Degree of freedom

Meaning of degree of freedom (df):

1. In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.[1]

2. Mathematically, degrees of freedom is the dimension of the domain of a random vector, or essentially the number of 'free' components: how many components need to be known before the vector is fully determined.

3. The number of degrees of freedom in a problem, distribution, etc., is the number of parameters which may be independently varied.

4. The concept of degrees of freedom is central to the principle of estimating statistics of populations from samples of them. "Degrees of freedom" is commonly abbreviated to df. In short, think of df as a mathematical restriction that we need to put in place when we calculate an estimate one statistic from an estimate of another.

5. In statistics, the number of degrees of freedom (d.o.f.) is the number of independent pieces of data being used to make a calculation. It is usually denoted with the greek letter nu, ν. The number of degrees of freedom is a measure of how certain we are that our sample population is representative of the entire population - the more degrees of freedom, usually the more certain we can be that we have accurately sampled the entire population. For statistics in analytical chemistry, this is usually the number of observations or measurements N made in a certain experiment.

6. For a set of data points in a given situation (e.g. with mean or other parameter specified, or not), degrees of freedom is the minimal number of values which should be specified to determine all the data points.

7. In statistics, the term degrees of freedom (df) is a measure of the number of independent pieces of information on which the precision of a parameter estimate is based.

Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom (df). In general, the degrees of freedom of an estimate is equal to the number of independent scores that go into the estimate minus the number of parameters estimated as intermediate steps in the estimation of the parameter itself.

The df can be viewed as the number of independent parameters available to fit a model to data. Generally, the more parameters you have, the more accurate your fit will be. However, for each estimate made in a calculation, you remove one degree of freedom. This is because each assumption or approximation you make puts one more restriction on how many parameters are used to generate the model. Put another way, for each estimate you make, your model becomes less accurate.

Another way of thinking about the restriction principle behind degrees of freedom is to imagine contingencies. For example, imagine you have four numbers (a, b, c and d) that must add up to a total of m; you are free to choose the first three numbers at random, but the fourth must be chosen so that it makes the total equal to m - thus your degree of freedom is three. Essentially, degrees of freedom are a count of the number of pieces of independent information contained within a particular analysis.

The maximum numbers of quantities or directions, whose values are free to vary before the remainders of the quantities are determined, or an estimate of the number of independent categories in a particular statistical test or experiment. Degrees of freedom (df) for a sample is defined as: df = n - 1 Where n is the number of scores in the sample.

The degrees of freedom for an estimate equals the number of observations (values) minus the number of additional parameters estimated for that calculation. As we have to estimate more parameters, the degrees of freedom available decreases. It can also be thought of as the number of observations (values) which are freely available to vary given the additional parameters estimated. It can be thought of two ways: in terms of sample size and in terms of dimensions and parameters.

Degrees of freedom are often used to characterize various distributions. See, for example, chi-square distribution, t-distribution, F distribution.

In case, the df was n-1, because an estimate was made that the sample mean is a good estimate of the population mean, so we have one less df than the number of independent observations.

In many statistical calculations you will do, such as linear regression, outliers, and t-tests, you will need to know or calculate the number of degrees of freedom. Degrees of freedom for each test will be explained in the section for which it is required.


Source:
http://en.wikipedia.org/wiki/Degrees_of_freedom_(statistics)
http://mathworld.wolfram.com/DegreeofFreedom.html
http://www.statsdirect.com/help/basics/degrees_of_freedom.htm
http://www.chem.utoronto.ca/coursenotes/analsci/StatsTutorial/DegFree.html
http://www.statistics.com/resources/glossary/d/degsfree.php
http://www.statemaster.com/encyclopedia/Degrees-of-freedom-(statistics)

Difference Quantitative and Qualitative Method - 3

3. Research characteristics



Main Points
a. Qualitative research involves analysis of data such as words (e.g., from interviews), pictures (e.g., video), or objects (e.g., an artifact).
b. Quantitative research involves analysis of numerical data.
c. The strengths and weaknesses of qualitative and quantitative research are a perennial, hot debate, especially in the social sciences. The issues invoke classic 'paradigm war'.
d. The personality / thinking style of the researcher and/or the culture of the organization is under-recognized as a key factor in preferred choice of methods.
e. Overly focusing on the debate of "qualitative versus quantitative" frames the methods in opposition. It is important to focus also on how the techniques can be integrated, such as in mixed methods research. More good can come of social science researchers developing skills in both realms than debating which method is superior.

