Statistics

Statistics

1. Chapter 1

   • Data: consist of information coming from observations, counts, measurements, or responses.
   • Statistics is the science of collecting, organizing, analyzing, and interpreting data in order to make decisions.
   • A population is the collection of all outcomes, responses, measurements, or counts that are of interest.
   • A sample is a subset, or part, of a population.
   • A parameter is a numerical description of a population characteristic.
   • A statistic is a numerical description of a sample characteristic.
   • Descriptive statistics is the branch of statistics that involves the organization, summarization, and display of data.
   • Inferential statistics is the branch of statistics that involves using a sample to draw conclusions about a population. A basic tool in the study of inferential statistics is probability.
   • Qualitative data consist of attributes, labels, or nonnumerical entries
   • Quantitative data consist of numerical measurements or counts.
   • Nominal level of measurement are qualitative only. Data at this level are categorized using names, labels, or qualities. No mathematical computations can be made at this level.
   • Ordinal level of measurement are qualitative or quantitative. Data at this level can be arranged in order, or ranked, but differences between data entries are not meaningful.
   • Interval level of measurement can be ordered, and meaningful differences between data entries can be calculated. At the interval level, a zero entry simply represents a position on a scale; the entry is not an inherent zero.
   • Ratio level of measurement are similar to data at the interval level, with the added property that a zero entry is an inherent zero. A ratio of two data values can be formed so that one data value can be meaningfully expressed as a multiple of another.
   • Observational study, a researcher observes and measures characteristics of interest of part of a population but does not change existing conditions.
   • Experiment, a treatment is applied to part of a population and responses are observed.
     • Control group, in which no treatment is applied.
     • In many cases, subjects (sometimes called experimental units) in the control group are given a placebo, which is a harmless, unmedicated treatment, that is made to look like the real treatment.
       • The placebo effect occurs when a subject reacts favorably to a placebo when in fact the subject has been given no medicated treatment at all. To help control or minimize the placebo effect, a technique called blinding can be used.
       • Blinding is a technique where the subjects do not know whether they are receiving a treatment or a placebo
       • In a double-blind experiment, neither the experimenter nor the subjects know if the subjects are receiving a treatment or a placebo. The experimenter is informed after all the data have been collected. This type of experimental design is preferred by researchers.
   • A simulation is the use of a mathematical or physical model to reproduce the conditions of a situation or process. Collecting data often involves the use of computers. Simulations allow you to study situations that are impractical or even dangerous to create in real life, and often they save time and money.
   • A survey is an investigation of one or more characteristics of a population. Most often, surveys are carried out on people by asking them questions. The most common types of surveys are done by interview, mail, or telephone. In designing a survey, it is important to word the questions so that they do not lead to biased results, which are not representative of a population.
   • A confounding variable occurs when an experimenter cannot tell the difference between the effects of different factors on a variable.
   • Randomization is a process of randomly assigning subjects to different treatment groups.
   • Completely randomized design, subjects are assigned to different treatment groups through random selection. In some experiments, it may be necessary for the experimenter to use blocks, which are groups of subjects with similar characteristics.
   • . To use a randomized block design, you should divide subjects with similar characteristics into blocks, and then, within each block, randomly assign subjects to treatment groups.
   • Matched-pairs design, where subjects are paired up according to a similarity.
   • Sample size, which is the number of subjects, is another important part of experimental design.
   • Replication is the repetition of an experiment under the same or similar conditions.
   • A census is a count or measure of an entire population.
   • .A sampling error is the difference between the results of a sample and those of the population.
   • A random sample is one in which every member of the population has an equal chance of being selected. A simple random sample is a sample in which every possible sample of the same size has the same chance of being selected.
   • Stratified Sample When it is important for the sample to have members from each segment of the population, you should use a stratified sample. Depending on the focus of the study, members of the population are divided into two or more subsets, called strata, that share a similar characteristic such as age, gender, ethnicity, or even political preference. A sample is then randomly selected from each of the strata. Using a stratified sample ensures that each segment of the population is represented. For instance, to collect a stratified sample of the number of people who live in West Ridge County households, you could divide the households into socioeconomic levels, and then randomly select households from each level.
   • Cluster Sample When the population falls into naturally occurring subgroups, each having similar characteristics, a cluster sample may be the most appropriate. To select a cluster sample, divide the population into groups, called clusters, and select all of the members in one or more (but not all) of the clusters. Examples of clusters could be different sections of the same course or different branches of a bank. For instance, to collect a cluster sample of the number of people who live in West Ridge County households, divide the households into groups according to zip codes, then select all the households in one or more, but not all, zip codes and count the number of people living in each household.
   • A systematic sample is a sample in which each member of the population is assigned a number. The members of the population are ordered in some way, a starting number is randomly selected, and then sample members are selected at regular intervals from the starting number. (For instance, every 3rd, 5th, or 100th member is selected.)
   • A type of sample that often leads to biased studies (so it is not recommended) is a convenience sample. A convenience sample consists only of available members of the population.

