Quantitative Reasoning

The Course

Quantitative Reasoning

Covers collection of data and analysis of everyday quantitative information using spreadsheets or statistical packages. Touches upon population vs. sample, parameter vs. statistic, variable type, graphs, measures of center and variation, regression analysis, and hypothesis testing.

Zeynep Teymuroglu

Assistant Professor of Mathematics Zeynep Teymuroglu received her Ph.D. at the University of Cincinnati. Her Ph.D. research was in applied and computational mathematics with an emphasis on mathematical biology. After graduation, she worked as an Associate Manager at the Research and Development department of a market research company. She also has her master’s in mathematics with an emphasis on Financial Mathematics.  Her research interests are mathematical biology, mathematical modeling and financial mathematics. She teaches Calculus, Differential Equations and Applied Mathematics courses.

Project Mosaic: Mississippi

Project Mosaic: Mississippi

By: Kathleen Boyd, Renee Fiorot, Brad Geisen, Kyle McCoy, and Ricci Prioletti

Our class worked in cooperation with Project Mosaic in order to “promote a synergistic dialogue among faculty and enhance student understanding of the Africa and African-American experience (http://www.rollins.edu/aaas/mosaic.html).” Our group worked with Mississippi data ranging from the year 1790 to1960 and dealt with information regarding both the slave population as well as free African-Americans.  We also analyzed data surrounding farm life and livestock, which gives us more insight into the lives of the people at this time.  Our classmates dealt with similar data from different states, which, in the spirit of Project Mosaic, will lead to more open dialogue between students on the matter.  We analyzed data using skills learned in our quantitative reasoning class and specifically utilized our knowledge of Microsoft Excel technology to create graphs and spreadsheets that will help further our understanding. 

The first graph in our series examines the total general population of Mississippi in contrast to the total slave population of Mississippi.  In 1820, both the general population and the slave population were relatively low and the slave population only equaled about half that of the general population.  Within the next ten years, both populations grew slightly but remained relatively the same in relation to each other.  In 1840, there was a drastic increase in population, nearly doubling both the general population and the slave population.  At this point, the slave population is still roughly half that of the general population.  In 1850 and 1860, the general population and the slave population steadily increased while the relationship between the two remained roughly the same.  In 1870 there is no longer a slave population whatsoever.  The general population continued to increase (with the exception of a slight decrease in 1920) until our data ends in the year 1950.  Our data begins in 1820 where the general population consists of roughly 125,000 people.  In 1880, the population was over 1,000,000 and by 1930 the population hit 2,000,000.  The median number of people, or the middle number of all the people in Mississippi within the time frame of our data, was 979,759.5.  Over the span of the years that our data covers, the average number of people living in Mississippi was 1,051,352.

The next graph includes data dealing with the total number of farms and the acres of improved land. In regards to agriculture, the term “improved land” refers to any area of land that has been prepared for farming.  Both continued to increase in number even after slavery was abolished and, as the population was also increasing, it would be necessary for the size of these farms to increase as well.  As the value of acres of improved land increases, the total number of farms will increase by 0.431.  According to the data, when the value of acres of improved land is 0, there would be a total of 131,194 farms. This statistical measure is less useful to our hypothesis however, because, if there were no acres of improved farmland, there would be no farms. The graph shows us that there is a positive regression line with a value close to one, which means our information is accurate and will continue to increase within the next ten years.

Collectively, our data displays a pattern of positivity or increase.  This is similar to the way our world generally works.  As time goes on, population increases.  As population increases, demand, as well as consumption, increases.  It is not any wonder, then, that, as Mississippi’s population increased between the years of 1790 and 1960, the state’s people had a higher demand for life’s necessities.  A human being needs to nourishment, provisions, and living space.  A man running a farm could feed himself, provide income for himself, and live on the farm.  Mississippi, a hot agricultural state of the south, would provide ideal opportunities for a man to meet the needs of him and his family.  Life is survival and, in an early, agricultural world, running a farm was the way to survive. 