Difference Quantitative and Qualitative Method - 2

2. Research process

a. Quantitative method
Process in quantitative method have linear character, it have clear step: formula of problem, theoretical, hypothesis, collecting data, data analysis, and make summary. Some method in quantitative method: survey method, ex post fact, experiment, evaluation, action research, etc.

Flow in quantitative method:
Source of problem (empiric-theoretic) -> formula of problem -> relevant theory concept or relevant invention -> test of hypothesis -> prejudice to relation between variable -> compiling research instrument - method / strategy approach of research -> collecting and analyzing data -> summary

b. Qualitative method
Divide in some steps:
1. Description step
In this step, researcher describes what he look, hear, and feel. Researcher know less to obtained information from data. Data quite a lot, varying and not yet lapped over clearly.
2. Reduction step
In this step, researcher reduces all information in previous step. Researcher only focus in certain problem. Researcher choose problem from data which interesting data, important, useful and new information. Only chosen data will be used.
3. Selection step
In this step, researches analyses data deeper than previous step to obtain new information. So that researcher can be find themes by construct data become hypothesis.

End result in qualitative method not only get information which can not find in quantitative method but also get new information and meaningful information, and also create new hypothesis to solve problem.

Different research process in quantitative method and qualitative method
-. Quantitative method have deductive, start from theoretical frame work, focus on formal theory, middle range theory, substantive theory, and then formulate it in hypothesis, hypothesis test, and get empirical social reality.
-. Qualitative method have inductive, start from observe in empirical social reality, then develop it in substantive theory, middle range theory, formal theory, and then result became theoretical frame work.

*. Formal theory is developed for board conceptual area in general theory
*. Substantive theory is developed for specific area of social concern
*. Middle range theory can be formal or substantive, middle range theory are slightly more abstract the empirical generalization or specific hypotheses.

Difference Quantitative and Qualitative Method - 1

Difference their methods covering in three things: difference in axiom, research process, and research characteristics

1. Difference in axiom
Axiom [is] elementary view. axiom in quantitative and qualitative cover axiom about reality, relation between researcher and object research, variable, possibility of generalizing, and value role.


Source:
http://uk.geocities.com/balihar_sanghera/ipsrmehrigiulqualitativequantitativeresearch.html
http://wilderdom.com/research/QualitativeVersusQuantitativeResearch.html

For Today’s Graduate, Just One Word: Statistics

from: http://www.nytimes.com/2009/08/06/technology/06stats.html?_r=2&scp=1&sq=today%27s%20graduate&st=cse


by STEVE LOHR
Published: August 5, 2009

MOUNTAIN VIEW, Calif. — At Harvard, Carrie Grimes majored in anthropology and archaeology and ventured to places like Honduras, where she studied Mayan settlement patterns by mapping where artifacts were found. But she was drawn to what she calls “all the computer and math stuff” that was part of the job.

“People think of field archaeology as Indiana Jones, but much of what you really do is data analysis,” she said.

Now Ms. Grimes does a different kind of digging. She works at Google, where she uses statistical analysis of mounds of data to come up with ways to improve its search engine.

Ms. Grimes is an Internet-age statistician, one of many who are changing the image of the profession as a place for dronish number nerds. They are finding themselves increasingly in demand — and even cool.

“I keep saying that the sexy job in the next 10 years will be statisticians,” said Hal Varian, chief economist at Google. “And I’m not kidding.”

The rising stature of statisticians, who can earn $125,000 at top companies in their first year after getting a doctorate, is a byproduct of the recent explosion of digital data. In field after field, computing and the Web are creating new realms of data to explore — sensor signals, surveillance tapes, social network chatter, public records and more. And the digital data surge only promises to accelerate, rising fivefold by 2012, according to a projection by IDC, a research firm.

Yet data is merely the raw material of knowledge. “We’re rapidly entering a world where everything can be monitored and measured,” said Erik Brynjolfsson, an economist and director of the Massachusetts Institute of Technology’s Center for Digital Business. “But the big problem is going to be the ability of humans to use, analyze and make sense of the data.”

The new breed of statisticians tackle that problem. They use powerful computers and sophisticated mathematical models to hunt for meaningful patterns and insights in vast troves of data. The applications are as diverse as improving Internet search and online advertising, culling gene sequencing information for cancer research and analyzing sensor and location data to optimize the handling of food shipments.

Even the recently ended Netflix contest, which offered $1 million to anyone who could significantly improve the company’s movie recommendation system, was a battle waged with the weapons of modern statistics.

