Mean: 1/n * Σ(x). Median: sort(x) and gets the value in the middle. Mode: Most frequent value. Useful for Bimodal or Multimodal (several bumps in a chart) data.

Bar chart: Useful to see global trend where individual points have too much noise.

Difference between Bar Chart and Histograms: Bar Charts can show 2D whereas Histograms show only 1D, which is the frequency of data between ranges.

Pie chart: Good for relative data and comparing.

Statistics is deep and often manipulated, we need to be careful when translating raw data into decisions/conclusions.

If the probability of P(H) = 0.5, then P(H, H) would be: P(H) * P(H) = 0.25

For simple probability problems, a truth table helps finding the answer.

Probability of an event (P), an opposite event (1-P) and a composite event (P*P...*P) are independent.

Conditional Probability is where one event depends on another e.g. P(test | disease).

Bayes Rule: P(A|B) = P(B|A) * P(A) / P(B) Pior Probability (Cancer) * Test Evidence (Pos|Neg) = Posterior Probability

Bayes Rule describes the probability of an event, based on prior knowledge of conditions that might be related to the event e.g. if cancer is related to age, then, using Bayes Rule, a person’s age can be used to more accurately assess the probability that they have cancer.

In continuous distributions, every outcome has a probability of 0 e.g. spin a bottle in a 360 panel to hit an exact angle.

Data is correlated if knowledge about one variable tells you something about another variable.

Studies and news often confuse correlation and causation.

A correlation between variables does not automatically mean that the change in one variable is the cause of the change in the values of the other variable.

Causation indicates that one event is the result of the occurrence of the other event; i.e. there is a causal relationship between the two events.

Maximum Likelihood Estimator: 1/n * Sum(Appearances), e.g.: HHHT => 1/4 * Sum(H) => 1/4 * 3 = 0.75

When there isn't much data... fake it!

Variance measures how far the data is from the mean value, in other words, it show how spread the data is.

Variance can be calculated as the average of the squared difference for each element from the mean.: σ² = i/n * Σ(x-μ)²

You can think of variance as a measure of how wide the tree is from which the apples fell down.

Standard Deviation (SD) is also a measure of spread and it is the σ = sqrt(σ²). The scale of SD is the same as the mean.

The standard score is the signed number of standard deviations by which the value of an observation or data point is above the mean value of what is being observed or measured. Observed values above the mean have positive standard scores, while values below the mean have negative standard scores. z = (x - μ)/σ

Quartiles are the three points that divide the data set into four equal groups, each group comprising a quarter of the data. We could remove the lower and higher quartiles in order to get rid of outliers.

A percentile is a measure used in statistics indicating the value below which given percentage of observations in a group of observations fall. For example, the 20th percentile is the value (or score) below which 20% of the observations may be found.

The binomial distribution is used when there are exactly two mutually exclusive outcomes of a trial. The binomial distribution is used to obtain the probability of observing k successes in N trials, with the probability of success on a single trial denoted by P.

The Central Limit Theorem establishes that, in most situations, when independent random variables are added, their properly normalized sum tends toward a normal distribution (informally a "bell curve" or Gaussian) even if the original variables themselves are not normally distributed.

The Central Limit Theorem is a key concept in probability theory because it implies that probabilistic and statistical methods that work for normal distributions can be applicable to many problems involving other types of distributions.

The normal (or Gaussian) distribution is a very common continuous probability distribution. Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not known.

If you're a doctor and you have 1 patient, use simple probability like a coin flip, if you have few patients use Binomial Distribution, if you have hundreds of thousand of patients (such as in testing a new drug) use Normal Distribution.

The more data we have and the more we trust the data the smaller the Confidence Interval will be.

Standard Deviation is not affected by the sample size whereas the Confidence Interval is.

Confidence Interval can be calculated by 1.96 * sqrt(σ²/N) 1.96 is a constant that depends on the sample size.

Hypothesis Test: Given a hypothesis H0 (or null hypothesis) (P = 0.8), we test if it is plausible or not. First, we define the Alternate Hypothesis H1 (P < 0.8), calculate the critical region, then we check if the sample result falls into the critical region.

Hypothesis Test: Checking the critical region -- one-side tail goes until 5% is reached, two-sided tail goes until 2.5% is reached on both ends. If yes, the we can reject H0 with a 95% confidence.

If the Null Hypothesis falls down into the Confidence Interval, we accept it, otherwise we reject it.

In order to calculate the Confidence Interval we need: 1) calculate the mean μ, 2) calculate the variance σ², 3) get the probability constant according to the number of samples, 4) calculate the CI: constant * sqrt(σ²/N).

In Linear Regression, we try to find a linear equation (y = bx + a) that best fits the given data. Where b = Σ((x-μx)*(y-μy))/Σ(x-μx) and a = μy - b*μx

Linear Regression: "b" is also known as Slope and "a" as Intercept. b = cov(x, y)/var(x) and a = μy - b*μx

Correlation coefficient is a value between -1 and 1 and it tells us how "related" 2 variables are. Both -1 and 1 stand for perfectly linear data.

Correlation formula is: r = Σ((x-μx)*(y-μy)) / sqrt(Σ(x-μx)² * Σ(y-μy)²)

Here is where our real selfhood is rooted, in the divine spark or seed, in the image of God imprinted on the human soul. The True Self is not our creation, but God's. It is the self we are in our depths. It is our capacity for divinity and transcendence.

All too often, the conversation about appropriate and balanced environmental stewardship gets caught up in partisan politics. Yet, this conversation is key to the preservation of our great country for generations to come, as important as ensuring we have fiscally responsible policies to secure our future.

What do you do when life blindfolds you and spins you around? We think it's our fault, that we're to blame, when really we should be focused on being gentle with ourselves.

I enjoy having my privacy and not seeing my face on the cover of everything, y'know?

Looking for the perfect day is not going to make us happy, because that day isn't going to come.

Knowing is the most profound kind of love, giving someone the gift of knowledge about yourself.

New Yorkers have been fortunate to have Andrew Cuomo as our Attorney General - protecting working New Yorkers against the banks, insurance companies and big corporations.

Adversity is simply part of earth life. From it we can grow and progress if we choose to. Yes, some trials come because of our own disobedience, but many trials are simply part of life.