A quick way to organize data and visualize distributions

Stem plots (sometimes "stemplots") or stem-and-leaf plots are a quick way of sorting and organizing data, and a quick way of visualizing the shape of a distribution — or of comparing two distributions. We'll learn them by example using a sample data set. Consider this set of hypothetical test scores between 0 and 100:

How to do it

To make a stem plot of this data, follow these procedures.

1. Write the stems in order

First, we have to decide what the stems are. These (except for the 100) are all two-digit numbers, so we'll choose the first digit as the stem. We have first digits of 3, 5, 6, 7, 8, 9 and 10. Here they are:

         3|
         4|
         5|
         6|
         7|
         8|
         9|
        10|
        

2. Now add the leaves

the leaves are our set of second digits, each associated with its own first digits. For example, the first digit of 3 has an 8 that goes with it. The stem value of 4 has nothing associated with it – there are no scores in the 40's. It looks like this:

         3|8
         4|
         5|6
         6|78
         7|82137
         8|034559
         9|7430
        10|0
        

3. Order the leaves

Finally, put all of the leaves in numerical order, and add a key for the reader:

         3|8
         4|         Key: 7|2 means a
         5|6             score of 72
         6|78
         7|12378
         8|034559
         9|0347
        10|0
        

Splitting stems

Sometimes when we create a stem plot from data we find that a few stems have most of the leaves, perhaps distorting our view of the shape of the underlying distribution. To fix that we can split some or all of the stems. We have to take great care when doing this, however, lest we mislead ourselves or confirm a bias. Here's how it works. First a data set, then the stem plots:

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Pie charts

Pie charts – they look like a sliced pie – are a common way of showing proportion among categories. Here is a pie chart, using data from 2022, that shows the broad categories of the federal budget of the United States. About 62% of all federal funds are spoken for each year including this one. These are things like salaries, social security, medicare, etc. 8% of the budget has to go to "debt service," that is, interest on borrowed money, and the rest, 30%, can be reallocated by congress and other agencies with authority to do so.

The pie chart is a nice way to display this data. It's understood that the full circle represents 100% of the budget, and we can see right away that mandatory spending takes up the largest portion of it, followed by discretionary spending and interest payments. Not only do we get a good look at which categories are bigger and smaller, but how much bigger or smaller.

There are plenty of options for pie charts. For example, we can pull a pie wedge out for emphasis, as in the chart below.

Bar charts

Bar charts are ubiquitous in nearly all fields that involve data. The reader of a bar chart compares the relative heights of the bars against a scale to make judgements about the data. They can be used just like pie charts to show proportion, but more often they're used for comparison and sometimes to show growth. Here's an example using data on the sales of personal computers in 2021.

It is a little easier to see the ordering of total sales (most to least, right-to-left) in the bar chart, but to get a sense of how much of the "pie" each complany owns, nothing really beats a pie chart. You can roll over or tap the image to see the percentages of total personal computer sales.

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TI-84 calculator

Statistics-related buttons

Here is a shot of a TI-84 Plus calculator. Some important keyboard areas are hilighted in red. These buttons are of particular use in entering and manipulating statistical data. In following sections, we'll do a couple of video examples of how to accomplish various tasks.

LIST / STAT

The LIST / STAT button ( LIST is 2^nd+[STAT] ) opens up ways of entering and manipulating lists of data. There is a bit of redundancy between these two sets of functions, but in general, if you'd like to enter new data into a list, clear a list to accept new data, or perform math operations on lists, use [STAT]. LIST is a way to select lists of data for use in other functions. For example, if you wanted to add items in list L₁ to those in list L₂ and store the result in list L₃, your key sequence could be:

2^nd [STAT]   L₁   [+]   2^nd [STAT]   L₂   STO→   2^nd [STAT]   L₃

TI-84: Entering lists

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TI-84: Finding the correlation coefficient

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A programming example

Here is an example of how to code something like a correlation coefficient calculation. This is a simple Python program that calculates the correlation coefficient for the same data as the spreadsheet example above. The data is in the arrays x and y. You can click/tap on the code to download a text copy to run yourself.

__author__ = 'Your name'

# program: correlation.py for xaktly.com/Correlation.html (this is a comment)

#!/user/local/bin/python
#Include "Python.h"

import math   # Needed for the square-root function

nData = 38    # Number of data points in each array

x = {20,21,23,28,29,29,31,33,33,34,36,40,40,41,42,43,44,
     44,44,45,47,48,50,51,52,55,58,61,62,66,67,67,72,74,77,80}
     
y = {204,189,200,191,198,184,188,188,189,195,178,186,178,184,172,
     171,183,174,170,181,169,175,165,164,173,174,176,163,156,155,
     159,149,160,156,142,139,149,134}
     
xsum  = 0.0       # sum of all x[i]
ysum  = 0.0       # sum of all y[i]
x2sum = 0.0       # sum of all (x[i] - xavg)^2
y2sum = 0.0       # sum of all (y[i] - yavg)^2
xavg  = 0.0       # average value of x's
yavg  = 0.0       # average value of y's
xysum = 0.0       # sum of (x[i] - xavg)(y[i] - yavg)

i = 0

for i in range (0, nData):
    xsum += x[i]
    ysum += y[i]
    
xavg = xsum / nData
yavg = ysum / nData

for i in range (0, nData):
    x2sum += (x[i] - xavg) * (x[i] - xavg)
    y2sum += (y[i] - yavg) * (y[i] - yavg)
    xysum += (x[i] - xavg) * (y[i] - yavg)
    
# calculate r, remembering to take roots:

r = xysum / ((math.sqrt(x2sum)) * (math.sqrt(y2sum)))

print 'correlation coefficient: r = ', r
      

More complicated data sets

Often we work with data sets in which an dependent variable depends on more than one independent variable. Maybe it's a data set that models, say, a six dimensional function,

$$y = f(x_1, x_2, x_3, x_4, x_5, x_6)$$

In such a case, we would want to get an idea about the independence of the variables x₁ ... x_n. To do that we can calculate (I won't say how here) a matrix of correlation coefficients. It might look like this one (which is made up).

Notice that x₁ is perfectly correlated with x₁, and so on, those coefficients lying along the diagonal (upper left to lower right) of our matrix.

In analyzing such a matrix, we're making sure that (1) the positive and negative coefficients are more or less randomly distributed, i.e. that there's no significant bias one way or the other.