In statistics, the degrees of freedom (DF) indicate the number of independent values that can vary in an analysis without breaking any constraints. It is an essential idea that appears in many contexts throughout statistics including hypothesis tests, probability distributions, and regression analysis. Learn how this fundamental concept affects the power and precision of your statistical analysis!

In this blog post, I bring this concept to life in an intuitive manner. I’ll start by defining degrees of freedom. However, I’ll quickly move on to practical examples in a variety of contexts because they make this concept easier to understand.

Definition of Degrees of Freedom

Degrees of freedom are the number of independent values that a statistical analysis can estimate. You can also think of it as the number of values that are free to vary as you estimate parameters. I know, it’s starting to sound a bit murky!

What Are Degrees of Freedom?

Degrees of Freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample.https://919619a599ea8cd98fd19e7706ffe8f0.safeframe.googlesyndication.com/safeframe/1-0-38/html/container.html

KEY TAKEAWAYS

• Degrees of Freedom refers to the maximum number of logically independent values, which are values that have the freedom to vary, in the data sample.
• Degrees of Freedom are commonly discussed in relation to various forms of hypothesis testing in statistics, such as a Chi-Square.
• Calculating Degrees of Freedom is key when trying to understand the importance of a Chi-Square statistic and the validity of the null hypothesis.

Understanding Degrees of Freedom

The easiest way to understand Degrees of Freedom conceptually is through an example:

• Consider a data sample consisting of, for the sake of simplicity, five positive integers. The values could be any number with no known relationship between them. This data sample would, theoretically, have five degrees of freedom.
• Four of the numbers in the sample are {3, 8, 5, and 4} and the average of the entire data sample is revealed to be 6.
• This must mean that the fifth number has to be 10. It can be nothing else. It does not have the freedom to vary.
• So the Degrees of Freedom for this data sample is 4.

The formula for Degrees of Freedom equals the size of the data sample minus one:

begin{aligned} &text{D}_text{f} = N – 1 \ &textbf{where:} \ &text{D}_text{f} = text{degrees of freedom} \ &N = text{sample size} \ end{aligned}​Df​=N−1where:Df​=degrees of freedomN=sample size​

Degrees of Freedom are commonly discussed in relation to various forms of hypothesis testing in statistics, such as a Chi-Square. It is essential to calculate degrees of freedom when trying to understand the importance of a Chi-Square statistic and the validity of the null hypothesis.

Chi-Square Tests

There are two different kinds of Chi-Square tests: the test of independence, which asks a question of relationship, such as, “Is there a relationship between gender and SAT scores?”; and the goodness-of-fit test, which asks something like “If a coin is tossed 100 times, will it come up heads 50 times and tails 50 times?”

For these tests, degrees of freedom are utilized to determine if a certain null hypothesis can be rejected based on the total number of variables and samples within the experiment. For example, when considering students and course choice, a sample size of 30 or 40 students is likely not large enough to generate significant data. Getting the same or similar results from a study using a sample size of 400 or 500 students is more valid.

History of Degrees of Freedom

The earliest and most basic concept of Degrees of Freedom was noted in the early 1800s, intertwined in the works of mathematician and astronomer Carl Friedrich Gauss. The modern usage and understanding of the term were expounded upon first by William Sealy Gosset, an English statistician, in his article “The Probable Error of a Mean,” published in Biometrika in 1908 under a pen name to preserve his anonymity.

In his writings, Gosset did not specifically use the term “Degrees of Freedom.” He did, however, give an explanation for the concept throughout the course of developing what would eventually be known as Student’s T-distribution. The actual term was not made popular until 1922. English biologist and statistician Ronald Fisher began using the term “Degrees of Freedom” when he started publishing reports and data on his work developing chi-squares.

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1. Columbia University

Columbia University has an Applied Mathematics Master program that made our list of the best online masters in statistics. In the program, elearners become applied mathematicians that solve problems driven by outside forces. Applied mathematicians deal with modeling and abstract solutions. With a faculty to student ratio of 6:1, Columbia presents an ideal learning environment for students with highly qualified faculty member teachers one of the country’s best online statistics masters.

