Stats Cumulative IN ORDER (11/3/25)

Population

ALL that we are examining

• Be specific, #’s, species, etc

• It’s the WHOLE GROUP, not just the ones being experimented on

Sample

Portion of the population that we are actually examining

• Smaller group within population

• Usually the group that’s being experimented on

Cencus

Method for collecting data from every individual in the population

• Kinda like population (the experimentees)

• Individuals info

Sample design

Method used to pick the sample from the population

Voluntary response sample

“Do you want to volunteer in this experiment?”

• When someone or a group decides themselves to take part in/respond to the suvey/experiment

• BIASED

Convenience sample

Choosing people that are easy to reach

• Your sample survey should be your unbiased estimate of your parameter population

• Always underrpresentative of the population

Bias

Systematically favoring certain outcomes

• Deviates away from true population parameter

Systematic random sample

Choosing every nth person

• Using a chance process

• Random bias (bias is questioned)

Probability sample

Each member of the population is given a known chance to be selected

The “chance” isn’t necessarily equal

EX) Choose a sample that consists of 20% cats and 80% dogs

Sample survey

A study that chooses a sample that represents a specific population

Simple random sample

A method of selecting participants for a study where every member of the larger population has an equal chance of being chosen for the smaller sample

• A sample of size n is chosen from a population such that every group of n individuals in the population has an equal chance to be selected as the sample

  • In theory, this should be an equal shot, but it’s about the sample size

  • Every 10th person, not every single person, had a chance

How to choose an SRS

• Names out of a hat (small group)

  • Avoids bias

• Random Digits

Random Digits

Used to randomize a list of individuals before selection for a sample

1. Give everyone in the group a number based on name (alphabetically)

2. The line will already be given to you, so say the first numbers to come up will go in group one, and continue until no more spots, then go to the next groups

3. Make sure to say no repeated numbers and the numbers you will omit

   EX) Will choose numbers 01-30, will omit any repeated numbers as well as 00 and 31-99

Stratified random sample

A sampling that separates population into groups (stratas) then chooses a separate SRS from each group

Those that are selected in each group are then combined to form one complete sample

EX) Countries can be stratified into rural, suburban, and urban. From each we can find an SRS and then combine our results into one larger sample

• This will always have someone in each group, an SRS has the possibility of having everyone in one big group

Multistage sample

• Samples chosen in stages

1. Find an SRS of states in the US

2. Find an SRS of high schoolers in those states

3. From these high schoolers, find an SRS of students from each school

   NOT A COMMON SAMPLE MAKER

Cluster sample

Classify the population into groups of individuals that are located near each other (clusters). Choose an SRS of the clusters. All individuals in the clusters are included in the sample

• Cluster samples are often chosen for ease, and thus they may have as much variability as the population itself

What’s the difference betwen CLUSTER + STRATIFIED?

1. Each cluster looks like the population but on a smaller scale

2. Each stratum contains similar individuals but there are larger differences between strata

Undercoverage bias

When people are left out of the sampling process

EX) Homeless people, college students, people without phones, etc

Non-response bias

Occurs when people can’t be contacted (away from phone, etc) or refuse to participate

Response bias

A systematic pattern of incorrect responses in a sample survey, Sometimes thought of as a liar or being influenced by the interviewer

Poorly worded questions bias

EX) Should people slaughter cute baby seals for their fur?

Observable study

Measurements representing a variable of interest are observed and recorded without controlling any factor that night influence their values

• No treatment is imposed

EX) Didn’t stay up late cuz Mr. Collins gave them a pill

• No testing

Experimental study

Observations after imposing some treatment on a group of subjects to measure their response

What’s the difference between OBSERVABLE + EXPERIMENTAL studies?