Though at the fore, statisticians are only a small part of an army of experts using modern statistical techniques for data analysis. Computing and numerical skills, experts say, matter far more than degrees. So the new data sleuths come from backgrounds like economics, computer science and mathematics.

They are certainly welcomed in the White House these days. “Robust, unbiased data are the first step toward addressing our long-term economic needs and key policy priorities,” Peter R. Orszag, director of the Office of Management and Budget, declared in a speech in May. Later that day, Mr. Orszag confessed in a blog entry that his talk on the importance of statistics was a subject “near to my (admittedly wonkish) heart.”

I.B.M., seeing an opportunity in data-hunting services, created a Business Analytics and Optimization Services group in April. The unit will tap the expertise of the more than 200 mathematicians, statisticians and other data analysts in its research labs — but that number is not enough. I.B.M. plans to retrain or hire 4,000 more analysts across the company.

In another sign of the growing interest in the field, an estimated 6,400 people are attending the statistics profession’s annual conference in Washington this week, up from around 5,400 in recent years, according to the American Statistical Association. The attendees, men and women, young and graying, looked much like any other crowd of tourists in the nation’s capital. But their rapt exchanges were filled with talk of randomization, parameters, regressions and data clusters. The data surge is elevating a profession that traditionally tackled less visible and less lucrative work, like figuring out life expectancy rates for insurance companies.

Ms. Grimes, 32, got her doctorate in statistics from Stanford in 2003 and joined Google later that year. She is now one of many statisticians in a group of 250 data analysts. She uses statistical modeling to help improve the company’s search technology.

For example, Ms. Grimes worked on an algorithm to fine-tune Google’s crawler software, which roams the Web to constantly update its search index. The model increased the chances that the crawler would scan frequently updated Web pages and make fewer trips to more static ones.

The goal, Ms. Grimes explained, is to make tiny gains in the efficiency of computer and network use. “Even an improvement of a percent or two can be huge, when you do things over the millions and billions of times we do things at Google,” she said.

It is the size of the data sets on the Web that opens new worlds of discovery. Traditionally, social sciences tracked people’s behavior by interviewing or surveying them. “But the Web provides this amazing resource for observing how millions of people interact,” said Jon Kleinberg, a computer scientist and social networking researcher at Cornell.

For example, in research just published, Mr. Kleinberg and two colleagues followed the flow of ideas across cyberspace. They tracked 1.6 million news sites and blogs during the 2008 presidential campaign, using algorithms that scanned for phrases associated with news topics like “lipstick on a pig.”

The Cornell researchers found that, generally, the traditional media leads and the blogs follow, typically by 2.5 hours. But a handful of blogs were quickest to quotes that later gained wide attention.

The rich lode of Web data, experts warn, has its perils. Its sheer volume can easily overwhelm statistical models. Statisticians also caution that strong correlations of data do not necessarily prove a cause-and-effect link.

For example, in the late 1940s, before there was a polio vaccine, public health experts in America noted that polio cases increased in step with the consumption of ice cream and soft drinks, according to David Alan Grier, a historian and statistician at George Washington University. Eliminating such treats was even recommended as part of an anti-polio diet. It turned out that polio outbreaks were most common in the hot months of summer, when people naturally ate more ice cream, showing only an association, Mr. Grier said.

If the data explosion magnifies longstanding issues in statistics, it also opens up new frontiers.

“The key is to let computers do what they are good at, which is trawling these massive data sets for something that is mathematically odd,” said Daniel Gruhl, an I.B.M. researcher whose recent work includes mining medical data to improve treatment. “And that makes it easier for humans to do what they are good at — explain those anomalies.”

Andrea Fuller contributed reporting.

Quantitative and Qualitative Method

Many labels have been used to distinguish between traditional research methods and these new methods: positivistic versus post positivistic research; scientific versus artistic research; confirmatory versus discovery – oriented research; quantitative versus interpretive research; quantitative versus qualitative research. The quantitative-qualitative distinction seem most widely used. Both quantitative researchers and qualitative researchers go about inquiry in different ways (Borg and Gall, 1989).

The others name of quantitative methods are traditional, positivistic, scientific and discovery methods, and qualitative methods are new methods, post positivistic, artistic and interpretive research.

Quantitative research is the systematic scientific investigation of quantitative properties and phenomena and their relationships. The objective of quantitative research is to develop and employ mathematical models, theories and/or hypotheses pertaining to natural phenomena. The process of measurement is central to quantitative research because it provides the fundamental connection between empirical observation and mathematical expression of quantitative relationships.

