Statistics

Nominal data

Incomparable. Yes/No. Healthy/Sick. These are qualitative measurements that categorize data without a natural order or ranking.

Ordinal data

Relatively comparable, like size. Example: categories or stages (of cancer). Refers to data that can be ranked or ordered, but the intervals between the ranks are not equal. Examples include rating scales, such as satisfaction levels.

Discrete data

Integer numbers. Have an order, can be compared arithmetically. -1, 0, 1, 2.

Continuous data

Real numbers. Have an order, can be compared arithmetically. Size, weight, concentration, altitude.

Frequency distribution

A summary of data that shows how often each value or range of values occurs in a dataset.

Binning

Process of grouping continuous numerical values into smaller, discrete intervals - bins. Example: x axis from 0 to 20 and we create 4 bins (0-5, 6-10, 11-15, 16-20).

Relative frequency distribution

displays the proportion, percentage, fraction of observations falling into each category or class interval of a dataset, rather then the raw count.

Cumulative frequency distribution

It’s the sum of the class and all classes below it. (are not meaningful for nominal (unordered data)).

Cumulative relative frequency distribution

Shows the running total of percentages or proportions for all data values up to a specific point. Adds up everything and the last value is 1.

Arithmetic mean

Just add everything and divide on n. Best for additive quantities, linear difference.

Geometric mean

Multiply everything and put in a ^1/n (take a root). log(GM(𝒙))=AM(log(𝒙))
For exponentially growing data, when we have extreme outliners.

Harmonic mean

Emphasis on small values. Best for rates and rations. Use to calculate speed mean.

What is smaller/bigger? AM, GM, HM?

Inequality works for positive x_i values.

Median

Middle point of the dataset. Depends on parity of n. Median isn’t sensitive to extreme outliners.

Mode

The most frequently occurring observation. Makes more sense for categorical data.

Range

Range(x) = Max(x) - min(x). Highly sensitive to the extreme values of outliners.

Quantiles

The 𝑝-th quantile of 𝒙, 𝑞𝑝(𝒙), is the value such that at least a fraction 𝑝 of the observations 𝑥𝑖 are less than or equal to it. Commonly used quartiles: Q1 (25%), Q2 (50%, median), Q3 (75%). IQR (interquartile range) - Q3-Q1.

Sample Variance

Measures how far spread out the numbers are in a small group (a sample) taken from a larger group (a population).

Sample standard deviation

Measures how much data values in a subset scatter around the sample mean. Same units as the mean and data.

The Coefficient of Variation

also called relative standard deviation (RSD)—is a unitless, standardized measure of how spread out data points are around the mean. Is unstable when mean = 0.