1st Week

Science: use of evidence, testable explanations and predictions

Theory: comprehensive explebation support by vast body of evidence

Facts: tested and confirmed so many times, so that there is no doubt

Inductivism: Bottom-up reasoning → from observations to general laws (creates theories, but not certain).

-> infer laws from examined cases

Problem: no non-circular way to justify predictions

Deduction: Top-down reasoning → from general laws to specific cases (tests/applies theories, certain if premises are true).

guess → predict → test → try to disprove.

1. Formulate research question

2. Theory & hypotheses

3. Method selection

4. Data gathering

5. Data cleaning

6. Exploratory data analysis

7. Modelling

• (new, interesting, relevant)

• Clarity

• Focus

• Feasibility

• Testability

• Should not be leading

• 10% has a satiscaction (st) of 1

• 23% has an axact st of 5

• 25% has a st of 2 or less

• 40% watches 1 hour or less

• 40% watches between 1 and 2 hours

• 80% watches 2hours or less

• the small spike at the end means that all values >10 were combined

• “Durchschnitt”

• sum of all deviations is zero

• good intuition if the data is normally distributed

• Caution! Outliers can severely bias this intuition

compared to the mean, the median is more robust to outliers

Mean and median are similar if the data is symmetrically

distributed.

If the data has more than one center, neither mean

nor median have meaningful interpretations.

Mean and median may differ if the data is skewed.

If the data has outliers it is better to use the median

as it is robust to outliers.

To compare the distribution of two variables

1. Luigi is slower than domenico

2. Mario and Salvadore deliver equally fast -> variables have the same distribution

value that occurs most compared to other values

As the distibution is right/ positively skewed the median is > than the mean, as all small values count equally the mean get’s “influences”

• measue of asymetry

right/ positively skewed

• “Where the tail is”, so the following chart is right-skewed

Why positive? Because in a right-skewed distribution:

• The mean is pulled to the right (towards larger, positive deviations from the median/mode).

• When you calculate the skewness formula, those big positive deviations dominate, so the result is > 0.

1. Absolute deviation

2. Mean squared error

3. Variance

4. Standard deviation

We take the absolute as otherwise negative values might bias the deviation metric

Pros:

• less sensitive to outliers than the MSE or Variance

• easy to interpret

Cons:

• MSE of the mean

• Pros: Mathematically convenient

• Cons: hard to interpret and sensitive to outliers

• square root of variance

• Pros: Interpretable, widely used

• Cons: Sensitive to outliers

• var is standardized if its mean is 0 and the variance is 1

• To standardize a variable, we subtract its mean

  and divide it by its standard deviation

• makes values better comparable

• faster convergence ( convergense means reaching the optimal solution in iterative algorithms.)

• comuting distanves more appropriate

• IQR (Interquartile Range): Quite large → high variability, NOT Variance, as IQR focuses only on the middle 50%

  
  

• No extreme outliers visible for this month

• The median being closer to Q1 means the lower half of the data (Q1 → median) is more concentrated (shorter spread)

• If this were shown as a density curve: The longer spread above the median suggests a right/positive skew

  
  

  
  

  
  

  
  

• measures the tailedness of a distribution (skewness measured the asymmetry)

• A portfolio with a low kurtosis value indicates a more stable and predictable return profile which may indicate lower risk. Investors may intentionally seek investments with lower kurtosis values when they're building safer, less volatile portfolios

• Measures the degree to which two variables vary

  together (values from -inf until inf)

• The unit of the covariance is expressed in the product

  of the units of both variables → difficult to interpret

• positive covariance, increase or decrease together

• negative cov, if one increases the other decreases

• cov zero -> no linear relationship

For example you want to see the covariance between hours watched and income, at the end you get a number that has hours and € included -> difficult to interpret

• Correlation measures both the strength and direction of

  the linear relationship between two variables

• normalized and unitless version of covariance (-1 to 1)

Interpretation:

• correlation close to +1 or −1 indicates a strong linear

  relationship

• A correlation close to 0 indicates a weak or no linear

  relationship

• While covariance measures only the direction,

  correlation also indicates the strength of the relationship

• The values of R (unitless) lie between −1 and +1 and measures the degree of correlation between the ranks of X and Y.

Interpretation:

• R = 1 → all observations are assigned the same rank

• R = -1 → all observations are assigned the opposite rank

• While covariance measures only the direction,

  correlation also indicates the strength of the relationship

• Pearson’s correlation looks at raw values → best for linear, normally distributed data.

• Spearman’s correlation looks at ranks → useful for ordinal data, non-linear monotonic trends, robust to outliers.

Example: If exam scores increase with study hours but in a curved (non-linear) way, Pearson might be low, but Spearman will still be high, since the ranking order is preserved

• The cumulative density function (cdf) describes the probability that a random variable takes a variable smaller than a given number

• CDF deals with continuous and discrete random variables

Two random variables, X and Y are independent if and only if:

• We are usually interested in how one random variable is related to one or more other random variables.

• The conditional distribution of Y given X tells us the distribution of Y conditional on us having information about X

Ex. • Probability of blue eyes given that a person is Portuguese

1. Measures of central tendency

2. Measures of variability or spread

3. Measures of association between two random variables

• The expected value of a random variable X, E(X), is a weighted average of all possible values of X.

• The variance of a random variable X is a measure of its dispersion around the expected value, μ ≡ E(X):

• The standard deviation of a random variable, sd(X), is simply the square root of the variance:

• It is useful to have summary measures of how two random variables vary with one another

  
  

The correlation coefficient is a measure of the degree of linear relationship between X and Y

• Covariance = raw measure of joint variability, scale-dependent.

• Correlation = normalized covariance, scale-independent, bounded between -1 and 1.

-> Think of correlation as the unit-free, standardized version of covariance

• Zero correlation only rules out linear association.

• independence always implies uncorrelated.

• Uncorrelated does not imply independence. (counterexample: Y=X^2).

  
  

  
  

• uniform distribution

• normal “

• Exponential

• Waiting time for a bus (if buses arrive every 10 minutes, and you arrive at a random time → Uniform(0,10)).

• Randomly selecting a decimal number between 0 and 1 → Uniform(0,1).

  
  

• N(0,1) -> standard normal distribution N(μ,σ2)

• μ = mean (where the pak is, shifts <->) (verschiebt)

• σ2 = variance -> how high the curve is (staucht oder streckt)

• The Central Limit Theorem states that when you take a sufficiently large sample size n from a population with any shape of distribution (with a finite mean and variance), the distribution of the standardized sample mean will approximate a standard normal distribution. This approximation improves as the sample size increases, regardless of the original distribution's shape.

There are 2^3=8 equally likely outcomes. I enumerated them and computed the sample mean for each one (you’ll see the full table titled “n=3 – All 8 outcomes and sample means” just above).

n=10 -> more like a bell curve