Stats Exam 1

vocab and formulas + examples

Treatment group

the group/subject that received the treatment

ex: drug for sleep; receive a drug that shows real effect 

Control group

the group/subject that did not receive the treatment

ex: drug for sleep; receive a placebo (sugar pill) instead of real drug

Randomization

randomly assigning participants to treatment or control groups

Why is randomization important?

Helps reduce bias in both control and treatment groups 

confounding factors

a variable that influences both the independent variable (what you’re testing) and the dependent variable (the outcome), making it hard to tell if the observed effect is real

ex: people who carry lighters have higher rates of lung cancer, CF = smoking

basically other factors that explain the results of the experiment

placebo effect

fake treatment that has no actual effects

double blind

neither the participants nor the researchers know who is in the treatment group and who is in the control group;

actual vs placebo no one knows until the end

explanatory variable

the subject that is changed/controlled in a study; often manipulated by researchers 

response variable

the result/subject that is observed

experimental study

researchers control and randomly assign participants (treatment vs control), where variables clearly affect another variable 

ex/key words: randomly, measure, experiments, “effect of..”, “applied/given”

observational study

researchers observe and collect data

ex/key words: study, survey, observe, “association”

association

Apply to: observational and experiment

Def: two variables are associated/one doesn’t necessarily effect the other 

ex: A and B happen together, but A doesn’t necessarily make B happen; glass is broken and water is spilled

causation

Apply to: experiments that are random and controlled; CANNOT be OBSERVATIONAL

Def: directly affects another variable 

ex: A actually makes B happen (cause effect); push glass and it’ll break

confounding variable examples

sunscreen can increase risk of skin cancer; A and B are associated 

CF: genetics of skin cancer

Variable A: Using sunscreen

Variable B: risk of skin cancer

population/population of interest

the entire group of the study

average number of hours high school student sleep

ex: all high school students

sample

the group that is being studied

average number of hours high school student sleep (150 students)

ex: the number of students in this study = 150 students

representative

choosing sample (participants in study) at random; will be more likely to be representative

Identify population, sample, explanatory and response variable, population of interest, type of variables (numerical, ordinal, categorical) in the study:

Average number of hours all 150 high school students in the U.S. sleep per night

Population: All high school students in the U.S.

Sample: 150 high school students

Explanatory variable: none, because it’s observational.

Response variable: Number of hours of sleep per night

Types of variables:

Number of hours of sleep: Numerical (quantitative, continuous)

Grade level (if collected): Ordinal (9th, 10th, 11th, 12th)

Gender (if collected): Categorical (male, female, other)

simple random sampling 

every member has a chance of being selected

convenience sampling

individuals that are selected who are easily accessible/convenient

selection bias

when sample (individuals) collected not representative of the population (topic of study)

In a data set, row is____, the column is _____?

row = observational unit/case

column = variable

Numerical variable — include what two types? (Quantity)

discrete and continuous 

Definition: measure/record numerical data

ex: Number of hours students sleep per night — 7.5, 8 hrs

Age of people → 18, 25, 40 years

Height in cm → 160, 175, 182 cm

Test scores → 85, 92, 78

Categorical variable — consist of what two types? (Quality)

ordinal and nominal

Definition: represent categories/groups NO NUMBERS

ex: Gender → male, female, non-binary

Eye color → blue, brown, green

Type of pet → dog, cat, bird

Yes/No responses → yes, no

ordinal variable def/examples

meaningful order

ex: Education level → High School < Bachelor’s < Master’s < PhD

Satisfaction rating → Unsatisfied < Neutral < Satisfied < Very Satisfied

nominal variable def/ex

no natural order

ex: Eye color → Blue, Brown, Green

drinks —> pepsi, sprite

Type of pet → Dog, Cat, Bird

discrete def/ex

numerical value that is countable/separate values 

ex: Number of siblings → 0, 1, 2, 3…

Number of cars in a household → 0, 1, 2…

continuous def/ex

def: any value in range (measured)

ex: height, weight 

numerical vs categorical + subsections

Numerical — can be measured (quantity)