2. Chapter 2

   • Important characteristics to look for when organizing and describing a data set are its center, its variability (or spread), and its shape.
   • A frequency distribution is a table that shows classes or intervals of data entries with a count of the number of entries in each class. The frequency f of a class is the number of data entries in the class.
   • Each class has a lower class limit, which is the least number that can belong to the class, and an upper class limit, which is the greatest number that can belong to the class.
   • The class width is the distance between lower (or upper) limits of consecutive classes.
   • The difference between the maximum and minimum data entries is called the range.
   • The midpoint of a class is the sum of the lower and upper limits of the class divided by two.
   • The relative frequency of a class is the portion or percentage of the data that falls in that class.
   • The cumulative frequency of a class is the sum of the frequencies of that class and all previous classes.
   • A frequency histogram is a bar graph that represents the frequency distribution of a data set.
     • The horizontal scale is quantitative and measures the data values.
     • The vertical scale measures the frequencies of the classes.
     • Consecutive bars must touch.
   • Class boundaries are the numbers that separate classes without forming gaps between them. If data entries are integers, subtract 0.5 from each lower limit to find the lower class boundaries. To find the upper class boundaries, add 0.5 to each upper limit. The upper boundary of a class will equal the lower boundary of the next higher class.
   • A frequency polygon is a line graph that emphasizes the continuous change in frequencies.
   • A relative frequency histogram has the same shape and the same horizontal scale as the corresponding frequency histogram. The difference is that the vertical scale measures the relative frequencies, not frequencies.
   • A cumulative frequency graph, or ogive, is a line graph that displays the cumulative frequency of each class at its upper class boundary. The upper boundaries are marked on the horizontal axis, and the cumulative frequencies are marked on the vertical axis.
   • Stem-and-leaf plots are examples of exploratory data analysis (EDA), which was developed by John Tukey in 1977.
     • In a stem-and-leaf plot, each number is separated into a stem (for instance, the entry’s leftmost digits)
     • A leaf (for instance, the rightmost digit)
   • A pie chart is a circle that is divided into sectors that represent categories. The area of each sector is proportional to the frequency of each category.
   • A Pareto chart is a vertical bar graph in which the height of each bar represents frequency or relative frequency.
   • When each entry in one data set corresponds to one entry in a second data set, the sets are called paired data sets.
   • One way to graph paired data sets is to use a scatter plot, where the ordered pairs are graphed as points in a coordinate plane. A scatter plot is used to show the relationship between two quantitative variables.
   • A data set that is composed of quantitative entries taken at regular intervals over a period of time is called a time series.
     • You can use a time series chart to graph a time series.
   • A measure of central tendency is a value that represents a typical, or central, entry of a data set. The three most commonly used measures of central tendency are the mean, the median, and the mode.
   • The mean of a data set is the sum of the data entries divided by the number of entries. To find the mean of a data set, use one of the following formulas.
     • The median of a data set is the value that lies in the middle of the data when the data set is ordered.The median measures the center of an ordered data set by dividing it into two equal parts
   • The mode of a data set is the data entry that occurs with the greatest frequency
   • If two entries occur with the same greatest frequency, each entry is a mode and the data set is called bimodal.
   • An outlier is a data entry that is far removed from the other entries in the data set.
   • A data set can have one or more outliers, causing gaps in a distribution.
   • A weighted mean is the mean of a data set whose entries have varying weights.
   • A frequency distribution is symmetric when a vertical line can be drawn through the middle of a graph of the distribution and the resulting halves are approximately mirror images. A frequency distribution is uniform (or rectangular) when all entries, or classes, in the distribution have equal or approximately equal frequencies. A uniform distribution is also symmetric. A frequency distribution is skewed if the “tail” of the graph elongates more to one side than to the other. A distribution is skewed left (negatively skewed) if its tail extends to the left. A distribution is skewed right (positively skewed) if its tail extends to the right.
   • The range of a data set is the difference between the maximum and minimum data entries in the set. To find the range, the data must be quantitative.
   • The deviation of an entry x in a population data set is the difference between the entry and the mean of the data set.
   • When you add the squares of the deviations, you compute a quantity called the sum of squares, denoted In a population data set, the mean of the squares of the deviations is called the population variance.
   • The population standard deviation of a population data set of N entries is the square root of the population variance.
   • Fractiles are numbers that partition, or divide, an ordered data set into equal parts. The three quartiles, and approximately divide an ordered data set into four equal parts. About one quarter of the data fall on or below the first quartile About one half of the data fall on or below the second quartile (the second quartile is the same as the median of the data set). About three quarters of the data fall on or below the third quartile Q3 .
   • The interquartile range (IQR) of a data set is a measure of variation that gives the range of the middle 50% of the data. It is the difference between the third and first quartiles. The standard score, or z-score, represents the number of standard deviations a given value x falls from the mean To find the z-score for a given value, use the following formula

