Statistics Study Guide – Flashcards

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Descriptive statistics
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A branch of statistics that uses numerical and graphical methods to look for patterns in a dataset and to present that information in a convenient form, but does not use analysis
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Inferential statistics
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A branch of statistics that uses sample data to make estimates, predictions, and decisions about a population
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Population
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A collection of individuals or objects that is under study
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Sample
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A subset of the population
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Variable
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A characteristic of interest about each individual element of a population or sample
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Data
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Numbers or information with a context
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Quantitative data
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Variables that can assume numerical values (in the true sense)
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Qualitative data
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Variables that are not numerical (in the true sense) but are categorized into various groups
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Simple random sample
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A sample in which each element in the population has an equal chance of being selected
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Systematic sample
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A sample in which the first element is picked at random, and then every kth element is picked
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Stratefied random sample
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A sample obtained by stratifying the sampling frame and then selecting a fixed number of elements from each stratum by simple random sampling
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Cluster sample
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A sample obtained by sampling some of, but not all of, the possible subdivisions within the population
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Frequency distribution
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A tabular summary of data showing different classes and the frequency of items in each of several non-overlapping classes
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Relative frequency distribution
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A tabular summary of data showing both the frequency and the relative frequency of items in each class
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Relative frequency
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The proportion of the total number of observations belonging to the class
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Bar chart
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A graphical summary of data that uses bars of fixed width and varying height to show the frequency or relative frequency of items in each class
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Pie chart
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A graphical summary of data that uses a circle partitioned into sectors to show the relative frequency of items in each class
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Distribution
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How the observations are spread over the range of the data
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Positive distribution
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Skewed to the right; a distribution with a high frequency of observations in the lower end of the range
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Symmetric distribution
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A mound- or bell-shaped distribution with a high frequency of observations in the middle of the range
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Negative distribution
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Skewed to the left; a distribution with a high frequency of observations in the high end of the range
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Stem and leaf plot
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A summary of data that divides each observation into two parts, with the leaves grouped on a stem
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Leaf
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The right-most digit of each observation (in a stem and leaf plot)
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Stem
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The digits of each observation, excluding the leaf (in a stem and leaf plot)
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Leaf unit
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The unit used to separate the leaf from a stem in a stem and leaf plot, assumed to be 1
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Split stem plot
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A stem and leaf plot in which some stems are split into two parts to reduce the size of the plot
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Histogram
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A graphical representation of frequency distribution in which observations are divided into classes
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Center
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Measure of central tendency
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Variability
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Spread of the data
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Mean
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The average of a group of numbers
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Sample-mean
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x-bar; The mean of a sample data
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Median
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The middle-most observation
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Mode
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The most frequent observation
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Range
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The difference between the minimum and maximum observations
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Variance
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The average squared distance between all the observations and the mean
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Formula for population variance
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δ²=∑(X-µ)²/N
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Formula for sample variance
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s²=∑(xi-x-bar)²/n-1
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Standard deviation
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Square root of variance, or average between a typical observation and the mean of a dataset
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Formula for population standard deviation
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δ=√∑(X-µ)²/N
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Formula for sample standard deviation
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s=√∑(xi-x-bar)²/n-1
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The empirical rule
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For data with a bell-shaped distribution, approximately 68% of the observations will be within one standard deviation of the mean, approximately 95% of the observations will be within two standard deviations of the mean, and approximately 99.7% of the observations will be within three standard deviations of the mean
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Chebyshev's rule
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For any number k>1, at least (1-1/k²) fraction of the data will lie within k standard deviations of the mean
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Z score
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Measures the relative position of an observation compared to the mean, expressed in terms of standard deviation
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Formula for sample z-score
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z=x-xbar/s
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Formula for population z-score
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z=x-µ/δ
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Outlier
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An extreme observation that does not match the general pattern of a dataset
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Percentile ranking
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A way to express where an observation falls in a dataset using percentiles
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Quartile
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One of three numbers which partition a dataset into four parts
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Interquartile range
