Statistics by: Cabot Smith Wilson

Statistics by: Cabot Smith Wilson

1. Chapter 1

   • An Overview of Statistics
     • 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
   • Data Classification
     • Qualitative data consist of attributes, labels, or nonnumerical entries
     • Quantitative data consist of numerical measurements or counts.
     • Data at the 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.
     • Data at the 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
     • Data at the 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.
     • Data at the 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
   • Data Collection and Experimental Design
     • A confounding variable occurs when an experimenter cannot tell the difference between the effects of different factors on a variable
     • 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.
     • Randomization is a process of randomly assigning subjects to different treatment groups.
     • Replication is the repetition of an experiment under the same or similar conditions.

2. Chapter 2

   • Frequency Distributions and Their Graphs
     • 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.
     • Constructing a Frequency Distribution from a Data Set 1. Decide on the number of classes to include in the frequency distribution. The number of classes should be between 5 and 20; otherwise, it may be difficult to detect any patterns. 2. Find the class width as follows. Determine the range of the data, divide the range by the number of classes, and round up to the next convenient number. 3. Find the class limits.You can use the minimum data entry as the lower limit of the first class.To find the remaining lower limits, add the class width to the lower limit of the preceding class.Then find the upper limit of the first class. Remember that classes cannot overlap. Find the remaining upper class limits. 4. Make a tally mark for each data entry in the row of the appropriate class. 5. Count the tally marks to find the total frequency f for each class.
     • The midpoint of a class is the sum of the lower and upper limits of the class divided by two.The midpoint is sometimes called the class mark.
     • The relative frequency of a class is the portion or percentage of the data that falls in that class.To find the relative frequency of a class, divide the frequency f by the sample size n.
     • The cumulative frequency of a class is the sum of the frequencies of that class and all previous classes. The cumulative frequency of the last class is equal to the sample size n.
     • 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. (ojive)
   • More Graphs and Displays
     • a newer way to display quantitative data, called a stem-and-leaf plot. 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) and a leaf (for instance, the rightmost digit). You should have as many leaves as there are entries in the original data set and the leaves should be single digits
   • Measures of Central Tendency
     • 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. If the data set has an odd number of entries, the median is the middle data entry. If the data set has an even number of entries, the median is the mean of the two middle data entries.
     • The mode of a data set is the data entry that occurs with the greatest frequency. A data set can have one mode, more than one mode, or no mode. If no entry is repeated, the data set has no mode. 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 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
   • Measures of Variation
     • 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.
     • The population standard deviation of a population data set of N entries is the square root of the population variance
     • The sample variance and sample standard deviation of a sample data set of n entries are listed below
   • Measures of Position
     • The three quartiles Q1, Q2 and Q3 approximately divide an ordered data set into four equal parts. About one quarter of the data fall on or below the first quartile Q1 About one half of the data fall on or below the second quartile Q2 (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

   • Basic Concepts of Probability and Counting
     • 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.
     • If one event can occur in m ways and a second event can occur in n ways, the number of ways the two events can occur in sequence m*n is This rule can be extended to any number of events occurring in sequence
     • Classical (or theoretical) probability is used when each outcome in a sample space is equally likely to occur. The classical probability for an event E is given by
     • 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.
     • As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event
     • As an experiment is repeated over and over, the empirical probability of an event approaches the theoretical (actual) probability of the event
     • 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 E' and is read as “E prime
   • Conditional Probability and the Multiplication Rule
     • 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. P1B ƒ A2 = P1B2 P1A ƒ B2 = P1A2.
     • The probability that two events A and B will occur in sequence is If events A and B are independent, then the rule can be simplified to This simplified rule can be extended to any number of independent events. P1A and B2 = P1A2 # P1B2. P1A and B2 = P1A2 # P1B ƒ A2.
   • The Addition Rule
     • Two events A and B are mutually exclusive if A and B cannot occur at the same time.
     • The probability that events A or B will occur, is given by If events A and B are mutually exclusive, then the rule can be simplified to This simplified rule can be extended to any number of mutually exclusive events.
   • Additional Topics in Probability and Counting
     • A permutation is an ordered arrangement of objects.The number of different permutations of n distinct objects is n!.
     • The number of permutations of n distinct objects taken r at a time is nPr
     • The number of distinguishable permutations of n objects, where are of one type, are of another type, and so on
     • A combination is a selection of r objects from a group of n objects without regard to order and is denoted by The number of combinations of r objects selected from a group of n objects is nCr = n! 1n - r2!r! . nCr

4. Chapter 4

   • Probability Distributions
     • 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 of the random variable can assume, together with its probability. A discrete probability distribution must satisfy the following conditions. 1. The probability of each value of the discrete random variable is between 0 and 1, inclusive. 2. The sum of all the probabilities is 1
     • 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 standard deviation is s = 2s2 = 2g1x - m22P1x2
     • The expected value of a discrete random variable is equal to the mean of the random variable.
   • Binomial Distributions
     • 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
     • Symbols
       • n =The number of times a trial is repeated
       • p=P(s) = The probability of success in a single trial
       • q=P(F) = The probability of failure in a single trial
       • x = The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, Á , n.
     • In a binomial experiment, the probability of exactly x successes in n trials
     • Mean: m = np Variance: s2 = npq Standard deviation: s = √npq
   • More Discrete Probability Distributions
     • A geometric distribution is a discrete probability distribution of a random variable x that satisfies the following conditions. 1. A trial is repeated until a success occurs. 2. The repeated trials are independent of each other. 3. The probability of success p is constant for each trial. 4. The random variable x represents the number of the trial in which the first success occurs
     • The Poisson distribution is a discrete probability distribution of a random variable x that satisfies the following conditions. 1. The experiment consists of counting the number of times x an event occurs in a given interval. The interval can be an interval of time, area, or volume. 2. The probability of the event occurring is the same for each interval. 3. The number of occurrences in one interval is independent of the number of occurrences in other intervals