   24 | 
##   25 | 
##   26 | 3
stem(mf_return_5Yr_Value, scale=2)
## 
##   The decimal point is at the |
## 
##    2 | 8
##    3 | 
##    4 | 48
##    5 | 45
##    6 | 466688
##    7 | 05669
##    8 | 12224444588
##    9 | 0344445689
##   10 | 0012467999
##   11 | 012789
##   12 | 0111222345555699
##   13 | 02335
##   14 | 0289
##   15 | 0122446678
##   16 | 357
##   17 | 017
##   18 | 256
##   19 | 7
##   20 | 
##   21 | 44
##   22 | 
##   23 | 
##   24 | 
##   25 | 0

You see that its not visually easy to compare the two plots.

  1. The stem and leaf plots are fine for small data sets. Its better to use histograms for larger datasets.
  2. For large data sets the center of the distribution is not clearly visible.

Histogram

When there are a large number of observations, its useful to summarise the data using a histogram. A histogram is used to describe numerical data that have been grouped into frequency, relative frequency or percentage distributions. The variable of interest is plotted along the x axis grouped in intervals and the number of or percentage or proportion is plotted along the y axis.

Lets now construct a histogram of the 5 year returns of mutual funds data.

library(ggplot2)
## Warning: package 'ggplot2' was built under R version 3.2.3
ggplot(mf_return, aes(X5.Yr.Return)) + geom_histogram()
## `stat_bin()` using `bins = 30`. Pick better value with `binwidth`.

Fig1

Lets change the bin width to make things a bit clearer

ggplot(mf_return, aes(X5.Yr.Return)) + geom_histogram(binwidth=10)

Fig1-1

Barplot

What about a bar plot? The bar plot will plot each and every point of the data rather than binning the data in ranges.

ggplot(mf_return, aes(X5.Yr.Return)) + geom_bar()

Fig2

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