Basic Formulas

Distribution

Uniform Distribution - all outcomes are equally probable; e.g., the probability of rolling a 1, 2, 3, 4, 5, or 6 on a die

Z-score - (or standard score) tells you how far the data point is from the mean; formula is z = (x – μ) / σ

Standard Error (SE) - measure the standard deviation of the samples mean to the population mean; z-score formula is (new mean - population mean) / (standard deviation / (square root of number of the new number of items)); z = (x – μ) / (σ / √n)

Student’s T-test - a method for testing the significance of the differences between sample groups when the population standard deviation is unknown; i.e. “did this happen by chance?”

‣ Independent Samples t-test - compares the means for two groups
‣ Paired sample t-test - compares means from the same group at different times
‣ One sample t-test - tests the mean of a single group against a known mean

T-score - (or T-value) the ratio between the difference between two groups and the difference within the groups

‣ The larger the t score, the more difference there is between groups
‣ The smaller the t score, the more similarity there is between groups
‣ T score of “3” means that the groups are 3x as different from each other as they are within each other
‣ When running t-test, the bigger the t-value, the more likely that the results are repeatable
‣ Every t-value has a p-value to go with it

z-score vs. t-score

• Use t-score when sample size < 30 and standard deviation is unknown

• Use z-score when sample size > 30 and standard deviation is known

Confidence Interval

• a range of values, above and below the statistic’s (sample) mean, that would contain an unknown population parameter; the probability that the confidence interval would contain the true population parameter when you draw a random sample many times

• degree of uncertainty associated with a sample estimate of a population; most often, researchers choose 90%, 95%, or 99%

• Not to be confused: 95% confidence interval does not mean 95% chance that the population parameter falls between a and b. It means that 95% of the interval estimates would include the parameter.

• conducted during statistical methods like the t-test

• formula is sample statistic + margin of error

Margin of error - the range of values above and below the sample statistic in a confidence interval; formula is critical value * standard deviation of statistic or critical value * standard error of statistic

Critical value - a cut-off value that tells us how far from the sample mean we can vary and remain confident—usually one standard deviation from the mean; this is usually the t-score or z-score

Hypothesis & Testing

Significance (α) - the measure of whether the results of research were due to chance; significance level generally fixed at 0.05 (this is just a statistical benchmark)

P-value - “probability value”; the way in which significance is quantified and reported statistically

‣ defines the probability of getting a result that is either the same or more extreme (rare/special) than other actual observations; “did the result from the sample data occur by chance?”
‣ used as an alternative to the rejection point to provide the least significance for which the null hypothesis would be rejected (α < 0.05)
‣ the smaller the p-value, the stronger is the evidence that is in favor of the alternative hypothesis given observed frequency and expected frequency

P > 0.10 (>10%) - No evidence against Null Hypothesis; Nothing special
P > 0.05 (>5%) - Weak evidence against H0
P = 0.05 (5%) - Moderate evidence against H0; Statistically significant
P < 0.05 (<5%) - Good evidence against H0
P < 0.01 (<1%) - Strong evidence against H0
P < 0.001 (<0.1%) - Very strong evidence against H0; Highly significant

P-value approach to hypothesis testing - P-value of the hypothesis test is reported and readers can interpret the statistical significance themselves

Statistical Relationship

Correlation - the degree to which two factors appear to be related (not the same as causation)

Causation - a relationship where one factor causes the other

Regression - a statistical method that determines the relationship between a dependent variable (Y-axis) and a series of other independent variables (X-axis)

Coefficient of Determination - or R-squared (R2); the percentage variation in y explained by x-variables; i.e. how the differences in one variable can be explained by the differences in another variable; expressed as a percentage or between 0 to 1

Anscombe’s Quartet - an example where 4 datasets have similar statistical metrics (e.g. mean, variance, correlation) but significant differences when data is plotted on a graph. Don’t rely on statistical metrics alone; use data visualization to get the full picture.

Probability

Experiment - (in probability theory) a procedure that involves chance of probability and can be infinitely repeated; experiment is random if there is more than one possible outcome, and deterministic if it has only one

Outcome - the result of the experiment

Conditional Probability - the likelihood of an outcome based on the occurrence of a previous outcome; calculated by multiplying the probability of the preceding event by the updated probability of the succeeding event

Bayes’ Theorem - ​mathematical formula for determining conditional probability; formula is P(A∣B) = P(A⋂B) / P(B) ​= P(A) x P(B∣A) / P(B)

• named after 18th-century mathematician Thomas Bayes

• allows one to update the predicted probabilities of an event by incorporating new information

• often used in finance for calculating or updating risk evaluation

• useful when implementing machine learning​

Formulas and Symbols

• Probability of ‘A’ and ‘B’ = P(A ∩ B)

• Probability of ‘A’ or ‘B’ = P(A ∪ B)

• Probability of ‘A’ given event ‘B’ occurred = P(A | B)

Combination - order does not matter; e.g. ingredients for a salad; formula is C(n, r) = n! / (n - r)! r!

Permutation - order matters; there are more possible combinations when order matters; e.g. phone number or lock; formula is P(n, r) = n! / (n - r)!