5 | Introduction To Statistics

Question 1

(POLL)

Two sections of one hospital have very different survival rates for patients with heart problems. Station 2 (second floor) has much better performance, than station 1 (ground floor)? Who is most likely responsible?

• The doctors, on station two, they are just better!
• The porter, who asks the patients usually: “Feel you fit to walk the stairs into the second floor?”.
• The patient, who just feel better in higher floors?
• None of them

Answer 1

The porter, who asks the patients usually: “Feel you fit to walk the stairs into the second floor?”.

Question 2

(POLL)

Looking at the relation between black schokolade consumption and IQ is:

• correlational research
• experimental research
• making me hungry
• none of these suggestions

Answer 2

correlational research

Question 3

(POLL)

Double blind trials are clinical trials where neither the patient nor the doctor knows the medication, select true statements:

• they are correlational research
• they are experimental research
• they are worse than observational studies because they increase selection bias
• they are better than observational studies because they decrease selection bias
• they have no selection bias
• they still can have selection bias

Answer 3

• they are experimental research
• they are better than observational studies because they decrease selection bias
• they still can have selection bias

Question 4

(POLL)

To evaluate an outcome of a patient after a virus infection the following boxes were prepared for a survey: asymptomatic, common cold, long term suffering, dead … What type of variable is this?

• discrete numerical 0, 1, 2, 3 etc for the levels
• continuous numerical 0.0 1.0, 2.0, 3.0 for the level
• nominal categorical, 0 (asymptomatic), 1 (cold), 2 (suffer a lot), 3 (dead)
• ordinal categorical, 0 (asymptomatic), 1 (cold), 2 (suffer a lot), 3 (dead)

Answer 4

ordinal categorical, 0 (asymptomatic), 1 (cold), 2 (suffer a lot), 3 (dead)

Question 5

(POLL)

Which of the following measures are robust against outliers?

• mean
• trimmed mean
• Median

Answer 5

• trimmed mean
• Median

Question 6

(POLL)

Which measures give information about the spread of the data

• mean
• trimmed mean
• median
• IQR
• sd
• z-score

Answer 6

• IQR
• sd

Question 7

Name some terms from statistics which are deceptive when compared with the terms in the context of science

Answer 7

significant, error, hypothesis

Question 8

Explain the difference between dependent and independent variables and give an example

Answer 8

dependent:
– depends on another –> an outcome variable

independent
– variable influences another (dependent) variable –> is a predictor variable
– might be manipulated

example:
let’s assume we can predict weight based on sex
and height (machine learning!)
– weight is the outcome variable (dependent)
- sex and height are predictor variables (independent)

Question 9

Define descriptive statistics.

vs inferential?

Answer 9

📊descriptive statistics:
- describe main features of data (sample)
- in quantitative terms

vs inferential statistics:
- used to support inferential statements
- data (sample👥) –> population 👥👥👥

Question 10

Define inferential statistics

Answer 10

Statistical inference or statistical induction comprises the use of statistics and random sampling to make inferences concerning some unknown aspect of a population.

Question 11

Name the different sample data centers

Answer 11

• Modus: most frequent value
• Median: value where 50% of data are smaller and 50% of
  data are larger (robust against outliers)
• IQR (interquartile range) = 3.Quart-1.Quart = mid 50%
• Mean

Question 12

Which sample data center can be used for nominal datatypes?

Answer 12

• modus

Question 13

Which sample data center can be used for ordinal datatypes?

Answer 13

• modus
• (median ?)
• (mean ?)

Question 14

Which sample data center can be used for numerical datatypes?

Answer 14

• median
• mean

Question 15

How can one describe the sample distribution?

Answer 15

with:
- max, min
- quantile
- IQR (interquartile range)
- standard deviation
- CV (coefficient of variation)

Question 16

Do the following describe the sample distribution?

• SEM
• CI
• P-value

Answer 16

They describe more the population

Question 17

SEM:

What does this stand for?
What does it measure?
How is it calculated?

