Estimation I: One Variable Statistics

We must be careful not to confuse data with the abstractions we use to analyze them.

• William James, 1907

In this lab, we will use Python to calculate sample statistics and graphical representations for a dataset that represents the length of each Roman Emperor’s reign. we will use these statistics and graphs to draw conclusions about the distribution of Roman Emperor reigns.

Instructions

1. Download the .csv dataset below and place it in the Linux Files folder where you saved your .py scripts.

2. In the same folder, create a Python .py script named NAME_project_three.py in your Linux Files folder on your file system. You can do this by opening an IDLE session, creating a new file and then saving it. Replace NAME with your name.

3. Create a docstring at the very top of the script file. Keep all written answers in this area of the script.

4. Read the Background section.

5. Read the Imports section.

6. Read the Functions section.

7. Read the Loading In Data section.

8. Load in the data from the .csv file using the technique outlined in the Loading In Data section.

9. Perform all exercises and answer all questions in the Project section. Label your script with comments as indicated in the instructions of each problem.

10. When you are done, zip your script and the csv file in a zip file named NAME_project_three.zip

11. Upload the zip file to the Google Classroom Project Three Assignment.

Background

In 27 BC, Julius Caesar’s nephew, Octavian Caesar, was given the title Augustus by the Roman Senate. With this event, the city-state of Rome ceased to be a republic. Augustus Caesar became the first emperor of the Roman Empire.

The Roman Empire would last for several centuries, absorbing large parts of what is now Europe, Africa and Asia into its boundaries.

Evolution of the Roman Empire

Rome from the city-state by the Tiber to the fall of the Western Roman Empire

Source: Roman Empire

Many people would go on to claim the title of Roman Emperor after Augustus Caesar died.

Some of them, such as Hadrian and Marcus Aurelius had long and peaceful reigns. Others, such as Marcus Otho and Didius Julianus, only ruled for several months and their reigns were fraught with political upheaval and violence.

In this project, we are going to look at a dataset that represents the length of each Roman Emperor’s reign. We will look at the distribution shape and various descriptive statistics of this dataset in order to draw conclusions and tell a story about the leadership of the Roman Empire using empirical evidence.

Imports

To complete this lab, you will need to import the following packages. Add these lines to the top of your .py script.

import matplotlib.pyplot
import math

Functions

Note

We will do this part in class together.

Important

Refer to Creating Functions section for a more information on creating your own function in Python.

We have been only been using functions up to this point. In order to complete this lab, we will need to create a few functions that will calculate sample statistics. In particular, we are going to create four functions: sample_mean, sample_std_deviation, sample_median and sample_percentile. These functions will accept a sample of data as an argument (or input) and then calculate a sample statistic and return its value.

We can create the sample_mean function as follows,

def sample_mean(data):
    n = len(data)
    sumx = sum(data)
    xbar = sumx / n
    return xbar

Take note of the indentation. Each line of the function is on the same indentation level. This is how Python separates functions from the commands you are executing. For more information about the syntactical components of a function defintion, refer to the Creating Functions section.

Now that we have defined our sample mean function, we can call it using its name,

some_data = [ 1, 2, 3, 4, 5 ]
xbar = sample_mean(some_data)
print("the sample mean is ", xbar)

Output

the sample mean is 3.0

In this project, we will create two more functions:

• A function to calculate the sample standard deviation

• A function to calulate a sample percentile.

Loading In Data

The following code snippet will load in a CSV spreadsheet named example.csv, parse it into a list and then print it to screen, assuming that CSV file is saved in the same folder as your script. Modify this code snippet to fit the datasets in this lab and then use it to load in the provided datasets in Datasets section.

import csv

# read in data
with open('example.csv') as csv_file:
    csv_reader = csv.reader(csv_file)
    raw_data = [ row for row in csv_reader ]

# separate headers from data
headers = raw_data[0]
columns = raw_data[1:]

# grab first column from csv file and ensure it's a number (not a string)
column_1 = [ float(row[0]) for row in columns ]

print(column_1)

Project

1. Write a function that accepts a list of data as an argument and computes the following sample statistics. Write a separate function for each exercise and label it with a comment. Name the function appropriately.

   1. The sample mean of a dataset.

   2. The sample percentile of a dataset.

   3. The sample standard deviation of a dataset.

Tip

#1b will require two arguments, the list of data and the percentile you wish to find.

Note

We will do this part in class together.

2. Load in the data from the Dataset section. Note the length of a reign is separated in a Years column, a Months column and a Days. To clean the data and compute the total length of a Roman Emperor’s reign, apply the formula to each row of data,

Save the cleaned data in a new list. Label the list with a comment.

3. Using the functions created in #1, find the following statistics using the Dataset. Label each computation with a comment.

   1. The mean length of a Roman Emperor’s reign.

   2. The median length of a Roman Emperor’s reign.

   3. The 25 ^th percentile length of a Roman Emperor’s reign.

   4. The 75 ^th percentile length of a Roman Emperor’s reign.

   5. The sample standard deviation of a Roman Emperor’s reign length.

4. Compare the answers to #2a and #2b. What do these two answers tell you about the skew of this distribution? Interpret the skew in terms of Roman Emperors and the length of their reign, i.e. what does the skew tell you about Roman Emperor’s and the length of their reigns? Save your answer in the docstring.

5. Construct a relative frequency histogram and a cumulative relative frequency using 10 classes for this sample of data. Label the code for creating the plots with a comment. What type of distribution shape does this dataset have? Does this agree with your answer to #4? Explain. Save your answer in the docstring.

6. Construct a boxplot for this sample of data. Label the code for creating the plot with a comment. Based on the boxplot, are there any potential outliers in this dataset? Are the outliers Emperors who had long rules or short rules? Save your answer in the docstring.

6. Find the coefficient of variation for this dataset. What does this statistic tell you about the distribution? Interpret the coefficient of variation in terms of Roman Emperors and the length of their reign. Save your answer in the docstring.

7. Summarize the conclusions you can draw about Roman Emperors and the length of their reign in your docstring. Answer the following questions in your summary.

   1. What percentage of Roman Emperors had reigns longer than 30 years?

   2. What percentage of Roman Emperors had reigns shorter than 1 year?

   3. Interpret the results of #a and #b. What does this tell you about the distribution of Roman Emperors?

8. Based on the graphs and descriptive statistics calculated in the previous problems, write a few sentences in the docstring describing what the distribution of Roman Emperor reigns tells us about the Roman state.

Dataset

You can download the full dataset here.

The following table is the a preview of the data you will be using for this project.

Roman Emperor Reigns

Emperor	Years	Months	Days
TheodosiusII	42	2	27
Augustus	41	7	3
ConstantineI	30	10	27
ValentinianIII	29	7	22
Honorius	30	7	8

The meaning of the columns is as follows:

• Emperor is the name of the Roman Emperor.

• Years is the number of years in the reign.

• Months is the number of months in the reign.

• Days is the number of days in the reign.