Where our data became interesting is in the development of the state’s land in the twenty-year span between 1860 and 1880.  What we expected from a southern state in the years both before and following this two-decade span was for an increase in improved land, agricultural production, and farm development.  Our graphs illustrate that well.  What our graphs also illustrate is the correlation between the slave population’s sudden disappearance and Mississippi’s brief drop in the high level of agricultural productivity.  Between 1850 and 1860 the acres of improved farm land increased by nearly 2,000,000.  The slave population then ceases to exist.  There is no increase but rather an approximately 1,000,000 acre decrease in improved farm land between 1860 and 1870.  The same can be seen in the amount of actual farms being established during those years.  It should be obvious, then, that the abolition of slavery was a minor setback in Mississippi’s agricultural economy.  However, we noted that after the year 1870, the 1,000,000 acre deficit was replaced entirely and only increased from there.  Thus a one-decade minor setback was all Mississippi experienced from the abolition of slavery in terms of the states ability to produce and develop. 

Using the information gathered from our graphs and statistical data, we were able to analyze facts about the historical, agricultural, economic, and general human growth (or demise) between 1790 and 1960.  The conclusions we drew are relevant to our understanding of the world that we live in: it is only ever getting bigger and more populated.  As population increases, demand also increases, which causes development to increase, which is followed again by an increase in population.  In other words, it is a never-ending circle of human life and productivity.  The data we gathered following Mississippi’s establishment and development as a U.S. state between 1790 and 1960 illustrates that point perfectly.

 

 

Descriptive Statistics Slave Population

   
Mean

130024.125

Standard Error

58996.97598

Median

49236.5

Mode

0

Standard Deviation

166868.6471

Sample Variance

27845145398

Range

436631

Minimum

0

Maximum

436631

Sum

1040193

Count

8

 

 

Descriptive Statistics of Total Population

   
Mean

1051352.444

Standard Error

198318.6535

Median

979759.5

Mode

0

Standard Deviation

841394.7884

Sample Variance

7.07945E+11

Range

2183796

Minimum

0

Maximum

2183796

Sum

18924344

Count

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Descriptive Statistics Improved Land

Descriptive Statistics of Total Farms

Mean

5957955.857

Mean

125752.1429

Standard Error

779741.7616

Standard Error

35077.93703

Median

5216937

Median

101772

Mode                      N/A Mode

#N/A

Standard Deviation

2063002.788

Standard Deviation

92807.49789

Sample Variance

4.25598E+12

Sample Variance

8613231664

Range

5881319

Range

240422

Minimum

3444358

Minimum

33960

Maximum

9325677

Maximum

274382

Sum

41705691

Sum

880265

Count

7

Count

7

 

Project Mosaic: Missouri

By: Tayler Mitchell, Claire Talley, Ashton Lange, Shelby Stephenson, Jenny Skarda

Intro:

By looking through data provided by Project Mosaic we were able to discover and compare the general population of Missouri with their slave population between 1820 and 1960. We viewed this data through different aspects such as education and manufacturing establishments. When we first began this project at the beginning of the term we were unfamiliar with the population history of Missouri. We had picked Missouri because none of us knew much about it and thought it would be interesting to learn more about the history. As we researched Missouri, we found the slave population to be surprisingly high for the Midwest during the 1800’s because in 1830 Missouri slavery population peaked at 18%.
Analysis:

After reviewing Missouri’s data we found that the minimum population value was in 1820 at 66,586 people.  Between the years of 1850 and 1860 the 25th percentile of the population was 932,028 people.  The median of the population was 2,679,184 people in 1890.  Between 1920 and 1930 the 75th percentile of the population was 3,516,711 people.  The max amount of population was 4,319,813 people in 1960.  The average population between the years 1820 and 1960 was 2,301,081 people. Over the fifteen years the population varies by 1,496,018 people. Over the years the sum of the population adds up to 34,516,210 people.

In Missouri the average total population from 1820-1960 is 34,516,210 people. However, the median was 2,679,184 people, meaning it was lower than the average population because of possible outliers. This shows when one looks at the range, which is 4,253,227 people, the range is significantly higher than the mean and median suggesting there are outliers. The lowest population number is 66,586 people, occurring in 1820, while the highest population number is 4,319,813 people, occurring in 1960. Throughout the span of 1820-1960 the population deviated from the mean by 14,960,818 people, meaning how far away it varied from the mean each year. This data represents that 25% of the population is under 932,028 people (25th percentile) and 75% of the population is under 4,319,813 people (75th percentile).