• discrete: countable numbers (whole numbers) ; ex: number of siblings

• continuous: any value of range; ex: height, weight, fraction, decimal

Categorical — category of group; no numbers, quality

• nominal: no order; ex: eye color, gender, pet

• ordinal: has order; ex: ratings (satisfied neutral bad), educational level (freshman soph junior senior)

Dot plot question examples

1. fewest total number in data set = lowest value on chart;  ex: 1-4, one is the lowest

2. largest total number = highest value on chart

3. most frequently observed total number = has the most/highest values on chart; ex: column 3 has the most people voted

4. least frequently observed = the shortest/least value on chart

intervals (x axis)

intervals = range of data set

500-1000

1000-1500

frequency (y axis)

number/count in the intervals

500-1000 has 2

1000-1500 has 5

symmetric vs right vs left vs bell/no bell

1. symmetric = data clustered in middle w/ bell

2. right skewed = tail on right

3. left skewed = tail on left

4. not bell but symmetric = two bell, split in half

parameters

values calculated from population

ex: average number of hours student sleep

statistics

values calculated from samples

ex: 150 students randomly selected from diff schools

Identify population vs sample vs parameter vs stat example

population: all high school students

sample: 150 students randomly selected

parameter: average hours for all student

statistic: average hours for 150 students

mean formula

x bar = sum of all values/total number of values

ex: 1 2 3

x=1+2+3/3

median formula

middle value

odd = middle single value; ex: 123, median = 2

even = middle two values/2; ex: 123456, median = 3+4/2

Reading the mean vs median on a histogram

symmetric diagram: mean = median

right skewed: mean greater than median

left skewed: mean less than median

• mean is the tail on chart

range formula

range = max - min value

IQR Interquartile Range Formula

IQR = Q3-Q1

Percentile Ranges: 25, 50, 75

25 = median of lower half data, Q1

50 = median (middle line on boxplot)

75 = upper half after the median, Q3

Finding IQR Examples

50, 51, 56, 61, 70, 71, 80, 84

Q1: 51,56/2

Q3: 71,80/2

IQR: Q3-Q1

Notations for Parameters and Statistic 

Parameter

Mean: u

variance: o²

SD: o

proportion: p

Statistics

Mean: x bar

variance: s²

SD: s

proportion: p^

deviation formula

Sample deviation: x-x bar

Population deviation: x-u

squared deviation formula

(x-x bar)² OR (x-u)²

variance formula

measure average square deviation

Population (o²) = sum of (x-u)²/n

sample variance (s²) = sum of (x-x bar)²/n-1

SD formula; square root variance

Population SD: square root population variance

sample SD: square root sample variance

proportion formula

cases in category/total number of cases

ex: 20 out of 50 students have blue eyes

p=20/50

Proportion tyoes: observed sample vs population

sample = p hat

population = p

probability formula

number of desired outcomes/total number of possible outcomes

ex: chances of rolling a 4 ; 123456 = 1/6

Probability is always between 0 and 1

0 = unlikely

0.5 / 50% = half likely/not likely

1 = positive will occur

sensitivity vs specificity 

true positive vs true neg

specificity = 1-value

base rate def

proportion of population 

independent conclusion 

two variables A and B, is independent if A is not affected by B

null hypothesis (Ho)

no change, difference, or relationship between variables

if null is true = due to chance

null value = zero means independent

alternative hypothesis (Ha)

there is change, a difference, or relationship between variables

= not due to chance

observed difference

P (value A) - P (value B)

p-value

a null model used to calculate probability

• helps to decide whether to reject the null hypothesis

P-Value Chart

greater than 0.10 = little evidence

between 0.05 and 0.10 = some evidence

between 0.01 and 0.05 = strong evidence

between 0.001 and 0.01 = very strong evidence

less than 0.001 = extreme strong evidence