1. Descriptive Statistics 2. Inferential Statistics

Topic 1 Flashcards by LJ Alingasa

Why study Statistics?

To critically evaluate and interpret numerical data presented in reports, articles, and everyday life.
To conduct and understand statistical analyses in your field of work, helping you make data driven decisions.
To draw meaningful conclusions about a population based on insights gathered from a sample, improving the accuracy of predictions and decision-making processes.

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A field of study concerned with the collection, analysis, and interpretation of data to make
decisions, solve problems, and design products and processes in the presence of variability or
uncertainty

Statistics

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Simply the science of data or the art of learning from data

Statistics

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Information, either numerical or categorical, collected through observation or experimentation for the purpose of analysis.

Data

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Classification of Data

Quantitative (Numerical)
Qualitative (Categorical)

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2 Types of Quantitative

Discrete - or counted
Continuous - or measured

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has distinct interval or meaningful difference between values but there is no true zero

Interval

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has distinct interval or meaningful difference between values, and there is true zero

Ratio

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It refers to a point where there is no presence or amount of the variable being
measured, meaning the quantity is completely absent.

True zero

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labels, names, or classification and does not imply order or ranking

Nominal

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labels, names, or classification and does imply order or ranking

Ordinal

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The basic idea behind statistical methods of data analysis

make inferences about a
population by studying a relatively small sample

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Engineering/Scientific Method

Develop a clear description of the problem.
Identify the important factors affecting the problem or may play a role in its solution.
Propose or refine a model using engineering knowledge of the phenomenon being studied.
Conduct an appropriate experiment to confirm that the proposed solution to the problem is both effective and efficient.
Refine the model on the basis of the observed data.
Manipulate the model to assist in developing a solution to the problem.
Confirm/Validate the solution by conducting an appropriate experiment
Conclusions and recommendations

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Role of Statistics in Engineering

Statistics provides tools and methods to analyze data, make informed decisions, and improve processes

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An inherent characteristic of data

Variability

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Repeated observations of a system or phenomenon do not yield identical results

Variability

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developed to address and manage variability in data. It provides a framework for describing this variability and for learning about which potential sources of variability are the most important or which have the greatest impact.

Statistical analysis tools

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What if there is no variability?

 Simplified Analysis: statistical analysis would be straightforward, relying only on simple descriptive statistics like the mean
 A single observation would tell us everything about the entire population
 Statistics would be reduced to basic arithmetic

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Types of Statistics

Descriptive Statistics
Inferential Statistics

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 Summarizes and Presents Data: Focuses on organizing and displaying data in a
meaningful way, providing insights into the central tendency, variability, and overall distribution of observations within the sample
 Visual Representation: utilizes graphs like histograms, stem and-leaf plots, scatter plots, dot plots, and box plots to visually capture the characteristics and “footprint” of the sample
 Key Measures: includes essential calculations such as means, medians, and standard
deviations to describe the data’s central location and spread

Descriptive Statistics

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 Draws Conclusion: utilizes techniques that enable us to make inferences about a larger
population based on the analysis of a smaller sample
 Prediction: leverages sample data to predict outcomes for the broader population.
 Key Methods: includes hypothesis testing and confidence intervals to assess claims and estimate population parameters

Inferential Statistics

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The first step in statistical analysis is collecting data. To make accurate inferences about a population, the sample must represent the population

Collecting Data

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a collection of all elements that possess a characteristic of interest

Population

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a portion of a population selected for study

Sample

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Key Concepts in Collecting Data

1. Population 2. Sample

Types of Reasoning

- deductive reasoning - inductive reasoning

Statistical Inference: From a Sample to Population

inductive reasoning

From Physical Laws to Product Design

deductive reasoning

refers to a process of drawing conclusions or making judgments based on available evidence, data, or reasoning rather than direct observation.

Statistical Inference

Three Basic Methods of Collecting Data

1. Retrospective Study 2. Observational Study 3. Designed Experiment

use existing records of the population

Retrospective Study

some crucial information may be unavailable, missing, there may be transcription or recording errors resulting to outliers, or data on other important factors may not have been collected and archived

Retrospective Study

refers to a data point that differs significantly from other observations in a data set or it is an unusual value. It can appear either much higher or much lower (extreme values) than the rest of the data.

Outlier

What causes an Outlier?

 Human error in transcription or recording  Variability in data  Unusual occurrences (by chance)

collect data by observing the population with as minimal interference as possible

Observational Study

conducted for a relatively short period of time

Observational Study

information of the population for some conditions of interest may be unavailable and some observation be contaminated by extraneous variables

Observational Study

the findings would obtain scientific rigorousness through deliberate control of extraneous variables

Designed Experiment

collect data by observing the population while controlling conditions on the experiment plan

Designed Experiment

makes an inference or decision about which variables are responsible for the observed changes in output performance

Designed Experiment

variables that are not of interest but could affect the outcome of a study

Extraneous variables

Types of Studies Defining the Use of Sample in Statistical Inference

1. Enumerative Study 2. Analytic Study

Makes an inference to the well-defined population from which the sample is selected

Enumerative Study

Example: A sample of three semiconductor wafers selected from a lot (a batch or collection of items) to evaluate the lot’s quality based on the sample.

