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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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