 

The graph above shows the illiteracy of colored males ages 21 and over versus the number of manufacturing establishments between 1850 and 1940.  By looking at the graph above one can ascertain that the linear relationship is positive. This can be proven by the positive slope (2.8078) in the regression line equation.  The slope also shows that if one unit were added to the equation then the dependent variable would increase by 2.8078.  If the independent variable was zero, then the amount of the y intercept would be -14385, however this is not possible during this time period there would never be zero illiterate colored males ages 21 and over. The regression coefficient (0.9902) shows us that the graph is a good representation of the data collected.  The reason it is a good representation of the data is because it is close to one.

The graph above shows us how manufacturing establishments between 1860 and 1940 change over the allotted time period. The regression coefficient is 0.02396 and this shows us that it is an extremely poor model of the data collected and would not be very accurate. If used it would be only 2% accurate. By observing the graph one can see that the linear relationship is negative making the slope negative. The slope is -27.799 in the regression line equation. Studying the regression line equation, the y-intercept is 62,231. To better understand what this equation means, for example, if the independent variable was zero then y would be 62,231, however this is impossible because we are measuring time in years so we would never be at zero years. Also if there were one unit change in x then the dependent variable (y) would decrease by 27.799 because as said before the slope is negative.

            The population of illiterate colored males, ages 21 and over, between 1850 and 1920 in the state of Missouri is shown in the graph above. By looking at the graph above one can ascertain that the linear relationship is positive. This can be proven by the positive slope (122.09) in the regression line equation. The slope also shows that if one unit was added to the independent variable (time) then the dependent variable (numbers of illiterate colored males ages 21 and over) increases by 122.09. If the independent variable is zero, then the y-intercept is -215,281 however, this is not possible because we are measuring time and there will never be a year zero. The regression coefficient (0.0611) shows us that the graph is a poor representation of the data, because it is less than 0.4.

 

 

Conclusion:

Throughout our research on Missouri we focused on the illiteracy of colored males and manufacturing establishments. We concluded that as the rate of illiteracy increased the rate of manufacturing establishments increased as well.  After looking at the history we determined that colored males who were not literate ages 21 and over were not continuing education and going after the higher end jobs. Therefore they resorted to working in factories. These males working in the factories increased the amount of manufacturing establishments that were created.

Project Mosaic: Virginia Statistical Analysis

Introduction

Our group looked at the total population, slave population, and agricultural data (number of farms) of Virginia for the years 1790-1960. Our group chose Virginia because we are aware of the states involvement in slavery, and its large role in agriculture in early America. Prior to conducting out research, we were also aware of the large role that slave labor played in agricultural production. We wanted to investigate statistical evidence to see if it supported our theory that slave population and the number of farms were correlated. We also wanted to see how agriculture progressed after the abolition of slavery and ultimately what effect slavery had on the agricultural progression in Virginia. In this paper, we used descriptive statistics, graphs, statistical analysis, and historical context to tell a story about Virginia from 1790 until 1960. This story tells the impact which each of our variables of population, slave population, and number of farms all had on each other. We learned the web in which each of these variables in intertwined and they inherent effect on each other.

 

Year

Total

Agriculture

Slave

1790

747,550

292,627

1800

885,171

346,671

1810

974,622

392,518

1820

1,065,379

425,153

1830

1,211,405

469,757

1840

1,239,797

449,087

1850

1,421,661

76,943

472,528

1860

1,596,318

86,468

490,865

1870

1,225,163

73,849

1880

1,512,565

118,517

1890

1,655,980

127,600

1900

1,854,184

167,886

1910

2,061,612

184,018

1920

2,309,187

186,242

1930

2,421,851

170,610

1940

2,644,250

1950

3,318,680

150,997

1960

3,966,949

Mean

1,784,018

134,313

371,023

Median

1,554,442

139,299

425,153

Mode

#N/A

#N/A

#N/A

Min

747,550

73,849

0

Max

3,966,949

186,242

490,865

Q1

1,214,844.5

94,480.25

346,671

Q3

2,247,293.25

169,929

469,757

 

The above collected data about Virginia’s total population, slave population, and number of farms comes from the University of Virginia Library Database. Their “Historical Census Browser” can be used to find a plethora of statistical information throughout American history. This data is valid because the source of information is credible and reliable. Unfortunately, there was no available agricultural data from the years 1790 to 1840, and there were also missing entries for agricultural data in 1940 and 1960. Later in this paper we use statistical methods to estimate those numbers, allowing us to paint a more holistic picture of Virginia during this time period.