Answer 17

stands for:
standard error of the mean.

measures:
likelihod of discr. in sample’s mean vs pop mean.

calculate:
SD / sqrt(N)

Question 18

CI:

What does this stand for?
What does it measure?
How is it calculated?

Answer 18

Confidence interval

Range of values estimate expected to fall between if test redone, within certain level of confidence.

Confidence, in statistics, is another way to describe probability.

CI = mean of estimate plus and minus variation in that estimate.

Question 19

P-value

What does this stand for?
What does it measure?
How is it calculated?

Answer 19

p stands for probability

P values used in hypothesis testing to help decide whether to reject null hypothesis (inferential statistics)

Describes how likely you are to have found a particular set of observations if the null hypothesis were true.

Calculated from a statistical test.

Question 20

IQR

Stands for?
Meaning?

Answer 20

Interquartile range.

In descriptive statistics: tells you spread of middle half of distribution.

Quartiles segment any distribution that’s ordered from low to high into four equal parts.

The interquartile range (IQR) contains the second and third quartiles, or the middle half of your data set

Question 21

What is meant by parameters vs statistics?

Answer 21

A parameter is a number describing a whole population (e.g., population mean)

A statistic is a number describing a sample (e.g., sample mean).

we use the sample to estimate the parameters of the
population

Parameters and statistics have the same name but mean different things (eg mean of sample ȳ ≈ μ, mean of the population)

Question 22

How can you tell which one is parameter and which one is statistic? eg ȳ, μ

Answer 22

Usually:
- use latin letters for statistics (sample)
- use greek letters for parameters (population

Question 23

How is deviation of samples and populations described?

Answer 23

• **deviation from mean: s, sd **
• (Sample variance s², population variance σ²)

Question 24

How is the uncertainty/quality of results described, in inferential statistics?

Answer 24

SEM
CI

Question 25

R: how do you get the median from a data object *survey* with a subset *cm*? What if there are some empty cells?

Answer 25

> median(survey$cm) > median(survey$cm,na.rm=T)

Question 26

R: what does this command do? > summary(survey$cm)

Answer 26

returns: - min - 1st qu - median - mean - 3rd qu - max - NA's of that dataset

Question 26

R: how do you get the mean from a data object *survey* with a subset *cm*, and we want it to be less sensitive to outliers? How does this work exactly?

Answer 26

> mean(survey$cm,na.rm=TRUE,trim=0.1) removes upper and lower 10%

Question 27

R: how do you get the min / max / standard deviation ?

Answer 27

min() max() sd()

Question 28

R: How do you get the quantiles for a data object *survey* with a subset *cm*? and also take care of any NA's

Answer 28

> quantile(survey$cm,c(0.25,0.5,0.75),na.rm=T)

Question 29

R: you have a data object *survey* with 4 subsets. how can you get a summary for all of them?

Answer 29

> summary(survey[,c(1:4)])

Question 30

R: mean, trimmed mean, median - what is sensitive to outliers?

Answer 30

- mean: too sensitive to outliers! - trimmed mean: less sensitive - median: not sensitive - most robust measure --> use trimmed mean or median if there are outliers

Question 31

R: what does the aggregate function do?

Answer 31

- compute summary stats for data subsets - very similar to tapply() but: - can also input formula or time series object - output is df

Question 32

R: how can you use the aggregate function on a data object *survey* with subsets *cm* and *gender* to get the median height for each gender?

Answer 32

> aggregate(survey$cm,by=list(survey$gender),median, na.rm=T)

Question 33

R: How can you get a graphic to illustrate the height statistics, showing the quartiles of two gender subsets of a data object *survey* with subsets *cm* and *gender* ? use red for female and blue for male.

