Probability is a mathematical language used to discuss uncertain events and probability plays a key role in statistics. Any measurement or data collection effort is subject to a number of sources of variation. Instead, we query a relatively small number of Americans, and draw inferences about the entire country from their responses.
The Americans actually queried constitute our sample of the larger population of all Americans. The mathematical procedures whereby we convert information about the sample into intelligent guesses about the population fall under the rubric of inferential statistics.
In the case of voting attitudes, we would sample a few thousand Americans, drawn from the hundreds of millions that make up the country.
In choosing a sample, it is therefore crucial that it be representative. It must not over-represent one kind of citizen at the expense of others. For example, something would be wrong with our sample if it happened to be made up entirely of Florida residents. If the sample held only Floridians, it could not be used to infer the attitudes of other Americans. The same problem would arise if the sample were comprised only of Republicans. Inferential statistics are based on the assumption that sampling is random.
We trust a random sample to represent different segments of society in close to the appropriate proportions provided the sample is large enough. Furthermore, when generalizing a trend found in a sample to the larger population, statisticians uses tests of significance such as the Chi-Square test or the T-test.
These tests determine the probability that the results found were by chance, and therefore not representative of the entire population. Linear Regression in Inferential Statistics : This graph shows a linear regression model, which is a tool used to make inferences in statistics.
Data can be categorized as either primary or secondary and as either qualitative or quantitative. Data can be classified as either primary or secondary.
Primary data is original data that has been collected specially for the purpose in mind. This type of data is collected first hand. Those who gather primary data may be an authorized organization, investigator, enumerator or just someone with a clipboard. These people are acting as a witness, so primary data is only considered as reliable as the people who gather it. Research where one gathers this kind of data is referred to as field research.
An example of primary data is conducting your own questionnaire. Secondary data is data that has been collected for another purpose. This type of data is reused, usually in a different context from its first use. You are not the original source of the data—rather, you are collecting it from elsewhere.
An example of secondary data is using numbers and information found inside a textbook. Knowing how the data was collected allows critics of a study to search for bias in how it was conducted. A good study will welcome such scrutiny. Each type has its own weaknesses and strengths. Primary data is gathered by people who can focus directly on the purpose in mind. This helps ensure that questions are meaningful to the purpose, but this can introduce bias in those same questions.
Stated another way, those who gather secondary data get to pick the questions. Those who gather primary data get to write the questions. There may be bias either way. Qualitative data is a categorical measurement expressed not in terms of numbers, but rather by means of a natural language description.
Collecting information about a favorite color is an example of collecting qualitative data. Although we may have categories, the categories may have a structure to them. When there is not a natural ordering of the categories, we call these nominal categories.
Examples might be gender, race, religion, or sport. When the categories may be ordered, these are called ordinal categories. Categorical data that judge size small, medium, large, etc. Attitudes strongly disagree, disagree, neutral, agree, strongly agree are also ordinal categories; however, we may not know which value is the best or worst of these issues.
Note that the distance between these categories is not something we can measure. Quantitative data is a numerical measurement expressed not by means of a natural language description, but rather in terms of numbers.
Quantitative data always are associated with a scale measure. Probably the most common scale type is the ratio-scale. Observations of this type are on a scale that has a meaningful zero value but also have an equidistant measure i. For example, a 10 year-old girl is twice as old as a 5 year-old girl. Since you can measure zero years, time is a ratio-scale variable. Money is another common ratio-scale quantitative measure. Observations that you count are usually ratio-scale e.
A more general quantitative measure is the interval scale. Interval scales also have an equidistant measure. However, the doubling principle breaks down in this scale. Quantitative Data : The graph shows a display of quantitative data. Statistics deals with all aspects of the collection, organization, analysis, interpretation, and presentation of data. It includes the planning of data collection in terms of the design of surveys and experiments.
Statistics can be used to improve data quality by developing specific experimental designs and survey samples. Statistics also provides tools for prediction and forecasting. Statistics is applicable to a wide variety of academic disciplines, including natural and social sciences as well as government and business.
Statistical methods can summarize or describe a collection of data. This is called descriptive statistics. This is particularly useful in communicating the results of experiments and research. Statistical models can also be used to draw statistical inferences about the process or population under study—a practice called inferential statistics.
Inference is a vital element of scientific advancement, since it provides a way to draw conclusions from data that are subject to random variation. Conclusions are tested in order to prove the propositions being investigated further, as part of the scientific method. Descriptive statistics and analysis of the new data tend to provide more information as to the truth of the proposition.