Quantitative research is widely used in both the natural sciences and social sciences, from physics and biology to sociology and journalism. It is also used as a way to research different aspects of education. The term quantitative research is most often used in the social sciences in contrast to qualitative research.

A quantitative attribute is one that exists in a range of magnitudes, and can therefore be measured. Measurements of any particular quantitative property are expressed as a specific quantity, referred to as a unit, multiplied by a number. Examples of physical quantities are distance, mass, and time. Many attributes in the social sciences, including abilities and personality traits, are also studied as quantitative properties and principles.

Quantitative research is research involving the use of structured questions where the response options have been predetermined and a large number of respondents is involved.

By definition, measurement must be objective, quantitative and statistically valid. Simply put, it’s about numbers, objective hard data.

The sample size for a survey is calculated by statisticians using formulas to determine how large a sample size will be needed from a given population in order to achieve findings with an acceptable degree of accuracy. Generally, researchers seek sample sizes which yield findings with at least 95% confidence interval (which means that if you repeat the survey 100 times, 95 times out of a hundred, you would get the same response) and plus/minus 5 percentage points margin error. Many surveys are designed to produce smaller margin of error.

Quantitative methods based on positivism philosophy, used to research in population and sample, collecting data by random sampling, and data analyze for test certain hypothesis.

Positivism philosophy looking into symptom/reality/phenomenon can be classified into: constant relative, perceived, measured and relation with character of causality.

Research – quantitative method, in general, conducted at certain sample or population which is representative. Quantitative method process have the character of deductively, where to answer formula of theory or concept till can be formulated by hypothesis. Hypothesis is tested through field data collecting. Then, it has quantitative analyzed by descriptive or inference statistics, so that can be conclusion for the hypothesis formulated, proven or not. Quantitative method have done in random sample, so that this result can be generalized in population where sample taken.

Method qualitative also known as naturalistic research method because its research conducted at natural condition, also as ethnography method. Collected data and its analysis have the character of qualitative.

Qualitative research is a field of inquiry applicable to many disciplines and subject matters. Qualitative researchers aim to gather an in-depth understanding of human behavior and the reasons that govern such behavior. The qualitative method investigates the why and how of decision making, not just what, where, when. Hence, smaller but focused samples are more often needed, rather than large random samples

Qualitative Research is collecting, analyzing, and interpreting data by observing what people do and say. Whereas, quantitative research refers to counts and measures of things, qualitative research refers to the meanings, concepts, definitions, characteristics, metaphors, symbols, and descriptions of things.

Qualitative research is much more subjective than quantitative research and uses very different methods of collecting information, mainly individual, in-depth interviews and focus groups. The nature of this type of research is exploratory and open-ended. Small numbers of people are interviewed in-depth and/or a relatively small number of focus groups are conducted.

Participants are asked to respond to general questions and the interviewer or group moderator probes and explores their responses to identify and define people’s perceptions, opinions and feelings about the topic or idea being discussed and to determine the degree of agreement that exists in the group. The quality of the finding from qualitative research is directly dependent upon the skills, experience and sensitive of the interviewer or group moderator.

This type of research is often less costly than surveys and is extremely effective in acquiring information about people’s communications needs and their responses to and views about specific communications.

Qualitative research method based on post positivism philosophy, used to research into naturalistic condition (opponent with experiment)। In naturalistic condition researcher as key instrument, sample taken with purposive and snowball technics, technics of gathering with triangulation ( alliance) methods, data analysis with inductive or qualitative method, result of qualitative research more emphasize meaning than generalizing.

Post positivism philosophy also conceived of interpretive paradigm and constructive, which look into social reality as intact something that, complex, dynamic, and having the reciprocal character. Research done at natural object, that is object can not influence by researcher and attendance of researcher does not so influence dynamics at the object.

In qualitative method, its instrument is researcher itself। For that researcher have to have circumstantial and wide of knowledge to problem of accurate. Technics of its data collecting is technics of triangulation, that is joining various is technics of data collecting by simultan. Data analysis have the character of inductive pursuant to found fact then construction become theory or hypothesis. Qualitative method used to get circumstantial data, that is meaning of data. In qualitative method does not emphasize at generalizing, but at meaning.

Source:
-. http://en.wikipedia.org/
-. http://uk.geocities.com/balihar_sanghera/ipsrmehrigiulqualitativequantitativeresearch.html