Statistics Midterm

population

total set of individuals that are of interest

parameter

summarizes the population (μ-mean, and σ-standard deviation)

sample

portion OF the total population (should lack bias)

stastistic

value summarizing the sample (Xbar-mean, s-standard deviation)

cases

individual items on which data is collected

respondents

people who answer survey

subjects/participants

people who are experimented on

experimental units

objects of an experiment when not a person

graphs for categorical data

• bar Chart

• pie Chart

graphs for quantitative data

• dot plot

• histogram

• box plot

histogram benefit

• seeing distribution of the data

• good to compare two or three groups

to describe quantitative data

Shape

Outliers (and other unusual features)

Center

Spread

shape

• modality/peaks

• symmetry and skewness

outliers (what it is, how to find)

• data value that is far above or far below the rest of the data

• upper outlier: Q3+1.5(IQR)

• lower outlier: Q1-1.5(IQR)

center

median and mean

median (what it is, how to find, when to use it)

• the middle of the data

• order the values and find the one that is positionally the middle value

• best for symmetric distributions

• resistant to outliers

mean (what it is, how to find, when to use it)

• the average

• ybar=total/n

• good for skewed data

• not resistant to outliers

what happens to the mean when the data is skewed

it will be further in the direction of the skewness (ex. right skewed data will have a higher mean than median)

spread (2 main kinds)

• standard deviation

• IQR

standard deviation (what is it, how to find, what different sd’s mean)

• distance of a value from the mean, how tightly packed the data are

• s = √(∑ (y−ybar)²/n−1)

• small sd= data values less spread out and closer to the mean

quartiles

• Q1 (median of lower half of data)= 25th percentile

• Q2 (median)= 50th percentile

• Q3 (median of upper half of the data)= 75th percentile

• Q4= max data in the value, 100th percentile

5 number summary

median, quartiles, min, max, IQR, and range

IQR (how to find, benefits, drawbacks)

• IQR=Q3-Q1

• reasonable summary of distribution spread

• not affected by outliers

• most people don’t know what it is

center/spread combos

• mean + standard deviation

  • both best with roughly symmetric data

  • based on magnitude and values

• median + IQR

  • both best for skewed data

  • based on order of the data

independence

distribution of one variable is the same for all categories of another

dependent variables

have an association between the two variables

observational studies

• look at sample of data to learn more about larger population

• often lead to contradictory results because nothing is controlled for or really conclusive

boxplots

• central box shows the middle half of the data

• height of box=IQR

• whiskers show skewness if they are not roughly the same length

• if median is centered=roughly symmetric middle half of data

• compare many groups

z-score (what is it, how to find it, what do small/large, +/- z scores mean)

• how far a value is from the mean in terms of standard deviations but is useful for re-expression of values

• z=(y-ybar)/s

• small/large: close/far from the mean (respectively)

• positive/negative: above/below the mean (respectively)

shifting data (adding or subtracting to all values) does what?

• changes only the position, not the spread

• mean, median, and quartiles (location) are changed

• standard deviation, range, and IQR (spread) remain unchanged

rescaling the data (multiplying it by a constant) does what?

changes points and spread

the normal model (notation, empirical rule)

• N ( μ , σ )

• 68-95-99.7 rule (1σ away, 2σ away, 3σ away)

percentile

the percent of data that falls at or below some value

ex) the 80th percentile HAS 80% of the data below

• how to get from values (variables) to z-scores

• how to get from z-score to area (%, proportion percentile)

• standardize: z= (x-μ)/σ

  • normcdf(lower z, upper z)

• invnorm (percentile or area below)

scatterplots (what do they show, what are they good for)

• relationship between two quantitative variables

• detect patterns, trends, relationships, extraordinary values

roles for variables (what’s on y and x axis)

• y axis= response variable, what we want to predict

• x axis= explanatory variable, what is providing info and helping to predict response variable

correlation coefficient (r) (how to find, what it means, conditions)

• in ti-84 go to STAT→CALC→8

• strength and direction of linear relationship (btw -1 and 1)

• need a nearly linear relationship, quantitative variables, and no strong outliers

• has no units

• changing x and y does not change correlation

correlation does not equal causation

• lurking variables

• correlation means a LINEAR relationship, an association is ANY relationship

residual (how to find it, what is it)

• y-ŷ (ŷ= predicted value)

• residual is the difference between the observed value and the predicted value

• points above the line have + residuals and below the line have - residuals

line of best fit (lease squares line, regression line)

(what is it, what to do with it, how its written)

• line for which the sum of the squares of the residuals is the smallest

  • squaring the residuals makes them all positive

• best fitting line will have small residuals

  • if you have small residuals, that means you predicted the results of your data well

• y=mx+b

line of best fit interpretation

• for every (slope) +/- we can expect to see an (intercept) +/- in the (y)

conditions for using regression

• quantitative

• straight enough

• no outliers

what is r²

• the fraction of the data’s variation accounted for by the model

• found by squaring the residual and subtracting 1 from it

  • ex) r²= 0.76² = 0.58

    1-r² =1-0.58 = 0.42= 42%

• interpreted as:

  • 42% of the variation of y is accounted for by the residuals

  • 58% of the variation of y is accounted for by x

extrapolation

• the farther away from the mean, the less trust should be put in the predicted value of y

when do values have leverage

• x-values that are far from the mean of the rest of the x-values

• extreme in y have large residuals

when are values influential

• if omitting it from the analysis changes the model enough to make a meaningful difference

• determined by:

  • residual

  • leverage

simple random sample

• guarantees that each person has an equal chance of being selected

• ensures that a non-representative sample is unlikely to occur

stratified random sampling

• divides the population into HOMOGENOUS groups where proportionate amounts from each group are randomly selected

• estimates will be more precise, but watch out for simpson’s paradox

cluster sampling

• dividing population into smaller groups

• less expensive and less time consuming

• some populations are naturally broken into clusters already

• HETEREROGENOUS

multistage sampling

• combination of several sampling methods

  • ex) for a college, select dorms as a cluster and then continue with other sampling methods such as a census, etc.

observational studies

• researchers do not assign choices, passively observe participants

• bad for cause-and-effect establishment

• tough to handle lurking variables

retrospective studies

• collect data on something that has already occurred

• similar pros and cons as obs. studies

prospective studies

• study where we identify subjects in advance and collect data as events unfold

• possible to isolate variables

• can be expensive and time-consuming

4 principals of experimental design

1. control

   • control what you can

2. randomize

   • randomize the rest

3. replicate

4. block

   • group similar individuals together and randomize within each of these blocks

   • helps account for variability due to the difference between blocks