Probability represents the area of mathematics that deals with uncertain events. Any measurement or data collection is affected by different sources of variation. That means that repeating a measurement in an experiment could result in a different outcome. Probability attempts to understand or control these sources of variability. Calculation of probability may be for independent events or when two events are related. Independent events mean that the probability of one event occurring does not depend on another event happening. The probability of an event is a numerical measurement of the likelihood of its occurrence. Probability measures range between 0 and 1. That is the percentage of the time that the event will happen. For example, the probability of getting heads when you flip a coin is .5, meaning that half the time you will get heads, and half the time you will get tails. Understanding the likelihood of an event aids in making predictions regarding its occurrence.

Major applications of statistics today include areas like business analytics that build statistical models to assist business professionals in examining data from past business performance to understand the current market and predict future business trends. Data analysis relies on modeling statistical data to gain information used in decision-making. The field of data science uses statistical methods to gain knowledge from data, including applications in data mining and big data. Machine learning utilizes data to develop computer algorithms to build statistical models that can be used in applications for artificial intelligence or speech recognition. Data mining uses statistics to discover patterns in large sets of data. Another area is data visualization that deals with the graphic representation of data to assist with understanding complex relationships in a simpler form.

A normal distribution is the most common continuous statistical distribution. It is a probability function that describes how a set of values get distributed or spread out. This bell-shaped curve represents the probabilities for a distribution with a mean of zero and a standard deviation of one. Most of the observations get gathered around this center, while other values will spread out equally. More likely, events will fall closer to the middle, while less likely occurrences will lie toward the tails of this distribution. The relevance of the normal distribution in statistics frequently occurs in daily life, and many natural phenomena fit this distribution. Parameters like height and weight follow this distribution. Knowing the characteristics of the normal distribution allows for making inferences about the population in question and predicting where a member of a sample will fall along this curve.

Excel offers a powerful way to analyze data contained in spreadsheets. It does this through data visualization and simple statistics to create graphics and with a Pivot table to create an aggregate of individual items in a spreadsheet. Python is a general purpose, high-level programming language that is often used for data mining and visualization. Its simple syntax makes it easy to learn and use in various statistical applications. SPSS is a frequently used statistical software in behavioral research that compiles descriptive statistics and graphical representations for advanced statistical analysis. Machine learning is a powerful tool that can be used to develop computer algorithms that can improve through experience with additional data to make predictions and decisions automatically and can be used in fields like medicine or computer science for developing artificial intelligence.

Bayesian statistics is a mathematical procedure that applies probability theory to statistical problems. It relies on conditional probability using the information to determine the likelihood of an event’s occurrence. Bayesian statistics update our beliefs when presented with evidence from new data and eliminates mistakes in our decisions. It uses A/B testing, which analyzes the performance of two groups to determine which is performing better and allows for leveraging multiple sources of data. Bayesian statistics relates a calculated quantity to one that can answer arbitrary questions-- used with a complex model with many measurement errors or multiple data sources. That is essential in marketing to determine which sales technique remains more effective. An example of this determines which internet page creates more sales of a product or service and permits effective webpage design to increase sales.