Besides having one of the best online masters in mathematics, Columbia receives an A+ scorecard for factors including an 87% faculty approval rating and high availability for needed courses. Although students agree that Columbia presents challenging coursework, that translates into a stronger branding for successful graduates. 

HIGHLIGHTColumbia has a 96% graduation rate, which is extraordinary among similar universities, and 89% find suitable employment following graduation

DEGREEApplied Mathematics Master 

2. Texas A&M University

The online Statistics MS at Texas A&M University has one of the best online masters in statistics in the country and allows learners to continue their careers while pursuing an advanced degree. Skills learned in the program include a capstone project spanning the last two semesters. Past projects include price modeling and ratings for Airbnb rentals, modeling NFL draft and combine results and modeling average airfare for the United States. This unique optional project gives graduates invaluable real-world experience and helps identify Texas A&M as one of the best online statistics masters degrees.

Texas A&M is a well-respected public school with an enrollment of nearly 50,000 students. While admission is competitive, 68% if applicants meet the school’s entry requirements. Further, 82% of students successfully finish their coursework and go on to earn respectable salaries averaging $48,600.

HIGHLIGHTTexas A&M is ranked #22 among American universities and is one of the best online masters in mathematics available.

DEGREEStatistics, MS

3. Oklahoma State University

One of the best online statistics masters, the Applied Statistics MS at Oklahoma State University gives students an opportunity to schedule coursework around family and other obligations. This degree is a terminal degree and not meant to prepare students for doctoral degrees. Courses include two classes each in analytics, experimental data analysis and applied mathematical statistics, as well as coursework in SAS and R computer programming with SAS and R. Its concentration on statistical analysis makes this OSU program one of the best online masters in statistics in the country.

Oklahoma State receives high marks for its focus on academics, diversity and value for money. With an acceptance rate of 74%, the university provides competitive entry requirements while remaining accessible to three-quarters of all applicants. Four out of five students give faculty members a thumbs up on preparation and content knowledge, and 91% of students get financial aid to help with the cost of tuition and other expenses. 

HIGHLIGHTIts concentration on statistical advanced mathematics puts OSU on our list of the best online masters in mathematics.

DEGREEApplied Statistics, MS

4. Colorado State University

If you would like to earn an MAS in Applied Statistics, consider Colorado State University, one of the best online masters in statistics available. Students choose from statistical science or data science concentrations and gain tools to help them design studies, make inferences from data and translate insights into meaningful contexts. Elearners find out how to interpret graphics and data and translate this into actionable points for decision-making laymen. 

Along with making our ranking of the best online statistics masters programs, Colorado State offers students diverse student body in both on-campus and online courses. It offers competitive tuition rates for the area as well as a sound educational experience, thanks to its qualified faculty team.

HIGHLIGHTAs one of the best online masters in mathematics, Colorado State’s Applied Statistics program prepares graduates to work in industry, government and education settings.

DEGREEApplied Statistics, MAS

5. North Carolina State University

North Carolina State University’s Master of Statistics program is in good company among the best online masters in statistics. This flexible online offering features an MS and graduate certificates in Applied Statistics, Data Management or Statistics Education. Graduates develop marketable skills that meet the needs of multiple industries, and the schools’ strong focus on working with diverse data types and various programming language increase the ROI, solidifying its ranking as one of the best online statistics masters.

Beyond NCSU’s best online masters in mathematics status, the school is ranked #80 in national universities surveyed by U.S News and World Report. This ranking is based on numerous factors contributing to an overall environment of educational excellence. 

HIGHLIGHTThe Raleigh area offers graduates unprecedented opportunities for internships and job prospects and has a reasonable cost of living.