Experiments are the only source of fully convincing data if our goal is to understand cause+effect

Observational evidence can NOT do this

Experimental units

Individuals that are being experimented on

EX) 300 male volunteers testing drugs A and B, the EU would be the 300 male volunteers

Treatments

The combo of the variables being used on the individuals

EX) 25g of sugar + drug A, 25g of sugar + drug B, etc

Factors

The explanatory variables in the experiment

EX) 0% sugar, 10% sugar, 25% sugar, the factor is the sugar concentration and there would only be 1 factor in this example

Levels

The specific values of the factors

EX) 0% sugar, 10% sugar, 25% sugar, those would be the levels

Confounding variables

Variables that may effect the experiment that are out of their control

EX) 400 men on weight loss drugs, outside factors like lifestyle, genetics, etc would be the confounding variables

Explanatory variables

Pretty much what factors are

EX) Sugar concentration instead of the actual percentages of sugar concentration

When do you use a blocked web design instead of just a normal web design?

When there may be differences in ages or gender that will effect how the treatments effect them

Why use block design (sentence)?

Blocking accounts for variability in -what’s being measured-that may arise due to the variable you are blocking

Matched pairs

Subjects are their own control groups

• Pair 2 people by a common characteristic - randomize who gets the treatment and who goes to the control group - Test - Compare - Replicate

• Each person is their own control. Randomize the treatment order for each person. Before and After study. Compare treatment and replicate

Why must treatments be randomized in matched pairs?

So one doesn’t have an effect on the other, no order bias

Variability

Simulations

Figure out what people are saying without asking

EX) At WHS, 20% take art, 30% take child lab, and 50% take band

Let 0-1 take art, 2-4 take child lab, and 5-9 take band

Single blind

Participants don’t know but data receivers do (placebo)

Double blind

Neither the participants nor the data receivers know (placebo)

What are the 4 principles of experiment design?

Compare, control, randomize, replicate (not only replicate experiment, but more importantly the sample sizes which must be large for more accuracy)

What is statistical inference?

Drawing a conclusion about a population from information obtained by a random sample of that population

What is the importance of “blinding” in an experiment

To reduce bias

When is the best time to use a cencus?

With a small population where the only data being collected is facts (like their age, gender, etc)

Quantitive data

Numerical data, can be measured

Categorical data

Records the group an individual belongs to

Distribution

Distribution of a variable tells us what values the variable takes, and how often it takes them

Outliers

An individual value that falls outside the overall pattern of data

Potential outlier

An outlier that you yourself didn’t calculate, no matter how obvious the outlier is

Inference

Drawing conclusions that go beyond the data at hand

When do you make the bars on a bar graph touch and vis versa?

Touch when data is in groups and related, don’t touch in all other circumstances

Mean

The average of all numbers, may not reflect the data well due to potential outliers

• It is nonresistant which means it is sensitive to the influence of extreme potential outliers

Median

The middle most number

• It is resistant meaning it’s not impacted by extreme values

• Can “eliminate” outliers

• Typically used when the data is skewed

Mode

Most frequently observed value

Bimodal

When there are 2 modes

Multimodal

Usually accepted for multiple modes

What’s the mean and median when data is symmetrical?

Mean and median are equal

What’s the mean and median when the data is skewed left?

Mean is less than the median

What’s the mean and median when the data is skewed right?

Mean is greater than the median

Marginal distributions

The totals (usually in percents)

Conditional distribution

Locked into a certain variable/group on the table

Assosiation

Two variables are related if knowing the value of one variabe helps predict the value of the other

How do you describe data plots?

SOCS

Shape, outliers, center, spread

• How does the data skew?

• Are there any outliers? Make sure to say there are no outliers if there aren’t any

• What is the middle most number?

• Smallest to largest number

• INCLUDE IQR!!!

Questions to ask yourself when looking at data

1. Who/what are the individuals the information is about?

2. What are the variables used to describe any characteristic of these individuals?

How do you find the IQR?

IQR = Q3-Q1 = # - # = IQR

What is the sentence for what an IQR is?

This means that the range of the middle half (what’s trying to be found) is (IQR #)

What is the sentence for what standard deviation tells us?

The (theme) typically vary from the mean by ABOUT (standard deviation)

What is the IQR related to?

Median (use if data is skewed)

What is standard deviation related to?

Mean (use if data is symmetrical)

Equation to find outliers

Q1 - 1.5(IQR) = __

Q3 - 1.5(IQR) = __

How to find a number from the 25th percentile in a bar graph

1. Add up total numbers from bar graph (how high the bar goes)

2. Find 25% of the number

3. Go the number over on the x-axis

What is a usual problem with a bar graph?