3. Chapter 3

   • A probability experiment is an action, or trial, through which specific results (counts, measurements, or responses) are obtained. The result of a single trial in a probability experiment is an outcome. The set of all possible outcomes of a probability experiment is the sample space. An event is a subset of the sample space. It may consist of one or more outcomes.
   • A tree diagram gives a visual display of the outcomes of a probability experiment by using branches that originate from a starting point. It can be used to find the number of possible outcomes in a sample space as well as individual outcomes.
   • An event that consists of a single outcome is called a simple event.
   • Classical (or theoretical) probability is used when each outcome in a sample space is equally likely to occur.
   • Empirical (or statistical) probability is based on observations obtained from probability experiments.The empirical probability of an event E is the relative frequency of event E.
   • The complement of event E is the set of all outcomes in a sample space that are not included in event E. The complement of event E is denoted by and is read as “E prime.”
   • A conditional probability is the probability of an event occurring, given that another event has already occurred. The conditional probability of event B occurring, given that event A has occurred, is denoted by and is read as “probability of B, given A.”
   • Two events are independent if the occurrence of one of the events does not affect the probability of the occurrence of the other event. Two events A and B are independent if or if Events that are not independent are dependent.
   • Two events A and B are mutually exclusive if A and B cannot occur at the same time.
   • A permutation is an ordered arrangement of objects. The number of different permutations of n distinct objects is n!.

4. Chapter 4

   • A random variable x represents a numerical value associated with each outcome of a probability experiment.
     • A random variable is discrete if it has a finite or countable number of possible outcomes that can be listed. A random variable is continuous if it has an uncountable number of possible outcomes, represented by an interval on the number line.
   • A discrete probability distribution lists each possible value the random variable can assume, together with its probability. A discrete probability distribution must satisfy the following conditions.
   • The mean of a discrete random variable is given by Each value of x is multiplied by its corresponding probability and the products are added.
   • The expected value of a discrete random variable is equal to the mean of the random variable.
   • A binomial experiment is a probability experiment that satisfies the following conditions. 1. The experiment is repeated for a fixed number of trials, where each trial is independent of the other trials. 2. There are only two possible outcomes of interest for each trial. The outcomes can be classified as a success or as a failure 3. The probability of a success is the same for each trial. 4. The random variable x counts the number of successful trials.
   • A geometric distribution is a discrete probability distribution of a random variable x that satisfies the following conditions.
   • The Poisson distribution is a discrete probability distribution of a random variable x that satisfies the following conditions.