Answer 33

``` > boxplot(survey$cm~survey$gender, col=c("red","blue")) ```

Question 34

R: Boxplot - what are the whiskers? how big are they?

Answer 34

extensions to IQR 1.5 x IQR. anything beyond this: outlier

Question 35

What is Z-Score Center-Deviation/Dispersion?

Answer 35

Normalisation procedure - data transformation - data now have mean of 0 - data now have sd of 1 - 95% of data are within a z-score of -1.96 and + 1.96 z = (x - x¯ ) / s

Question 35

R: You have a data object *survey* with subsets *gender*, *smoker*. How can you make a table showing the number per gender? eg: F M 297 175 How can you make a table showing the number per gender divided by whether they smoke or not? eg: N Y F 260 36 M 148 27 Do you need to do anything about NA's Can you apply functions to the resulting tables?

Answer 35

> table(survey$gender) > table(survey$gender,survey$smoker) NA's are removed automatically Yeas you can

Question 36

R: How can z-score be implemented?

Answer 36

Two ways: > summary((survey$cm - mean(survey$cm,na.rm=TRUE))/ sd(survey$cm,na.rm=TRUE)) > summary(scale(survey$cm))

Question 37

R: What is the *cut* function?

Answer 37

The cut function is used in R for cutting a numeric value into bins of continuous values numeric --> categorical (factor) 'poor mans start' to get first overview of data

Question 37

R: what is known as factors in R?

Answer 37

categorical data

Question 38

R: You have a data object *survey* with subset *cm*. How can you make a new object cSize which holds the *cm* info, but changed from numeric to factor? with three intervals . Give the divisions a name as well?

Answer 38

> cSize=**cut**(survey$cm,c(0,160,185,250)) (values are breakpoints so intervals will be 1- 160, 160 - 185, 185 - 250) > **levels**(cSize)=c("dwarfs","normals","giants")

Question 38

R: With cut , how can you make the result not just categorical nominal, but also ordinal?

Answer 38

``` > cSize=cut(survey$cm,c(0,160,185,250),ordered=TRUE) ```

Question 39

Aims of statistics?

Answer 39

* Describe and summarize data * Visualize trends for better understanding * Make conclusions about a population based on analysis of the sample

Question 40

What types of conclusions can we make about a population based on the sample using statistics?

Answer 40

* Decide whether two groups can be taken as different * Decide if sample is different to total population * Describe the relationship between two variables (proportional analysis) * Decide if data difference is just a random one

Question 41

Statistics and randomness?

Answer 41

* Statistics just estimates “degree” of randomness * Statistics can’t tell you how likely it is that there is really a difference

Question 42

What is important when sampling?

Answer 42

* Sample items must be randomly selected (not a subset but a random selection!) * Items must be independent from each other (selection of one item should not alter the chance of other items to be selected)? * Having a plan before entering the greenhouse * More samples are better * Groups: balanced sampling is better

Question 43

Example of when random sampling is difficult?

Answer 43

* Cats falling from windows: information was usually from veterinarians  just those that survived falling! (= convenience sample) * Roosevelt survey - telephone interviewing. Mostly just wealthy people had phones  republicans * Coronavirus: selective testing - no proper sampling of population!

Question 44

Common sampling problems

Answer 44

* your cohort and topic might change over time (cancer, virus) * true population is more diverse than the population you were sampling from * using a convenience sample rather than a random sample (falling cats) * your measured variable is just a proxy for another variable * imprecise measurements (misunderstandings, wrong scale for some people) * combination of different measurements required * even with clear results: there is room for interpretation * statistics can’t help against bad experimental design

Question 45

Name 4 sampling strategies

Answer 45

* Accidental sampling (close to hand) → often very biased * Simple random sampling * Systematic sampling (eg every 10th person) * Stratified sampling (making subgroups based on categories) * Cluster sampling

Question 46

What is simple random sampling? What problems can arise and how can this be improved upon?

Answer 46

* SRS * Every element has same chance of selection as sample (eg dice) * Randomness might be a problem, especially for large populations or small samples * Systematic or stratified sampling might overcome this * Example: balancing sexes 50/50 or 60/40 (UP ratio)

Question 47

What is systematic sampling? Aka? Example? What problem may arise?

Answer 47

* Arranging population by some ordering * Selecting elements in regular intervals * Aka interval sampling Example: * Every kth person, but don’t start at the beginning Issues: * Vulnerable to periodicities, every 10th house is on street crossings …

Question 48

What is stratified sampling?

