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1.1) Getting Started
Key Concepts:

Statistics- refer to either the science of gathering, describing, and analyzing data OR the actual numerical descriptions of sample data

Parameter-A numerical description of a population characteristic

Population- consist of all persons or things about which one is trying to make an inference or decision ( The particular group of interest)

Sample- A subset of the population from which data are collected (a group from within the population)

Sample statistics-Numerical descriptions of sample characteristics

Variables- Values that can change amongst members of the population

Data- the information gathered about a specific variable

Census- A study in which data are obtained from every member of the population

Difference between population parameter and sample statistics

Parameters are numbers that summarize data from an entire population while statistics are numbers that summarize data from a sample
Difference between Population and Sample
The population is the entire group you wan to draw conclusions about while the sample is the specific group you want to collect data from

Branches of Statistics

Descriptive statistics: gathers, sorts, summarizes and displays the data that has been collected

Inferential statistics: uses descriptive statistics to interpret information

1.2) Data Classification
Key Concepts:

Qualitative data- also known as categorical data, consist of labels or descriptions of traits of the sample

Quantitative data- Consist of counts or measurements

Continuous data- data that can take on any value in a given interval and are usually measurements such as length and weight

Discrete data-data that can take on only particular values and are usually counts. e.g; the number of children you have

Levels of Measurement

Nominal: it is a type of qualitative data that consist of labels or names

Ordinal: it is a type of qualitative data that can be arranged in a meaningful orderly manner.

Interval: it is a type of quantitative data that can be arranged in a meaningful order, and differences between data entries are meaningful and equal.

Ratio: it is a type of quantitative data that can be ordered, differences between data entries are meaningful, and the zero point indicates the absence of something

1.3) The Process of a Statistical Study

Key Concepts

Observational study- Observes data that already exist

Experiment- a procedure carried out under controlled conditions in order to generate data to help identify cause-and-effect relationships

Representative sample- Has the same relevant characteristics as the population and does not favor one group from the population over another

Treatment-Some condition applied to a group of people or things in an experiment

Subjects- People or things that are being studied in an experiment

Participants- People being studied in an experiment

Response variable- The variable that responds to the treatment

Explanatory variable- it explains the variable that causes the change in the response variable

Treatment group- A group of subjects that receive treatment in an experiment

Control group- A group of subjects to which either receives no treatment, a standard treatment whose effect is already known, or a placebo (fake treatment)

Confounding variables- Factors other than the treatment that cause an effect on the subjects in an experiment

Placebo effect- A response to the power of suggestion, rather than the treatment itself, by participants of an experiment

Placebo- A substance that appears identical to the actual treatment but contains no intrinsic beneficial elements

Single-blind- Subjects do not know if they are in the control group or the treatment group, but the people interacting with the subjects in the experiment know in which group each subject has been placed

Double-blind- Neither the subjects nor the people interacting with the subjects know to which group each subject belongs

Institutional Review Board (IRB)- A group of people who review the design of a study to make sure that it is appropriate and that no unnecessary harm will come to the subjects involved

Informed consent-Completely disclosing to participants the goals and procedures involved in a study and obtaining their agreement to participate

Types of Observational Studies

Cross-sectional study: Data are collected at a single point in time

Longitudinal study: Data are gathered by following a particular group over a period of time

Meta-analysis: Study that compiles information from previous studies

Case study: Looks at multiple variables that affect a single event

Conducting a Statistical Study

Determine the design of the study.

State the question to be studied

Determine the population and variables.

Determine the sampling method.

Collect the data.

Organize the data.

Analyze the data to answer the question.

Sampling Methods

Random sampling: Every member of the population has an equal chance of being selected

Simple random sampling: Every sample from the population has an equal chance of being selected

Stratified sampling: Dividing the population into subgroups that share similar characteristics and drawing a random sample from each group (the group is called strata)

Quota sampling: A type of group in which certain characteristics of the population are preserved in the sample

Cluster sampling: Dividing the population into groups that are each similar to the entire population and randomly selecting a whole group to sample (the group is called clusters)

Systematic sampling: Selecting every member of the population

Convenience sampling: This type of sample is "convenient" for the researcher to work with.

Principles of Experimental Design

Randomize the control and treatment groups.

Control for outside effects on the response variable.

Replicate the experiment a significant number of times to see meaningful patterns.

1.4) How to Critique a Published Study

Key Concepts:

Bias- Favoring of a certain outcome in a study

Sampling bias- occurs when the sample chosen does not accurately represent the population that is being studied

Dropouts- Participants who begin a study but fail to complete it

Processing errors- Errors that occur simply from the data being processed, such as typos when data are being entered

Non-adherents- Participants who remain in the study until the end but move away from the directions they were given

Researcher bias- Occurs when a researcher influences the results of a study

Response bias- occurs when a researcher’s behavior causes a participant to change his or her response or when a participant gives an inaccurate response.