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Another measure of spread, which gives the middle 50% of a dataset
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Box plot
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A graphical representation of the distribution using five numbers: minimum, lower quartile, median, upper quartile, maximum
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Probability
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Study of randomness and uncertainty; numerical measure of chance
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Empirical probability
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P(Event) = Relative frequency of the event = Number of occurrences of an event/Number of times experiment is repeated
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Law of Large Numbers
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When an experiment is repeated many times, then the relative frequency of a particular outcome approaches the actual probability of that particular outcome
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Subjective probability
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A probability assigned based on the subjective judgment of an individual
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Theoretical probability
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A probability in which basic outcomes of the process are defined, probabilities are assigned to the basic outcomes and probabilities of compound events are computed
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Experiment
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A process that yields a single outcome that cannot be predicted with certainty
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Sample point
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Basic outcome of an experiment
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Sample space
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Set of all possible outcomes of an experiment, denoted by S
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Event
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Any subset of the sample space
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Example of a sample space
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S = {HH, HT, TH, TT}
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Examples of sample points
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HH, HT, TH, TT
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Basic counting principle
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In an experiment done in two independent stages, where Stage I has m possible outcomes and Stage II has n possible outcomes, the experiment can be performed in (m)(n) ways
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Factorial
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n! = (n)(n-1)(n-2)...3.2.1
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Permutation
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Ordered arrangement
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Compound event
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A composition of 2 or more events, can be the result of a union or intersection of events
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Venn diagram
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Pictorial diagram in which the sample space is represented by a rectangle with sample points represented by solid dots inside the rectangle and events by circles within the rectangle
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Union
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The entire Venn diagram, including both circles and the intersection
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Intersection
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The part of a Venn diagram where the circles overlap
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Complementary event
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All sample points that do not belong to the event
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Mutually exclusive events
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Events that share no sample points
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Additive rule of probability
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For any two events A and B, P(AUB) = P(A) + P(B) - P(AB)
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Conditional probability
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Probabilities of events change when additional information is provided, in particular, if another related event is known to have occurred
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Multiplicative rule
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P(AB)=P(A)P(B|A)=P(B)P(A|B)
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Independence
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Events that do not depend on the outcome of another event
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Random variable
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A variable that assigns a unique value to each outcome of the Sample space, S
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Discrete random variable
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A random variable that takes either finitely many values or a countably infinite set of values
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Countable set
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A set in one to one correspondence with integers
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Continuous random variable
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A variable in which the set of possible values consists of one or more intervals on the number line
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Probability distribution
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Specification of the possible values and probability associated with each possible value of the discrete random variable
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Expected value (mean)
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The center of the distribution of a random variable
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Formula for expected value
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µ=E(X)=∑x p(x)=x₁*p(x₁)+x₂*p(x₂)...
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Formula for variance of a discrete random variable
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δ²=E(X-µ)²=∑(x-µ)²p(x)=∑x²p(x)-µ²
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Bernoulli
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A random variable that can assume only two possible values: 1 (success) and 0 (failure)
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Binomial experiment
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An experiment in which identical trials are repeated and we are interested in the number of certain outcomes
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Binomial random variable
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The number of successes in n trials
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Finite interval
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An interval of the form (a,b) or [a,b]
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Semifinite interval
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An interval of the form [0,∞)
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Real line
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An interval of the form (-∞,∞)
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Area under the curve
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How to determine probability using a probability histogram
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Probability density function (PDF)
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Height of the curve at x
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Normal distribution
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A distribution that models population distributions with a symmetric mound shaped distribution
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Parameter
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Numerical descriptive measure of the population
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Sample statistic
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Numerical descriptive measure of the sample
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Central limit theorem
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If the sample size n is large then the sampling distribution x-bar is approximately normal with mean µ and variance δ/√n
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Target parameter
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The unknown population parameter that we are interested in estimating.
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Point estimator
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A rule or formula that tells us how to use the sample data to calculate a single number that can be used as an estimate of the target parameter (example: sample mean).
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Interval estimator (confidence interval)
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A formula that tells us how to use the sample data to calculate an interval that estimates the target parameter.
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