Answer 48

* You know categories in your data eg male, female, old, young… * Sample from each category according to your distribution of those categories in your population * One of the methods above for sampling within the strata

Question 49

What two types of research can be done on the sample?

Answer 49

* correlational research * experimental research

Question 50

What is meant by correlational research?

Answer 50

* Just observe what happens in nature * we don’t manipulate a variable * does reading books improve learning * we just collect answers for reading behaviour

Question 51

What is meant by experimental research?

Answer 51

* we manipulate a variable * item divide our sample for reading in two groups * one group must read statistic books * other group is not allowed to read statistics books * after a month we summarize

Question 52

Correlational vs experimental - why would you choose correlational?

Answer 52

It's difficult to research experimentally for some reason: - ethical reasons - financial reasons

Question 53

What is the main distinction when it comes to types of data?

Answer 53

* Measurement levels * Categorical (qualitative) vs numerical

Question 54

What is meant by data dimensions?

Answer 54

* Uni-, bi-, or multivariate data

Question 55

How can categorical data types be further distinguished?

Answer 55

* Nominal: eg gender, smoker, protein structures, nucleotides etc * Ordinal: age (young, medium, old), grade, month etc

Question 56

How can numerical data be further distinguished?

Answer 56

* Discrete: eg age (1,2,3,4..), height (eg 100, 101, 102 cm), length of helices * Continuous: eg height (eg 100.100…cm), weight (79, 998…kg)

Question 57

Define this level of measurement and give an example: nominal

Answer 57

A categorical level of measurement that doesn't have any order. eg - gender: female, male (binary) - smoker: yes, no (binary) - protein structures: H, E, … - nucleotides: A,C, G, T, U

Question 58

Define this level of measurement and give an example: ordinal

Answer 58

A categorical level of measurement with a specified order. eg: - age: young,medium,old - grade: 1,2,3,4,5 - month: 01..12(?)

Question 59

Define this level of measurement and give an example: Discrete

Answer 59

A numerical level of measurement, of discrete numbers. eg: – age: 6, 8, 84 – height: 112, 176, 161cm – number of helices per 1000 AA – length of helices

Question 60

Define this level of measurement and give an example: Continuous

Answer 60

A numerical level of measurement, non-discrete eg: – weight: 79.99kg, 72kg,… – height: 12.2, 12.5, 15.0

Question 61

What can you ask about a datatype, in order to determine which datatype/level of measurement it is?

Answer 61

Can you calculate a mean? - Yes --> - is the mean always a possible value? - Yes --> continuous numerical - No --> discrete numerical - No --> - Is there a logical order of values? - Yes --> Ordinal categorical - No --> Nominal categorical

Question 62

Explain univariate, bivariate, multivariate

Answer 62

how many variables there are in the statistical problem

Question 63

What is important to consider regarding group number and distribution type when analysing categorical data ?

Answer 63

* Different group number and distribution  different statistical tests to use

Question 64

In inferential statistics, which test would you use if you have 2 groups and a normal distribution? Eg running times for female vs male.

Answer 64

* t test

Question 65

In inferential statistics, what test would you use if you have 3 or more groups and a normal distribution? Eg running time for young, medium, and old ages.

Answer 65

* ANOVA

Question 66

In inferential statistics, what test would you use if you have 2 groups and a non-normal distribution?

Answer 66

* Wilcox

Question 67

In inferential statistics, what test would you use if you have 3 groups and a non-normal distribution?

Answer 67

* Kruskal

Question 68

What are the different ways to find the center of the sample?

Answer 68

* Mean – sensitive to outliers * Trimmed mean – less sensitive to outliers * Median – not sensitive to outliers * Modus – more relevant to numerical data

Question 69

Regarding dependent and independent variables, which might be manipulated in an experiment?

Answer 69

* independent variable

Question 70

What is an example of categorical nominal data and which statistic would you use for the sample data center?