DEGREEMaster of Statistics

6. University of Delaware

The University of Delaware has an Applied Statistics MS that is among the best online masters in statistics. From the comfort of their favorite cafe or couch, students transform into data professionals and meet increasing demands for qualified data scientists. The 100% online offering gives elearners a theoretical foundation in statistics and advanced preparation in model building and data for real analysis. Students who desire long-term success and a flexible learning opportunity value the best online statistics masters, including this one at the University of Delaware.

Ranked #33 among U.S. News and World Reports public university survey, the University of Delaware receives an A rating for academics, diversity and return on investment for Niche.com. Two-thirds of applicants receive the nod from the admissions committee, and UD undergrads choose finance, liberal arts and marketing among other popular majors, some of which gear students up for the school’s best online masters in mathematics curricula

HIGHLIGHTTo attract accomplished faculty members, the University of Delaware offers competitive salaries, and student reviews indicate a high faculty approval rating.

DEGREEApplied Statistics, MS

7. Johns Hopkins University

Johns Hopkins University has an Applied and Computational Mathematics MS students can pursue in a 100% digital format. As a member of our best online masters in statistics, JHU offers a post-master’s certificate as well as an MS in the field. Concentrations available include applied analysis, information technology, operations, probability and simulations and modeling. Graduates cash in on the university’s branding through high-paying careers in business, defense technology, medicine and public policy, making this a valuable investment for one of the best online statistics masters programs.

One of the most renowned universities at home and internationally, Johns Hopkins University has a grueling admissions process and an 11% acceptance rate. However, besides having a recognized best online masters in mathematics, JHU is widely regarded as the #1 university in Maryland. 

HIGHLIGHTWith a student to faculty ratio of 7:1, Johns Hopkins offers students more facetime with teachers and advisors.

DEGREEApplied and Computational Mathematics, MS

8. University of Houston

The Master’s of Arts in Mathematics at the University of Houston requires 33 semester hours that includes 21 credits of mathematics. Students master analysis, probability and statistics, algebra and applied mathematics in one of the best online masters in statistics programs. Students can take courses online or take courses both online and on-campus, giving students total freedom in deciding where they need the most help and which courses they can navigate on their own. 

As well as boasting of one of the best online statistics masters programs, the University of Houston is an above-average school that admits 83% of applicants. This wide-access admissions policy leads to a diverse student body. Plus, 85% of students receive student aid. Combined with an affordable tuition program, this adds to the list of reasons that make it one of the best online masters in mathematics.

HIGHLIGHTThe University of Houston ranks respectably at #185 among U.S. News and World Reports 2020 college rankings.

DEGREEMathematics, MA

9. University of Kentucky

The University of Kentucky offers a Masters of Applied Statistics or MAS agree that’s on our pick of the best online masters in statistics. The MAS online program equips elearners with a variety of skills, including big data management, analysis and interpretation. Effective data analytic and computing skills are essential for statisticians employed by Intel, Google and Pfizer. Graduates of the best online statistics masters can seek employment on Wall Street as well as numerous state and federal government organizations.

The University of Kentucky is a highly reputable public university in Lexington, Kentucky. It has a large undergraduate student body that choose majors such as nursing, business and marketing. With an acceptance rate of 94%, the school provides easy access to online learners looking for the best online masters in mathematics program for them.

HIGHLIGHTThe University of Kentucky is well=positioned in Lexington where students can pursue internships and employment opportunities in a bustling city.

DEGREEMasters of Applied Statistics, MAS

10. University of Tennessee-Knoxville

The University of Tennesse-Knoxville has a Masters of Mathematics that’s among the best online masters in statistics. The department offers a traditional MS that students may use to move on to a Ph.D. Online students choose where and when they will complete the 30 hours of coursework, and the program has courses available in the summer to help elearners finish as quickly as possible. Full-time students can finish in two years, making this a convenient and comprehensive program in our list of the best online statistics masters degrees.

The University of Tennessee was founded in 1794, a full two years before Tennessee was even a state. With a strong sense of tradition, it proudly servers as Tennessee’s flagship campus and boasts eleven graduate colleges. Combining the latest technology with its tradition of excellence, Tennessee makes the grade among our best online masters in mathematics.