Enumerative Study

Makes an inference to a future (conceptual) population

Analytic Study

Example: Sample data from the current production line of semiconductor wafers is used to evaluate the quality

Analytic Study

Explaining the relationship between variables

Models

1. Mechanistic Model 2. Empirical Model

established based on the underlying theory, principle, or law of a physical mechanism

Mechanistic Model

theoretical model

Mechanistic Model

Example: Measuring the current flow in a thin copper wire: the model will be Ohm’s Law because it relates the variables of current (I), voltage (E), and resistance (R) in a thin copper wire.

Mechanistic Model

established based on the experience, observation, or experiment of a system (population) under study.

Empirical Model

actual, considering factors or variables not considered in the theory

Empirical Model

Example: If there is no underlying physical mechanism to explain a phenomenon, an empirical model will be utilized to explain the relationship between the variables of interest (regression model)

Empirical Model

Any characteristic, number, or quantity that can be measured, counted, or observed for record

Variable

Types of Variable

1. Response or Dependent Variable 2. Explanatory or Independent Variable 3. Confounding Variable

outcome or variable of interest that is being measured or observed in an experiment or study

Response or Dependent Variable

serve to explain changes in the response

Explanatory or Independent Variable

predictor variable

Explanatory or Independent Variable

variable that is manipulated or observed to determine its effect on the response variable

Explanatory or Independent Variable

an extraneous variable that is related to other variables, thus having an effect on the relationship between these variables

Confounding Variable

An important aspect of quantitative data that gives an estimate of a typical value

Measures of Central Tendency

average value of data

Mean

eliminating a certain percent of both the largest and smallest set of values, insensitive to outlier but not as more insensitive than the median

Trimmed mean

middle value of ordered data

Median

the value that occurs most often in the data

Mode

Effects of Outlier

 Mean is sensitive to outliers, meaning extreme values can significantly affect its result.  Median is resistant to outliers, meaning its result is not significantly affected by extreme values.

It helps determine the appropriate measure of central tendency

Data Distribution Shape

Data distribution shapes

1. Symmetric (Bell Curve or Normally Distributed) 2. Left Skewed or Negatively Skewed 3. Right Skewed or Positively Skewed

mean, median, and mode are all the same

Symmetric

 mean < median  long tail on the left

Left Skewed or Negatively Skewed

 mean > median  Long tail on the right

Right Skewed or Positively Skewed

If the data distribution is skewed

it is better to use the median as a measure of central tendency compared to the mean, as the median is resistant to outliers.

Analysis of Variance (ANOVA) and the t-Test, which will be discussed in more detail in later topics.

Parametric tests

Common Sense Notion of Sampling

1. Simple Random Sampling 2. Stratified Random Sampling 3. Systematic Random Sampling 4. Cluster Random Sampling

every member (sample) of the population has an equal and independent chance of being selected

Simple Random Sampling

the population is divided into subgroups (strata), and random samples are taken from each group

Stratified Random Sampling

a random starting point is selected, and then every nth member of the population is chosen

Systematic Random Sampling

the population is divided into groups (clusters), and entire clusters are randomly selected.

Cluster Random Sampling

the entire production line is selected, and every pipe in the selected line is inspected, regardless of size

Cluster Random Sampling

the engineer ensures that each size category is proportionally represented in the sample by selecting pipes from all size groups

Stratified Random Sampling

Why assign experimental units randomly?

Not randomly assigning experimental units to treatments can lead to biased results and could “camouflages” result.

the smallest entity in an experiment that can receive a treatment or be observed independently

Experimental Unit

Measures of Variability

1. Sample Range 2. Sample Standard Deviation 3. Sample Variance

difference between the largest and smallest value in the dataset

Sample Range

a measure of the amount of variation or dispersion from the mean

Sample Standard Deviation

a measure of the average squared deviation from the mean

Sample Variance

Parameter  a numerical value that describes a characteristics of an entire population  fixed and constant Population mean: “mew” Population variance: 𝜎^2 Population standard deviation:𝜎 Population proportion: p

Statistic  a numerical value that describes a characteristic of a sample  varies from sample to sample Sample mean: 𝑥̅ Sample variance: 𝑠^2 Sample standard deviation: 𝑠 Sample proportion: 𝑝̂

Describing Data Graphically

1. type of data (quantitative or qualitative) 2. size of data (n is small or large) 3. main characteristic of the data that you would like to show (skewness, spread)

represent each observation by a dot on a single numerical axis

Dot Plot

used for smaller data sets, n < 50

Dot Plot

a graphical way to display the frequency of data points within a particular data set

Histogram

used for larger data sets, n > 50

Histogram

condenses large data sets into manageable and readable graphs

Histogram

displays digits from the actual data values to denote the frequency of each value

Stem-and-Leaf Plot

used for medium-sized data sets, n = 50

Stem-and-Leaf Plot

graphical display of the five-number summary displaying the shape, center, spread, and extreme points of a data set

Boxplot

observations that are considered to be unusually far from the bulk of the data

Outlier

Five-number summary

1. minimum 2. first quartile (Q1) – 25% 3. median or second quartile (Q2) – 50% 4. third quartile (Q3) – 75% 5. maximum

a type of graph used to display and examine the relationship between two quantitative variables

Scatter Plot

Topic 1 Flashcards

Introduction to Statistics and Data Analysis