Quantitative/Numerical Univariate Analysis Descriptive Statistics

• a type of Univariate Analysis Descriptive Statistics whose variable is quantitative

Statistics Terminology

Some may argue that statisticians are not really interested in generalizing from a sample to a specified population but to an idealized super­population spanning space and time

best course on statistics: https://bolt.mph.ufl.edu/6050-6052/

Introduction & Terminology

The field of statistics exists because it is usually impossible to collect data from all individuals of interest (population). Our only solution is to collect data from a subset (sample) of the individuals of interest, but our real desire is to know the “truth” about the population. Quantities such as means, standard deviations and proportions are all important values and are called “parameters” when we are talking about a population. Since we usually cannot get data from the whole population, we cannot know the values of the parameters for that population. We can, however, calculate estimates of these quantities for our sample. When they are calculated from sample data, these quantities are called “statistics.” A statistic estimates a parameter.

• population distribution consists of all units of interest
• empirical distribution consists of observed units collected from the population
• population parameter (𝜽)
  • sometimes just called a parameter
  • is any variate analysis of population distribution (e.g. mean, variance, etc)
  • usually have an unknown value
• sample statistic (𝜽ˆ)
  • sometimes just called statistic
  • is a function of sample distribution as input
  • is any variate analysis of a sample distribution (e.g. sample mean, sample variance, etc)
  • is an estimate of the corresponding population parameter 𝜽
  • is a random variable because it is computed from a random sample distribution a subset of population distribution. Thus, this statistic has a sampling distribution
  • see methods estimating sample statistic
• Error

Random Process - Random Variables - Stochastic Model - Probability Distribution - Statistical Inference - Statistical Model - Exploratory Data Analysis - Estimator - Probability Model

Many times there are observable phenomena that are random in nature. We call it a Random Process (Random Experiment). The random process has outcomes, and subsets of these outcomes are called Events. We map these events to a numeric form using Random Variables.

We study and capture our knowledge about this random process by creating a Stochastic Model. The stochastic model predicts the output of an event by:

1. providing different choices (of values of a random variable)
2. the probability of those choices

These two elements are summarized as a Probability Distribution.

This distribution has some parameters (like mean, standard deviation, etc) which were inferred from the observable phenomena using Statistical Inference.

Before inference, the distribution had unknown (not inferred yet) parameters. It was, hence, a family of distributions, since each value of the parameter is a different distribution. This family is called a Statistical Model.

Usually, a statistical model is guessed (exponential, binomial, normal, uniform, Bernoulli, etc) using Exploratory Data Analysis, then its parameters are inferred (estimated) by applying statistical inference (say, algorithms involving loss function minimization) to arrive at a stochastic model (statistical model with known parameters) (a.k.a. Estimator) that captures our knowledge about the random process.

The term ‘Probability Model’ (probabilistic model) is usually an alias for stochastic models.

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Quantitative/Numerical Univariate Analysis Descriptive Statistics - Types

Central Tendency - what are the most typical values?

see Central Tendency

Statistic	population parameter notation	sample statistic notation	Description
Mode			most occurring value in the distribution
Median	𝑀	𝑀̅ or 𝑥̃	the middle value in the sorted distribution same as 0.5-quantile, 50th percentile, and 2nd quartile is a value 𝑚 that minimizes 𝐄[|𝑋 - 𝑚|]
Arithmetic Mean	𝜇	𝑋̅	average in distribution is a value 𝑚 that minimizes 𝐄[(𝑋 - 𝑚)2]
Harmonic Mean			average in distribution
Root Mean Square Quadratic Mean			ignores negative sign in computing the arithmetic mean
Geometric Mean			average in distribution
Mid Range			the average between min and max

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Dispersion/Variation - how do the values vary?

see Variation

Statistic	population parameter notation	sample statistic notation	Description
Variance	𝜎2	𝜎̂2or 𝑠2	measures how far a set of numbers are spread out from their average value calculation of variance uses squares because it weights outliers more heavily than data very near the mean. This calculation also prevents differences above the mean from canceling out those below, which can sometimes result in a variance of zero
Standard Deviation	𝜎	𝜎̂ or 𝑠	measures how far a set of numbers are spread out from their average value brings variance back to the original unit of data
p-Quantiles	𝑞𝑝	𝑞̂𝑝	generalizes median from 0.5 to a range [0,1] quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities variants: median, quartiles, percentile, etc
Coefficient of Variation			TODO

Max Min	For a subset 𝑆 of field 𝐹, 𝑠̃∊𝑆 is called the max of 𝑆 if: ∀𝑠∊𝑆: 𝑠≤𝑠̃ For a subset 𝑆 of field 𝐹, 𝑠̃∊𝑆 is called the min of 𝑆 if: ∀𝑠∊𝑆: 𝑠≥𝑠̃
Upper Bound Lower Bound	For a subset 𝑆 of field 𝐹, 𝑠̃∊𝐹 is called an upper bound of 𝑆 if: ∀𝑠∊𝑆: 𝑠≤𝑠̃ For a subset 𝑆 of field 𝐹, 𝑠̃∊𝐹 is called a lower bound of 𝑆 if: ∀𝑠∊𝑆: 𝑠≥𝑠̃
Supremum Infimum	For a subset 𝑆 of field 𝐹, 𝑠̃∊𝐹 is called the supremum of 𝑆 if: 𝑠̃ is an upper bound of 𝑆 𝑠̃≤𝑠 for any other upper bound 𝑠∊𝑆 For a subset 𝑆 of field 𝐹, 𝑠̃∊𝐹 is called the infimum of 𝑆 if: 𝑠̃ is a lower bound of 𝑆 𝑠̃≥𝑠 for any other lower bound 𝑠∊𝑆
Range	range = max - min

Statistics Involving Distances
Central Tendency of Deviation	Description
Mode Deviation	is the most occurring distance between each data point and the mean
Median Deviation	is the middle distance between each data point and the mean
Variation of Distances	Description
Variation	is the variance of the distances between each data point

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Distribution Shape - are the values symmetrically or asymmetrically distributed?

see Distribution Shape

Statistic	population parameter notation	sample statistic notation	Description
Skewness			measures the symmetry or asymmetry of a dataset about its mean
Kurtosis			measures whether the data are heavy-tailed or light-tailed relative to a normal distribution

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Outliers - are there values that represent abnormalities?

Statistic	population parameter notation	sample statistic notation	Description

Others

Statistic	population parameter notation	sample statistic notation	Description
size	𝑁	𝑛	number of members of dataset (sample or population)
kth Moments	𝑀𝑘		raw moments vs central moments

Resources

• http://uc-r.github.io/descriptives_numeric