Answer 70

* Gender * Modus: most frequent value

Question 71

What is an example of categorical ordinal data and what statistic would you use for the sample data center?

Answer 71

* Age (young, medium, old) * Modus is most appropriate * (Median, mean could also possibly be used if the data can be converted to numerical)

Question 72

When are median and mean most appropriate?

Answer 72

* Numerical data – both discrete and continuous

Question 73

How can we figure out the sample distribution?

Answer 73

* Max, min, IQR, standard deviation, CV * (To describe the population: SEM, CI, p-value)

Question 74

What are statistics vs parameters?

Answer 74

* Sample → characterised by statistic * Population → characterised by parameter * We use the sample to estimate the parameters of the population * parameters and statistics have the same name but mean different things * Usually: latin letters for sample, Greek letters for population

Question 75

What are uncertainty qualities? * SEM, CI * How much can we trust the statistics

Answer 75

How can we look at the dispersion of a sample? * SD * Quartiles * IQR = the range for 50% of around mean data (box of boxplot is the IQR)

Question 76

Boxplot: what do the whiskers represent? * Usually 1.5 x the IQR (outside this are outliers)

Answer 76

What is z-score? * A Transformation → Normalisation procedure * New mean: 0 * New sd: 1 * 95% of data are within a z-score of -1.96 and + 1.96 …

Question 77

What is the formula for Z-Score Center-Deviation/Dispersion?

Answer 77

z = (x - x̄) / s

Question 78

R: What does as.factor() do?

Answer 78

* converts a numeric or character vector into a factor with levels * converts a vector into a factor, preserving its categorical nature. * Factors are essential for handling categorical data in R.

Question 79

R: What does this do: mean(data, na.rm=TRUE, trim=0.1)

Answer 79

* Returns the mean of data, removing any NAs and trimming off outliers.

Question 80

R: Functions for median, minimum, maximum, standard deviation?

Answer 80

* median() * min() * max() * sd()

Question 81

R: What does quantile() function do?

Answer 81

* Returns median (value at 50%, which divides sample in 2) * Along with other interquartile values (25%, 75%)

Question 82

R: What does the summary() function do?

Answer 82

* Numerical data: returns min, max, quantiles, number of NAs * Categorical data: returns count of groups, NAs

Question 83

R: What is the modus function in R?

Answer 83

* Trick question – there isn’t one * But it can easily be implemented

Question 84

R: What can we use the aggregate() function for ?

Answer 84

* Syntax: aggregate(x, by, FUN, ...) * apply a function to subsets of data, typically grouped by one or more factors or variables. * The data in x is divided into groups based on the by variable(s). * The specified function FUN is applied to each subset of the data. * Result is a summary table (df) showing grouped values, computed summaries.

Question 85

R: What function can we use for z score normalisation?

Answer 85

* Scale()

Question 86

R: What does the table() function do / which type of data do we use it on?

Answer 86

* Tabulates data to give summary (no NAs) * Table is a matrix so relevant functions can be applied * useful mostly for categorical data * (numerical data can be transformed into categorical data and that way tabulated as well )

Question 87

R: What is a factor in R?

Answer 87

* Categorical data

Question 88

R: How can we transform data from numerical to categorical?

Answer 88

* cut() function, assign levels with function * Example: * csize = cut(survey$cm, c(0,160,185,250) [add “ordered=TRUE” to keep the order] * levels(csize)= c(“dwarfs”,”normal”,”giants”)

Question 89

R: Ordinal data, special type of categorical data – what’s the difference?

Answer 89

* nominal factors: can only use == or != * ordered factors: can use also numeric operands , <=

Question 90

(QUIZ) The goal of ______ statistics is to summarize and describe the ______ whereas the goal of ______ statistics is to conclude to the population ______. We use sample ______ to estimate unknown ______ of the population.

Answer 90

The goal of _descriptive_ statistics is to summarize and describe the _sample_ whereas the goal of _inferential_ statistics is to conclude to the _population_. We use sample _statistics_ to estimate unknown _parameters_ of the population.