MGSC 291 Exam 1 University of South Carolina

putting in a dataset

data<-c(81,85,93,93,99,76,75,84,78,84,81,82,89,81,96,82,74,70,84,86,80,70,131,75,88,102,115,89,82,79,106)

length(data)

tells you how may entries are in your vector

sort(data)

puts data smallest to largest

summary(data)

returns the five number summary and the sample mean

mean(data)

sd(data)

sum(data)

sample mean

sample standard deviation

adds all the elements of the vector

seq(1,100,1)

seq(2,100,2)

tells r to create the sequence of numbers from 1 to 100 by 1 (1,2,3...)

1:100 does the same thing

(2,4,6...100)

data<-read.table(file.choose(),header=TRUE)

read.cvs

calls this dataset into r and you can name it whatever you want

getwd()

creates the current working directory and calls in your data too

dim(data)

checks dimensions of data

data

type the name of the data to see it

data[1:5]

calls in the first five rows of the dataset if you are working with a large dataset

head(data)

shows the first few rows

data[,1:2]

all rows for columns 1 and 2

data[1:5,1:2]

first five rows and first two columns

str(data)

structure of the object

View(data)

(capital V) puts your data in a viewable popup window

data$shoes

calls in the column called shoes

attach(data)

attaches data so you can work with it and don't have to keep calling it in

(if you attach several datasets that have the same column names r will be confused so you have to detach before attaching again)

data<-subset(data,Type=="WT")

you can pick out certain rows/columns and you can enter more == and "" in order to be more specific

hist(data)

give you a histogram

hist(data,breaks=c(60,70,80,90))

creates a sequence and breaks them

xlab="percents of ..."

x axis name

ylab="Frequency"

y axis name

main="..."

title of the graph

boxplot(data)

gives a box plot

boxplot(...~...)

gives side by side boxplots

boxplot(Oil~Type,names=c("WIld Type","Mutated"))

creates your own label for the boxplot

plot(X,Y)

create a scatterplot

X1 <- c(8,5,14,13,29)

X2 <- c(13,8,6,18,4)

X1 is first line with the numbers you want

X2...

dbinom(j,n,p)

gives P(Y = j) discrete binomial probability

pbinom(J,n,p)

gives P(<=J) = P(Y = 0) + P(Y = 1) + ... + P(Y = J) exp binomial prob

dpois(j,lambda)

gives P(X=j) poisson discrete

ppois(J,lambda)

gives P(X<=J) poisson exponential

pexp(x,lambda)

gives P(Y<=j) exponential

qexp(p, lambda)

gives the pth percentile exponential

pnorm(x,mu,sigma)

Pr{X < x} for X~N(mu,sigma) .....so, 1-pnorm(x,mu,sigma) gives Pr(X > x)

qnorm(p,mu,sigma)

gives the value in the normal distribution (with mean mu and sd sigma) that has p to the left of it

norm prob

pt(t,df)

Pr{T < t} for T~t(df)

t distribution

qt(p,df)

gives the value of the t distribution with df=df that has p to the left of it

qqnorm(data)

normal QQplot of the data.

t.test(data,conf.level=0.95)

Using it for a 95% confidence interval for the population mean

descriptive statistics

collecting, organizing, and presenting the data.

inferential statistics

drawing conclusions about a population based on sample data from that population.

statistic

is a number calculated from a sample and is used to estimate the parameter.

Parameter

is a number used to describe a population.

time series data

A variable that is measured at regular intervals over time

cross-sectional data

When a characteristic is measured on many subjects at the same time point (or same time frame)

data warehouse

These data are recorded and stored electronically, in vast digital repositories

big data

describe data sets so large that traditional methods of storage and analysis are inadequate.

types of variables

quantitative, qualitative

categorical

arise from descriptive responses to questions like "What kind of advertising do you use?".may only have two possible values (like "yes" or "no").may be a number like a zip code

Qualitative

don't have a meaningful numerical value

aka categorical variables

nominal

Categorical variables used only to name categories that don't have order (grocery, clothing, hardware)

ordinal

When data values can be ordered (freshman, sophomore, junior, senior)

quantitative

have a numerical value that works like a number

discrete

there are jumps between the possible values

continuous

there is another possible value between any two values

identifier variables

a unique identifier assigned to each individual or item in a group (social security number, student ID number)

pattern of a distribution

skewed left is when the little tail is to the left and most of the data is on the right (

qualities of a good graphical display(and things to avoid)

Avoid 3-D

GooD:

good title

sample size

units

Frequency

Relative Frequency

number of times an allele occurs in a gene pool compared with the number of times other alleles occur

Pie Chart

Categorical (qualitative) data

Displays parts of a whole

Not good when there are too many categories

Don't ever make it 3-D or "tilted"!