• There isn’t a 0 on y-axis appropriately

• Weird range (cherry picking)

• Axis numbers are inconsistent

• Looks distorted

How do you know what you’re finding percents for in conditional distribution?

EX) Find the conditional distribution of pizza preferences for each movie type

Whatever comes after each will be what you’re finding percentages for

Percentiles

the nth percentile of a distribution is the value with n percent of the observations LESS THAN the value in question

• A way to describe someones location within a group (their relative standing)

Z-Scores

Another way to describe someone’s position within a distribution is to tell how many standard deviations above or below the mean their value is

• Called standardizing

• Only use if data is normal!!

What is the Z-Score formula?

z = (x - mean)/standard deviation

What does the Z-Score formula measure?

This value (z) measures the number of standard deviations above or below the mean

How are percentiles and z-scores related?

Percentiles and z-scores compare two or more values from different distributions

• Standardizing can more fairly compare these values

How do you use percentiles in context?

EX) Boy with 22 pairs of shoes is more odd because the 85th percentile which means 15% of boys have as many or more shoes than him. The girls value (25th percentile) is closer to the median

What is the difference between percentiles and percents?

60% means he got a 60 out of 100, 60th percentile means he did better than 60% of others

What happens to measures of center (mean, median, and mode) in linear transformations (adding/subtracting and multiplying/dividing)?

Adding and subtracting from the data affects the measures of center (if add 5 then all add 5)

Multiplying and dividing multiplies/divides measures of center by amount and measures of location and spread (SHAPE DOESN’T CHANGE)

What happens to measures of variability (IQR, standard deviation) in linear transformations (adding/subtracting and multiplying/dividing)?

Adding and subtracting from the data doesn’t change the measures of variability

Multiplying and dividing changes the measures of variability

How do you find cumulative frequencies?

Add previous frequencies to each other

EX) 2, 7, 13, 12

CF 1 = 2, CF 2 = 9, CF 3 = 22, CF 4 = 34

How do you find cumulative percentages?

Frequency divided by total cumulative frequency

EX) F = 2, Total CF = 44

Cumulative % = (2/44) x 100 = 4.5%

How do you make a histograph from a line graph?

1. Estimate cumulative frequencies (the dots)

2. Find differences between the cumulative frequencies

3. Corresponds with the right bar

The coordinates are (x-axis, C%)

• The C% is from the previous number

Empirical rule

1 sd = 68%

2 sd = 95%

3 sd = 99.7%

what rules must a density curve follow to be a density curve?

1. it exists above or below the x-axis (no negative)

2. the area under the curve equals 1 (100%)

3. the area under the curve to the left of a given point is equal to the proportion of values that fall below that given point

mean and median when skewed right

mean is greater than median

mean and median when skewed left

mean is less than median

mean and median when symmetrical

mean equals median

names for normal distributions

1. symmetric

2. single-peaked

3. bell-shaped

what is mu

the populations mean

what is sigma

the populations standard deviation

steps for normal distribution calculations

1. define x

2. define normality (x~N(m,sd))

3. write probability is applicable (P(x<#), etc)

4. standardize z (plug in #’s and use table A)

5. show “1 minus” when z>#

6. write conclusion in context (TTQA)

what does a graph look like when it’s normal

an approximately straight line

what does the graph look like when it’s not normal

curved

fun facts about z-scores

1. normal data? distribution of z-scores is too! z~N(0,1)

2. non-normal data? distribution of z-scores is NOT normal even though mean of z-scores still 0 and standard deviation is 1

cumulative frequency

add to previous numbers

cumulative percentages

frequency over total cumulative frequency

tools to turn cumulative frequency graph into histograph

1. plot using the percents, always use the previous percent when graphing

2. when histograph, subtract the dot by the previous dot

EX) What is the data above or below 8?

there is no data at a single point, only above or below (it would be 0)

when do you subtract a number from table A by 1?

when z is grater than the number

EX) find z when 85% observations fall above it

find 15% (.1500) on table A and that’s z

• If shading in graph to the right of z, subtract by 1