Binary Data Basics

– A discrete variable with only one state contains zero information.

– The bit, with two possible values, is the standard unit of information.

– The number of states in a collection of n bits is 2^n.

– The number of states in a collection of discrete variables depends exponentially on the number of variables.

– Ten bits have more states (1024) than three decimal digits (1000).

– Binary data consists of categorical data with two possible values.

– Binary data is often used to represent the outcome of an experiment or a yes-no question.

– Binary data is nominal data and cannot be compared numerically.

– Binary data can also represent the presence or absence of a feature.

– Binary data can be used to represent political party choices in elections.

Binary Variables

– Binary variables have two possible values.

– Independent and identically distributed binary variables follow a Bernoulli distribution.

– Total counts of i.i.d. binary variables follow a binomial distribution.

– Binary data need not come from i.i.d. variables.

– The distribution of binary variables may not be binomial if they are not i.i.d.

Counting and Conversion of Binary Data

– Binary data can be converted to count data by assigning 1 for a value that occurs and 0 for a value that does not occur.

– Grouping binary data allows for counting the occurrences of each value.

– Binary data can be simplified to a single count by considering one value as success and the other as failure.

– Count data with n=1 is binary data.

– Counts of i.i.d. binary variables follow a binomial distribution.

Binary Regression

– Binary regression analyzes predicted outcomes that are binary variables.

– Binomial regression can be used when binary data is converted to count data.

– Logistic regression and probit regression are common methods for binary regression.

– Multinomial regression models counts of i.i.d. categorical variables with more than two categories.

– Non-i.i.d. binary data can be modeled using more complex distributions like the beta-binomial distribution.

Binary Representation and Formats

– 1 and 0 represent two different voltage levels.

– Computers understand 1 as higher voltage and 0 as lower voltage.

– Different methods can be used to store two voltage levels.

– Magnetic tapes with a coating of ferromagnetic material can store 1 and 0 data.

– The orientation of magnetic domains determines whether it is interpreted as 1 or 0.

– Textual data can be represented in binary format, such as compressed or formatted files.

– Image data can sometimes be represented in textual format, like the X PixMap image format.

– Binary formats are more specific for representing data without interpretation.

– Textual formats may include formatting codes and other text-related elements.

– The choice between binary and textual formats depends on the nature of the data.

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**Binary data** is data whose unit can take on only two possible states. These are often labelled as 0 and 1 in accordance with the binary numeral system and Boolean algebra.

Binary data occurs in many different technical and scientific fields, where it can be called by different names including *bit* (binary digit) in computer science, *truth value* in mathematical logic and related domains and *binary variable* in statistics.