Bar Graph (frequency and relative frequency)

Categorical (qualitative) data

Can be horizontal or vertical

Can display parts of a whole or separate values

For nominal data, put bars in ascending or descending order

For ordinal data, put bars in order of categories

Pictograms

A pictorial symbol or sign representing an object or concept. Used by many non-alphabetic written scripts.(can be misleading with pictures like the people pictures if one is bigger and one is smaller)

Line Graph

Displays quantitative data changing over timeTime should go on the horizontal axisVariable should go on the vertical axisUse different lines to denote separate categories or groups

Boxplot

A graph of the five-number summary.

good at comparing two datasets next to each other

Histogram (frequency and relative frequency)

medium to large quantitative datasets•

Bins touch•

Choice of number of bins can distort features of the shape of the distribution

(Notice, a boxplot displays the same data as the histogram, but the histogram shows more details about the shape. )

Scatterplot

is used to depict two potentially related quantitative variables.-Each point is a pairing: (x1,y1), (x2,y2), etc.-Linear, curvilinear, or no relationships-Positive vs. negative relationships

Five Number Summary

Boxplot: An efficient way to communicate the measure of center and variation all at once

IQR

Q3-Q1

Range

Max-Min

Sample Mean (x bar)

average of the sample

balancing point

Sample Standard Deviation (s)(just its properties not how to compute)

sqrt s squared

Mode

Most frequently occurring value

According to the shape of the distribution know when to use:

Mean vs. Median

Sample Standard Deviation vs. Quartiles (IQR and Range)

Coefficient of variation and What it is and why it's used

Standard deviation expressed as a percent of the mean

Compare variation in datasets with different units or means

Estimating percent of observations within certain standard deviations

Chebychev's Inequality

Empirical Rule

K = 2

1 -1/22

= 1 -¼

= 0.75 so at least 75% of observations lie within 2 standard deviations

Chebychev's Inequality

For any population with a mean, μ, and standard deviation, σ, the percent of observations that lie within kstandard deviations of the mean is at least (1−1/ksquared)×100

Empirical Rule

For unimodal distributions that are roughly normal, approximately

68% of observations are within 1 sdof the mean

95% of observations are within 2 sdof the mean

99.7% of observations are within 3 sdof the mean

Z-scores - calculation, interpretation, properties and when to use

A Z-score represents the number of standard deviations an observation is above or below the mean

Beware of using Z-scores from skewed distributions (the Z-scores have the same shape distributions as the original observations)

Cannot compare a Z-score from a skewed distribution to a Z-score from a symmetric distribution

2 basic rules (probability is on scale from 0 to 1; sum of probability of all (disjoint) events in sample space = 1)

P(A) = 0 → Event A will not occurP(A) = 1 → Event A will surely occurP(A) = ½ → Event A will happen 50% of the time

Compute probabilities for complements, unions, intersections, conditional events

Complement:

The probability of the complement of an event, P(Ac), is equal to one minus the probability of the event

Union:

The probability that event A or B occurs (at least one of the events happens) is:P(A U B) = P(A) + P(B) -P(A ∩ B)

Intersection:

General intersection rulefor two events both occurring (always works): P(A ∩ B) = P(A)P(B|A)= P(B) P(A|B)

Disjoint events

Two events, A and B, are said to be disjoint(or mutually exclusive) if they share no outcomes in common. Disjoint events have no intersection.P(A ∩ B) = 0

independent events

Two events are independentif the occurrence of one event does not affect the probability of the occurrence of the other event.Examples: Two flips of a coin, two rolls of a die, two spins of a roulette wheel

tree diagram( how to use and determine independence)

Bayes' Theorem

go over some examples in the notes

a theorem describing how the conditional probability of each of a set of possible causes for a given observed outcome can be computed from knowledge of the probability of each cause and the conditional probability of the outcome of each cause.

mean for a discrete random variable

The mean or expected value of a discrete random variable X is muX= E(X)= Σxi P(xi)

variance(sd) for a discrete random variable

The variance of a discrete random variable Y is sigmaX2= Var(X) = Σ[(Xi-mux)squaredP (xi)]

Binomial (when to use, mean and variance(sd))

A fixed number, n,trials take place, where:

1.Each trial has only two possible outcomes ("Success" or "Failure")

2.Probability of "Success" is a constant pfor every trial

3.Trials are identical and independentThese are called Bernoulli trials.

Poisson (When to use

Lambda (λ) and its relation to the exponential distribution

1)The event cannot occur twice at exactly the same time.

2)No occurrence of the event being analyzed affects the probability of the event re-occurring (events occur independently).

3)The expected number of occurrences of the event during any such interval is a constant

Note: The Poisson Distribution is only designed to be applied to events that occur relatively rarely.

Normal distribution (Properties, Standard normal distribution, z-scores and their interpretation)

Properties:

Lengths and weights of newborn babies

Scores on SAT

Cumulative debt in college students

Advertising expenditure of firms

Standard Normal Distribution

has a mean of 0 and a standard deviation of 1

(normal can be transferred to standard by z=x-u/sd)

z-scores

is the number of standard deviations from the mean a data point is. But more technically it's a measure of how many standard deviations below or above the population mean a raw score is

u is the population mean and sigma is the population standard deviation

Parameter (ch. 7)

number used to describe a population

(usually do not know the value of a parameter; it is a fixed number)

Statistic (ch. 7)

is a number calculated from a sample and is used to estimate the parameter

(we know the value; it will change from sample to sample)

sample survey

when the respondents in a survey provide their own data

census

when a survey attempts to use the entire population as the sample

Central Limit Theorem

the sampling distribution of a sum or percentage will become approximately normal as the sample size gets larger