            The 5 number summary above, provide us with quick, accessible information about Virginia’s total population, slave population, and agriculture (number of farms). By using the excel formula to find the minimum, maximum, 1st quartile, 3rd quartile, mode, median, and mean, we can easily interpret a large table to data. For example, the minimum number for the column ‘total population’ is 747,550. One can then look at the table and see that number appears for the year 1790, meaning the lowest total population was during that year. This information also gives us the ability to make direct comparisons between each of our three variables. Particularly, by comparing which years the ‘max’ numbers of population occurred.

 

Claim #1:

Our first claim was an increase in total population would cause an increase in slave population. In the graph above, one can see that this is in fact the case. As Virginia’s total population steadily increased from 1790 to 1860, so did its slave population.

 

Claim #2:

Our original hypothesis was that as the total population of Virginia increased, so would the number of farms. Because R² is only .43036, the correlation between total population and number of farms is only a moderate one. This is because eventually no matter how much the population increases, there will only be so much farmland. Ultimately there are inherent limitations on this correlation given the physical geographic space Virginia occupies.

 Data Limitations:

Because of the lack of data on agriculture from 1790 to 1840 and the missing years of data for 1940 and 1960, we used a regression line based off of a table comparing time, to number of farms, presented below.

 

 

Year

Total

Agriculture

Slave

1790

747,550

22,163

292,627

1800

885,171

 33,460

346,671

1810

974,622

 44,757

392,518

1820

1,065,379

 56,054

425,153

1830

1,211,405

 67,351

469,757

1840

1,239,797

 78,648

449,087

1850

1,421,661

76,943

472,528

1860

1,596,318

86,468

490,865

1870

1,225,163

73,849

0

1880

1,512,565

118,517

1890

1,655,980

127,600

1900

1,854,184

167,886

1910

2,061,612

184,018

1920

2,309,187

186,242

1930

2,421,851

170,610

1940

2,644,250

191,618

1950

3,318,680

150,997

1960

3,966,949

 <150,997

 Fortunately when we used the regression model to estimate the missing entries, we were consequently able to compare agriculture (number of farms) to the slave population and total population.

 

  Claim#3:

Our claim was that as slavery increased, so would the number of farms. Because one of the primary jobs of slaves was to yield crops on Virginia’s fertile land, it would only be logical that these two factors would directly correlate. Because the regression line demonstrates that R² is .931, or very close to 1, we can conclude that there is a very strong correlation between these two factors.

 Virginia’s Agricultural Fluctuation and the Factors that caused it

As one can see, Virginia has had a consistent increase in total population every year since 1790, with the exception of the decades 1860-1870. This can be attributed to the abolition of slavery in 1860, and therefore the fleeing of blacks from this area to head out west and start a land of their own, away from their previous owners. Agriculture, (measured by the total number of farms) also steadily increased every year since 1850 with only two exceptions. The first exception is the decline in the number of farms between 1860 and 1870. This can be attributed to the decline and slavery and essentially the decline in free labor to run farms in Virginia. With the sudden loss of this free labor, the number of farms decreased by almost 10,000 between these decades. Eventually more sustainable practices of minimally paid labor were instilled and the number of farms began to grow again. It was not until the 1940s when the number of farms again declined. It is not a coincidence that this decrease in farms happened during the industrial revolution, leading many workers to factory jobs. With technology replacing the need for labor in the fields, the number of farms declined and many people sought jobs in the industrial sector.

 Honor Code: Electronically Signed by Virginia Group

Authors:
Randi Alberry
Emily Gildden
Andrea D’Alfonso
Rohan Shroff
Andrew Lesmes

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