Nature of Probability and Statistics

Statistics (according to Webster Dictionary)

2 parallel definitions

• The science of data (collecting, organizing and analyzing data)

• Plural of the word “statistics”

Parameter

Characteristic of a population

• Usually a numerical characteristic

• Ex. population mean, population variance, population standard deviation, etc.

Statistic

Characteristic of a sample

• Ex. sample mean, standard deviation

Data

Collection of information

• 2 types:

  • Categorical (qualitative)

    • Nominal- according to name

    • Ex. Data with names, genders, race, etc.

  • Numerical (quantitative)

Numerical (Quantitative) Data

• According to a ratio scale

  • Possible value of zero is an inherent zero → the number, zero, signifies nothing is there

  • Ex. Data with heights, weights, time durations, grades, etc.

• According to the interval scale

  • Zero is not inherently zero → the number zero signifies a value

  • Ex. Data containing temperatures

• Two types:

  • Discrete dataset

  • Continuous dataset

Discrete Dataset

One where measurements take a countable set of isolated values

• Ex. The number of___

Continuous Dataset

Measurements take any real value within a certain range

• Ex. amount of rainfall in Charlotte in Jan. of last 30 years; length of customer waiting times at a local bank

Graphical Descriptive Statistics

Describes set of data via graphs

• Ex. bar graphs, pie charts, histograms, scatter plots, etc

Numerical Descriptive Statistics

• Measures describing center of data- mean (arithmetic average), median, mode and weighted mean

• Measures describing spread/dispersion of data- range, variance and standard deviation of data

• Measures of location- tells where a particular measurement stands compares to the rest of the data (ex. z-score and percentile ranking)

• Measures describing shape of data- (ex. skewness and kurtasis)

Inferential Statistics

Process of utilizing one or more random samples to gain insight about population where the samples were selected from

Sample

Part/portion of a population

Population

Set of all individuals, objects or measurements

Variable

characteristic of population of interest

How does a data analyst use inferential statistics?

• Data analyst selects one or more samples from the population of interest → performs statistical analysis

• When sample characteristics are verified/revealed, the data analyst uses this to transform sample information into population information

3 Main Methods of Inferential Statistics

• Constructing confidence intervals- estimates population parameter to be within 2 limits (a lower limit and an upper limit)

• Perform hypothesis testing- verifies or rejects hypotheses or claims

• Modeling or Testing relationships between data sets

Sample

Subset of elements from set of individuals with one or more common features (a population) that have been selected for a study

• n= number of elements in a sample

• N= number of elements in the population

Use samples to learn about populations because many real-world examples make it impossible to measure a characteristic from every member of a population

Random Sampling

Subset of a population that’s chosen in a manner where every member of the population has an equal chance of being selected for the subset/sample → allows for the sample size to represent the population they were selected from

• n= number of members of a random sample

Statistical Estimation

Type of inferential statistical analysis where statistics calculated from data of other samples is used to estimate population parameters

• Also used to quantify uncertainty in estimates

• Ex. Predict % of voters who will vote for a specific candidate

  • Conduct a poll where a random sample is selected

  • Ask each member who they plan to vote for and how likely they are to vote in the election

  • Use the info to calculate percentage of voters who will vote for a specific candidate (called a sample statistic)

  • Use the sample statistic to estimate percentage of entire population who will vote for a candidate

    • Results are called a statistical estimation of population parameter

Statistical Notation

• X (bar)- sample mean

• u- population mean

• s²- variance of a sample

• O²- sigma squared→ variance of a population

• s- standard deviation of a sample

• o- standard deviation of a population