HIGHLIGHTUniversity of Tennessee graduates who move into industrial jobs may make well over $100,000, and academic careers normally command starting salaries of around $65,000.

DEGREEMasters of Mathematics 

11. University of Texas – Rio Grande Valley

The University of Texas-Rio Grande Valley has an online masters in Mathematics that’s rated one of the best online masters in statistics. Elearners who successfully complete the program go on to pursue Ph.D.’s or move into careers in industry. This best online statistics masters program helps students gain a conceptual understanding of mathematics through rigorous proof. Another key takeaway is the communication of mathematical concepts to non-mathematical audiences in multiple formats.

UT – Rio Grande Valley may have one of the best online masters in mathematics, but it’s also a diverse public university in Edinburg, Texas. It has an acceptance rate of 81%, and its most popular majors are criminal justice, psychology and biology.

HIGHLIGHTUT-RGV is ranked number #113 among the best big colleges in the nation, giving graduates a respectable branding when it comes to pursuing careers in the field.

DEGREEMathematics, MS 

12. Emporia State University

Emporia State University has an MS in Mathematics, which is among the best online statistics masters. Students earn six credit hours in each of these subjects: algebra, analysis, statistics and applied mathematics. There are opportunities for motivated grad students to teach undergraduate courses. The program’s focus on marketable skills makes in one of the best online masters in statistics.

Emporia State boasts more the 3,000 students and 95% of them receive financial aid. With a faculty to student ratio of 17:1, elearners can expect assistance from qualified faculty members when they need it the most. 

HIGHLIGHTEmporia has one of the best online masters in mathematics, and most classes have a weekly online chat for students and professors to discuss coursework and other topics.

DEGREEMathematics, MS 

13. Indiana State University

Indiana State University’s Mathematics MS program is one of the best online masters in mathematics available. Elearners see what on-campus students do in this flexible program. Online students log into classes from their computers to hear, see and interact with professors and other students. As one of the best online masters in statistics, the program prepares students for careers as accountants, actuaries, teachers, bankers and data analysts.

Overall, students give Indiana State University high marks for diversity and high-quality professors. Besides having the best online statistics masters program, ISU admits 90% of applicants, giving money students an opportunity to further their education. 

HIGHLIGHTThe school’s vision statement calls for a commitment to career readiness, community engagement and experiential learning.

DEGREEMathematics, MS 

14. Western Kentucky University

Western Kentucky University’s MA in Mathematics is one of the best online masters in mathematics in the state and in the nation. This online program is aimed at students who are secondary teachers with teacher certification. Students can earn a change in rank by getting their Master’s degree. This program isn’t designed for students looking to move onto a doctorate in mathematics. Graduate assistantships may be available offering one more reason we includes WKU in our list of the best online statistics masters.

As well as earning the best online masters in statistics, WKU graduates benefit from the university’s regional name recognition. Up to 97% of applicants are granted admission, which results in a diverse and eclectic student body on-campus and online. Additionally, 96% of students receive student aid.

HIGHLIGHTNiche.com assigned a grade of B to WKU based on faculty accomplishments and reviews from students.

DEGREEMathematics, MA

15. California University of Pennsylvania

The Applied Mathematics PSM degree offered by the California University of Pennsylvania gives online students a chance to attend one of the best online masters in statistics. Designed for busy professionals in nonprofit, government and business careers, it provides a flexible way to obtains a master’s degree. Other elearners are teachers looking for advanced math skills to pass along to their students. The 30-credit program provides a fast-track to one of the best online statistics masters that can be finished in 13 months. 

Students receive the full benefit of the best online masters in mathematics in the region and also benefit from CUP’s location near Pittsburgh, Pennsylvania, a center of industry and education. With a 95% acceptance rate, the school opens its doors to everyone who’s willing to learn and apply themselves.

HIGHLIGHTA small student body of around 5,000 students gives online and other students a chance to establish themselves and benefits from greater individual attention.

DEGREEApplied Mathematics, PSM