Statistical methods are used in government regulation on topics such as stock trading rules, air purity standards, and new drug approvals. Statistics also are cited in court proceedings, parliamentarian or congressional hearings, and lobbying arguments. Politics involve statistics in the form of approval-rating surveys, voter registration, campaigning, and election predictions. Statisticians participate in government agencies and assist in national, provincial, or state government decisionmaking and policymaking.
Statisticians work on surveys in government, the social sciences, education, law, forestry, agriculture, biology, medicine, business, and e-commerce.
A survey statistician might study efficient survey design, experimental methods for increasing response rates, accounting for nonresponse and undercoverage, or how to release data to the public while maintaining the confidentiality of respondents. Other important issues include questioning wording and design and deciding where and how to take samples that will include traditionally under-represented groups.
Independent statistical consultants work on many of the same projects as other statisticians, but they usually are hired on a temporary basis to solve a specific problem that requires statistical expertise not available within the hiring company.
Since the field of statistics is so broad, many statistical consultants specialize in some area, such as quality improvement or pharmaceuticals. Consultants may be hired with grant money to work on short-term projects in medicine, agriculture, engineering, or business.
Statistics are becoming more important as court cases address increasingly complex problems. Sometimes the statistician analyzes data that can help the jury or judge decide whether someone is guilty of a crime or must pay damages for causing injuries. Court cases involving statistical analyses include DNA testing, salary discrepancies, consumer surveys, and disease clusters. Statisticians have teamed up with experts in agriculture to study a number of challenging questions, including chemical pesticides, hydrogeology, veterinary sciences, genetics, and crop management.
Statisticians are involved in studies ranging from small laboratory experiments to large projects conducted over many hundreds or thousands of square miles. In practice, statistics is the idea we can learn about the properties of large sets of objects or events a population by studying the characteristics of a smaller number of similar objects or events a sample.
Because in many cases gathering comprehensive data about an entire population is too costly, difficult, or flat out impossible, statistics start with a sample that can conveniently or affordably be observed. Two types of statistical methods are used in analyzing data: descriptive statistics and inferential statistics. Statisticians measure and gather data about the individuals or elements of a sample, then analyze this data to generate descriptive statistics.
They can then use these observed characteristics of the sample data, which are properly called "statistics," to make inferences or educated guesses about the unmeasured or unmeasured characteristics of the broader population, known as the parameters. Descriptive statistics mostly focus on the central tendency, variability, and distribution of sample data. Central tendency means the estimate of the characteristics, a typical element of a sample or population, and includes descriptive statistics such as mean , median , and mode.
Variability refers to a set of statistics that show how much difference there is among the elements of a sample or population along the characteristics measured, and includes metrics such as range , variance , and standard deviation. The distribution refers to the overall "shape" of the data, which can be depicted on a chart such as a histogram or dot plot, and includes properties such as the probability distribution function, skewness, and kurtosis. Descriptive statistics can also describe differences between observed characteristics of the elements of a data set.
Descriptive statistics help us understand the collective properties of the elements of a data sample and form the basis for testing hypotheses and making predictions using inferential statistics. Inferential statistics are tools that statisticians use to draw conclusions about the characteristics of a population, drawn from the characteristics of a sample, and to decide how certain they can be of the reliability of those conclusions.
Based on the sample size and distribution statisticians can calculate the probability that statistics, which measure the central tendency, variability, distribution, and relationships between characteristics within a data sample, provide an accurate picture of the corresponding parameters of the whole population from which the sample is drawn.
Inferential statistics are used to make generalizations about large groups, such as estimating average demand for a product by surveying a sample of consumers' buying habits or to attempt to predict future events, such as projecting the future return of a security or asset class based on returns in a sample period.
Regression analysis is a widely used technique of statistical inference used to determine the strength and nature of the relationship i. The output of a regression model is often analyzed for statistical significance , which refers to the claim that a result from findings generated by testing or experimentation is not likely to have occurred randomly or by chance but is likely to be attributable to a specific cause elucidated by the data.
Having statistical significance is important for academic disciplines or practitioners that rely heavily on analyzing data and research. Descriptive statistics are used to describe or summarize the characteristics of a sample or data set, such as a variable's mean, standard deviation, or frequency.
Inferential statistics, in contrast, employs any number of techniques to relate variables in a data set to one another, for example using correlation or regression analysis. These can then be used to estimate forecasts or infer causality. Statistics are used widely across an array of applications and professions.
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