Summarizing data : The M&M Problem

M&M's is a registered trademark of Mars, Inc. For more information, see http://www.mms.com (as of 2004-01-11).

An M&M represents one data sample from one package of M&M's. Here, it is assumed that a package of M&M's contains 60 M&M's.

The M&M colors represent nominal data in that there is preferred ordering or numerical value that can be associated with the M&M's.

Here is a typical package contents in text form.

Red Yellow Brown Green Green Blue Green Brown Orange Red Yellow Red Brown Red Blue Yellow Green Yellow Yellow Blue Yellow Brown Orange Red Green Orange Green Red Green Red Brown Brown Orange Red Blue Red Red Yellow Green Orange Red Blue Orange Red Red Blue Red Red Red Blue Blue Yellow Yellow Red Red Blue Blue Red Orange Red

Here is the raw data for this problem as a list of colors.

Red Yellow Brown Green Green Blue Green Brown Orange Red Yellow Red Brown Red Blue Yellow Green Yellow Yellow Blue Yellow Brown Orange Red Green Orange Green Red Green Red Brown Brown Orange Red Blue Red Red Yellow Green Orange Red Blue Orange Red Red Blue Red Red Red Blue Blue Yellow Yellow Red Red Blue Blue Red Orange Red

Often, a TPS (Transaction Processing System) is used to automatically collect the data and put it into the computer.

A spreadsheet can be used as a TPS.
A database can be used as a TPS.
A program can be used as a TPS.

It is better to automate the collection of data rather than use a human to manually enter data into the computer. What is a TPS and why is a TPS important?

One could write a Python program to summarize the data by color.

There are some built-in ways to summarize the data. Here is the Python code [#1]

import matplotlib.pyplot as plt
import rsPlot
dataList1 = [
	"Red","Yellow","Brown","Green","Green","Blue","Green","Brown","Orange","Red",
	"Yellow","Red","Brown","Red","Blue","Yellow","Green","Yellow","Yellow","Blue",
	"Yellow","Brown","Orange","Red","Green","Orange","Green","Red","Green","Red",
	"Brown","Brown","Orange","Red","Blue","Red","Red","Yellow","Green","Orange",
	"Red","Blue","Orange","Red","Red","Blue","Red","Red","Red","Blue",
	"Blue","Yellow","Yellow","Red","Red","Blue","Blue","Red","Orange","Red",
	]
countDict1 = {}
for color1 in dataList1:
	if not color1 in countDict1:
		count1 = 0
	else:
		count1 = countDict1[color1]
	countDict1[color1] = count1 + 1
countList1 = []
for color1 in countDict1:
	count1 = countDict1[color1]
	countList1.append((count1,color1))
print("Count Color")
for count1,color1 in countList1:
	print("\t{0:d} {1:s}".format(count1,color1))
xData1 = []
yData1 = []
cList1 = []
sum1 = 0
for count1,color1 in countList1:
	xData1.append(color1)
	yData1.append(count1)
	cList1.append(color1.lower())
plot1 = rsPlot.plot("mm11-1",1,0,72,640,480)
plt.title("M&M counts by color")
plt.xlabel("Color")
plt.ylabel("Count")
plt.bar(xData1, yData1,color=cList1)
plot1.save()

Here is the output of the Python code.

(module rsPlot imported)
Count Color
	20 Red
	9 Yellow
	6 Brown
	8 Green
	10 Blue
	7 Orange
	./mm11-1-01.png : 594 x 462 , 24,067 bytes

The Pareto chart is named after Vilfredo Pareto, 1848-1923, an Italian economist.

Vilfredo Pareto (1848-1923) made the observation that 20% of the population owned 80% of the wealth.

This idea has been called Pareto's principle and has been applied in many other areas.

More: Pareto principle: 80-20 rules

A Pareto chart is a combination chart that shows an individual count or frequency using a column bar chart, ordered in decreasing order from left to right, and a cumulative distribution using a line chart.

To get a Pareto chart, the following changes need to be made.

The colors need to be sorted in descending order by count.
A cumulative line graph needs to be added.

Here is the Python code [#2]

import matplotlib.pyplot as plt
import rsPlot
dataList1 = [
	"Red","Yellow","Brown","Green","Green","Blue","Green","Brown","Orange","Red",
	"Yellow","Red","Brown","Red","Blue","Yellow","Green","Yellow","Yellow","Blue",
	"Yellow","Brown","Orange","Red","Green","Orange","Green","Red","Green","Red",
	"Brown","Brown","Orange","Red","Blue","Red","Red","Yellow","Green","Orange",
	"Red","Blue","Orange","Red","Red","Blue","Red","Red","Red","Blue",
	"Blue","Yellow","Yellow","Red","Red","Blue","Blue","Red","Orange","Red",
	]
countDict1 = {}
for color1 in dataList1:
	if not color1 in countDict1:
		count1 = 0
	else:
		count1 = countDict1[color1]
	countDict1[color1] = count1 + 1
countList1 = []
for color1 in countDict1:
	count1 = countDict1[color1]
	countList1.append((count1,color1))
countList1.sort(reverse=True)
print("Count Color")
for count1,color1 in countList1:
	print("\t{0:d} {1:s}".format(count1,color1))
xData1 = []
yData1 = []
cList1 = []
cData1 = []
sum1 = 0
for count1,color1 in countList1:
	xData1.append(color1)
	yData1.append(count1)
	cList1.append(color1.lower())
	sum1 += count1
	cData1.append(sum1)
plot1 = rsPlot.plot("mm11-2",1,0,72,640,480)
plt.title("M&M counts by color")
plt.xlabel("Color")
plt.ylabel("Count")
plt.bar(xData1, yData1,color=cList1)
plt.plot(xData1, cData1,color="#000000",linestyle="--", marker="o")
plot1.save()

Here is the output of the Python code.

(module rsPlot imported)
Count Color
	20 Red
	10 Blue
	9 Yellow
	8 Green
	7 Orange
	6 Brown
	./mm11-2-01.png : 573 x 462 , 27,412 bytes

The above has shows some ways of summarizing nominal data using the M&M problem.