stad 2300

Entities measured and studied

Individuals

The entire group to be studied

Population

Subset of pop. from which you collect data

Sample

Numerical summary of population

Parameter

Numerical summary of sample

Statistic

Methods for summarizing collected data via tables, graphs, numerical summaries (averages or percentages)

Descriptive statistics

Methods that take a result from a sample, extend it to the population and measure the reliability of the result. Always contains uncertainty

Inferential stats

Any characteristic of the individuals within the pop.

Variable

What values a variable takes and how often it takes those values

Distribution of variable

If observation takes on numerical values

Quantitative

If each observation belongs to a set of categories

Qualitative variable

Measurement if the values of the variable name, label, or category. Doesn’t allow for ranking

Nominal (qualitative)

Measurement if the variable has properties of nominal level but can be ranked

Ordinal (qualitative)

Measurement if the variable has the properties of the ordinal level but each difference in value has meaning. (A value of zero doesn’t mean the absence of the quantity)

Interval level (quantitative)

Measurement if the variable has the properties of the interval level but the ratios of the value has meaning. A value of zero means absence of quantity

Ratio level (quantitative)

When someone assigns individuals in a study to certain experimental conditions then observed outcomes

An experiment

When someone observes and records the behavior of the individual without imposing any conditions

Observational study

Explanatory variable that was not considered in the study but that affects the value of the response

Lurking variable

characteristics typical of those possessed by the population of interest

Representative sample

n subjects from a population of N size in which each possible sample size of n has the same chance of being selected (n = sample, N = pop.)

Simple random sample

When dividing the population into separate groups called strata then selects a simple random sample from each stratum. (Each stratum should be homogeneous in some way w respect to variable of interest)

Stratified sample

Dividing the population into large number of clusters then a simple random sample of clusters is selected and all individuals in the selected clusters are included

Cluster sample

Selecting every kth individual from a pop. first individual selected corresponds to a random number btween 1 and k

Systemic sample

When the technique used tends to favor one part of the pop. over another

Sampling bias

When individuals selected to be in the sample who do not respond have different opinions from those who do respond

Non response bias

Response bias

When the answers on a survey do not reflect the true opinions of the respondents

Subject do not know what treatment they’re receiving

Single blind experimen

Neither subject nor researcher knows what treatment they’re subject is getting

Double blind experiment

frequency distribution

listing each category of data and the number of observations in each

listing category of data together with the relative frequency and the proportion of observation in each category. found by taking the frequency for a particular category and dividing by total number of observations

relative frequency distribution

mean

sum of observation divided by number of observations (average)

μ - population

x̄ - sample

median

the middle observation when they’re listed in ascending order

M

mode

observation that happens most frequently

if median is bigger than mean

left skewed

median = mean

symmetric

median less than mean

right skewed

standard deviation

typical value for how far the data fell from the mean

s - sample

σ - population

always greater or equal to 0

not resistant

1 standard deviation of the mean

68%

2 standard deviations of the mean

95%

3 standard deviations of the mean

99.7%

z score

distance that a data value is from the mean in terms of the number of standard deviations

z score equation

(observation - mean)/standard deviation

approx. 68% of z score’s

fall between -1 and 1

approx. 95% of z scores

fall between -2 and 2

approx. 99.7% of z scores

fall between -3 and 3

pth percentile

means p% of the observations fall BELOW the pth percentile and only (100-p)% fall above

first quartile Q1

25th percentile

Second quartile, median (M)

50th percentile

third quartile Q3

the 75th percentile

finding quarterlies

• arrange data in order

• find the median of all of them

• then find the median of the numbers above and below the median

interquartile range IQR

Q3-Q1

Lower fence for outliers

Q1-1.5(IQR)

Upper fence for outliers

Q3+1.5(IQR)

for approx. symmetric distributions

use the mean and standard dev.

for distributions that are skewed or have outliers

use the median and IQR

five number summary

minimum, Q1, M, Q3, maximum

approx. symmetric distributions the median is

near the center of the box

distributions that are skewed right the median is

slightly left of the center of the box.

right whisker will be longer than left whisker

distributions that are skewed left the median is

slightly right center of the box

left whisker will be longer than the right whisker

response variable (y)

measures outcome of study

explanatory variable (x)

explains or influences change in the response variable

a value of r close to 0 indicates

that the relationship is not linear

weak correlation

0.2-0.5

moderate correlation

0.5-0.8

strong correlation

0.8-1

correlation coefficient, r is

not resistant

unitless

if the absolute value of r is greater than critical value

linear relation exists between the two variables otherwise, no linear relation exists

least-squares regression

y(hat)=b_o+b₁x

yhat is predicted value of y for value of x

b_o is the average value of y when x=0

b₁ is the change in the average value of y when x increases by 1 unit

the way we quantify uncertainty

probability

probability, experiment is

an act or process of observation with uncertain results that can be repeated

Multiplication Rule of Counting

Total outcomes for an experiment = product of the number of outcomes for each individual task

Repeating a task with n possible outcomes, r times → total possible outcomes

use when you can repeat and when order does not matter

n^r

Selection where order IS important, without replacement (e.g., 1st/2nd place finishers)

Permutation

Formula for number of permutations of r objects from n

nPr = n! / (n-r)!

Selection where order is NOT important, without replacement (e.g., picking a group of friends)

Combination

Formula for number of combinations of r objects from n

nCr = n! / [r!(n-r)!]

Any collection of outcomes from a probability experiment

Event

An event with only one outcome

Simple event (eᵢ)

The probability of any event A must satisfy

0 ≤ P(A) ≤ 1

An event with a low probability of occurring (often < 0.05)

Unusual event

Probability formula when outcomes are equally likely

P(A) = (# outcomes in A) / (total # outcomes)

Estimating probability using real data: (# times A observed) / (# repetitions)

Empirical approach

the median is (outliers)

resistant

Standard deviation (outliers)

not resistant

IQR (outliers)

resistant

w/out replacement

order is important

permutation (n!)

pareto, pie, bar charts

qualitative

histogram, stem and leaf

quantitative

The Complement Rule states

P(Aᶜ) = 1

so P and Ac must add up to 1

if (Q2 - Q1) is greater than (Q3 - Q2)

skewed left

if (Q2 - Q1) is less than (Q3 - Q2)

skewed right

higher IQR means

more varied range

probability refers to

what is expected in the long-term, not short-term.