statistics 121 test 1

population

population

the entire group of individuals that is the target of our interest; generally too big to actually measure or observe

sample

subgroup of the population which we can examine or observe, measure and collect data from

individual

single entity that is being observed

variable

characteristic measured on each individual

quantitative variable

variable whose possible values are meaningful numbers

categorical variable

variable whose possible responses are non-quantitative categories (words/labels/attributes)

measurement

value of a variable for an individual

data

measurements for a set of individuals (Goal of Statistics: convert this to useful information)

data set

data identified with contextual information (who was observed, what was measured, why is study done) often given in a table

EDA (exploratory data analysis) goals

• organize and summarize data

• discover features, patterns and striking deviations

• interpret patterns in context

• include visual displays and numerical values

single variable pattern

distribution of a variable: summary of data one variable at a time (all the possible values and how often they occur)

process of statistical problem solving

1. Collect data

2. Summarize data

3. Interpret data

parameter

numerical fact about the variable in the population

statistic

numerical fact about the variable in the sample

convenience sampling

select individuals in the easiest possible way

volunteer response sampling

individuals select themselves

quota sampling

force the sample to meet specified quotas

simple random sample (SRS)

every possible set of a specified size has an equal chance of being selected

cluster sampling

a random sample of clusters is taken and all individuals in selected clusters are included in sample

stratified random sample

select a random sample (SRS) from each stratum and combine these SRSs together

multi-stage sample

take a sample at each hierarchical level of the population

treatment

the condition applied to a subject in an experiment (one of the subcategories/values of the explanatory variable)

lurking variables

variables that affect both the explanatory and response variables but are not measured or included as a planned factor in the study

control

an effort to reduce the effects of lurking variables

confounding

situation in which effects of lurking variables cannot be distinguished from effects of factors

historical comparison experiments

study involving only one treatment, where treated subjects are compared to untreated subjects from some external source

unreplicated experiments

assigns one subject only to each treatment

confounded experiments

treatment groups are handled differently in some way OTHER than the treatment

undercoverage

some individuals have no possibility of being selected

non-response

some selected individuals choose not to be in the sample because they refuse to provide information or cannot be contacted

misleading response

people lie or give inaccurate answers (often about sensitive issues)

interviewer effect

person asking questions influences responses (for in-person/phone surveys)

question order effect

the order that questions are asked promotes certain responses

question wording

the way a question asked leads, misleads or confuses

open questions

allow for almost unlimited possible responses (short answer), less restrictive but more difficult to analyze

closed questions

limit response options (multiple choice), easier to analyze but may be biased by the options provided. should include "other/unsure" option

observational studies

individuals are not assigned to treatments, are self selected, cannot conclude causation

experiment

study where individuals are assigned to treatments, causation okay if valid

subject

individual to which treatment is applied

response variable

characteristic measure on each subject; outcome of interest

explanatory variable

characteristic/measurement that is use to predict or explain changes in the response variable; variable we think could help us know about the response (measured earlier or more easily); independent variable

factor

planned explanatory variable

comparison

two or more groups; controls lurking variables by including comparison treatments

randomization

randomly assign subjects to groups; neutralizes effects of lurking variables by assigning subjects to treatments using a random device

replication

two or more subjects in each group; assign more that one subject to each treatment to detect important effects

double blinding

neither subjects nor the researchers in direct contact with the subjects know which treatment is received

placebo effect

favorable response of a human subject to a placebo because of trust in the medical provider or belief that the treatment will work

diagnostic bias

diagnosis of subjects is biased by preconceived notions about the effectiveness of the treatment (person administering treatments expects certain responses)

lack of realism

realism is compromised by the conditions of the study

hawthorne effect

people in experiment behave differently than they would normal behave, not like real life

non-compliance

subjects fail to submit to the assigned treatment or refuse to follow the protocol of the experiment

principles of data ethics

• safety and well-being of the subjects must be protected • all individuals must give their informed consent before data are collected • individual data must be kept confidential

randomized controlled experiment

randomly assign subjects to treatments, grouped by treatment

randomized block design

randomly assign to treatments within blocks, grouped by treatment or by block

benefits of randomized block design (RBD)

• removes confounding of lurking variables

• reduces chance variation by removing variation associated with the blocking variable

• yields more precise estimates of chance variation

matched pairs

two treatments; matched individuals or two measurements per subject

three principles of experiments

• randomly assign two treatments to two individuals or randomize the order of treatment application to each individual

• replication = number of pairs

• compare the two treatments

analysis of distribution of quantitative data

• always plot data first

• look for an overall pattern and for striking deviations

• look at shape, center, spread of distribution

• add numerical summaries to supplement graph

• if pattern is regular, use mathematical model to describe data

symmetric and bell shaped distribution examples

blood pressure, IQ, biological factors

symmetric and bell shaped distribution

mean, median, and mode are the same

right skewed distribution

concentration of data on left, tail extends to the right; mean > median

right skewed distribution examples

salary, home price, children, economic variables

left skewed distribution

concentration of data on right and the tail on the left; median > mean

left skewed distribution examples

test scores, olympic high jump

bimodal distribution

a distribution with two modes

bimodal distribution examples

speed limits, restaurant patrons

flat or uniform distribution

relatively equal across graph

flat or uniform distribution examples

rolling a die, day of the month born

center

typical, middle value; half of data to each side

spread

consistency/inconsistency of data; look for maximum and minimum

outliers

values that are far outside most of data

• is data point miscoded?

• unusual conditions?

• should data point be excluded?

mode

most frequently occurring score, corresponds to a peak

median

the middle score in a distribution; half the scores are above it and half are below it

mean

center of gravity; the arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores

mean vs median

• construct graph to evaluate skewness and outliers

• use median if distribution is markedly skewed or outliers are present

• use mean if distribution is roughly symmetric

range

maximum - minimum

interquartile range (IQR)

the difference between the first and third quartiles

standard deviation

average distance of values from the mean

first quartile (Q1)

a number for which 25% of the data is less than that number; same as the median of the data which are less than the overall median

second quartile (Q2)

median

third quartile (Q3)

a number for which 75% of the data is less than that number; same as the median of the part of the data which is greater than the median

5 number summary vs 2 number summary

use 5 number for skewed, and 2 number for symmetric

5 number summary

minimum, Q1, median, Q3, maximum

random phenomenon

individual outcome unpredictable, but outcomes from large number of repetitions follow regular pattern

sample space

the set of all possible outcomes

event

a collection of possible outcomes

probability of an outcome

The proportion of times that an outcome occurs in many, many repetitions of the random phenomenon

probability rules

• 0<P(A)<1

• summation of all probabilities is 1

• if two events cannot occur simultaneously, the probability of one or the other equals the sum of separate probabilities

• probability of event not occurring equals one minus the probability of event occurring

theoretical probability

number of favorable outcomes divided by total number of possible outcomes

empirical probability

number of outcomes divided by total of repetitions

law of large numbers

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the theoretical probability of the outcome

probability

the long-run relative frequency with which an event will occur

probability distribution

all possible events and their associated probabilities

random variable

a variable whose value is a numerical outcome of a random phenomenon

continuous random variable

a variable that can take on any possible value, all values cannot be listed

discrete random variable

variable whose possible values are a list of distinct values

𝜇

mean of a population

x-bar

mean of a sample

s

standard deviation of a sample

𝜎

standard deviation of a population