Participation bias- Occurs when there is a problem with the participation in the study

Nonresponse bias- occurs when there is no participation in a self-selected sample from a certain group of a population, when a person refuses to participate in a survey, or when a respondent omits questions when answering a survey.

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2.1) Frequency Distributions
Key Concepts:

Ordered array- An ordered list of data from largest to smallest or vice versa

Distribution- A way to describe the structure of a particular data set or population

Frequency distribution- A display of the values that occur in a data set and how often each value or range of values occurs

Probability distribution-A mathematical function that is used to predict the probabilities of occurrence of different possible outcomes for an experiment

Ungrouped frequency distribution- A frequency distribution in which data is given as a single value

Frequency- The number of the data point in a category of a frequency distribution

Class- A category of data in a frequency distribution

Grouped frequency distribution- A frequency distribution in which data are given as intervals

Class width-The difference between the lower class limits of two consecutive classes of a frequency distribution

Lower-class limit- The smallest number that can belong to a particular class

Upper-class limit- The largest number that can belong to a particular class

Sample size- The sum of all the frequencies in a frequency distribution

Constructing a Frequency Distribution

1. Decide how many classes should be in the distribution.

2. Choose an appropriate class width.

3. Find the class limits.

4. Determine the frequency of each class.

2.2) Graphical Display of Data
Key Concepts:

Pie chart- Shows how large each category of qualitative data is in relation to the whole; uses relative frequencies to divide the "pie" into wedges

Bar graph- Bars are used to represent the amount of data in each category; one axis displays the categories of qualitative data and the other axis displays the frequencies

Stacked bar graph- A bar graph that compares the same categories for different groups and shows category totals

Time-series graph- A line graph that is used to display a variable whose values change over time

Cross-sectional graph- Displays information collected at only one point in time

Pictograph- A bar graph that uses pictures of objects instead of bars

Outlier-A data value that falls outside the shape of the distribution

Histogram- A bar graph of a frequency distribution of quantitative data; the horizontal axis is a number line

Frequency histogram: A histogram in which the heights of the bars represent frequencies

Relative frequency histogram: A histogram in which the heights of the bars represent relative frequencies in either decimals or percentages

Frequency polygon-A visual display of the frequency of each class of quantitative data that uses straight lines to connect points plotted above the class midpoints

Ogive-A graph that displays the cumulative frequency of each class of quantitative data by using straight lines to connect points plotted above the upper-class boundaries

Stem-and-leaf plot- Retains the original data; the leaves are the last significant digit in each data value and the stems are the remaining digits

Dot plot- Retains the original data by plotting a dot above each data value on a number line

Line graph- Uses straight lines to connect points plotted at the value of each measurement above the time it was taken

2.3) Analyzing graphs
Key Concepts:

Uniform: The frequency of each class is relatively the same; the graph has a rectangular shape

Symmetric: The data lie evenly on both sides of the distribution

Skewed to the right: The majority of the data fall on the left side of the distribution

Skewed to the left: The majority of the data fall on the right side of the distribution

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3.1) Measures of Center
Key Concepts:

Mean- The "average" of recorded data, used for quantitative data as long as it is not skewed and there are no outliers. The mean will be in the direction of any outliers when graphed.

Sample Mean- The mean taken from a sample

Population Mean-  The average taken from a population

Median- The middle value taken from a set of data (the formula can vary based on if the number of data points is even or odd). The median is found in the middle of the graph and is used when there are outliers or skewed data.

Mode- This is the number that occurs most frequently in a data set. (A data set can have no mode, be unimodal, bimodal, or multimodal). The mode is typically used for qualitative data. The mode has the highest peak on the graph.

3.2a) Measures of Dispersion
Key Concepts:

Range- The difference between the largest and smallest number in a data set.

Standard Deviation- This tells you how much you can expect the a data point to differ from the mean. The greater the standard deviation, the more the data points are spread out.

Variance- Variance is the square of standard deviation

3.2b) Applying the Standard Deviation
Key Concepts:

Coefficient of Variation- This represents the relationship between the standard deviation and the mean as a percentage

Empirical Rule- Used for bell-shaped data to help determine how far data points are from the mean

Chebyshev's Theorem- This measures the same thing as the empirical rule, but it is less accurate because it is not used for the symmetrical bell-shape.

3.4) Measures of Relative Position
Key Concepts:

Percentiles- Percentiles give you a rough estimate of what data points lie at or below a certain value

Quartiles- This is when you divide data into four parts to group it by it's position on the graph.

Five Number Summary- This includes the minimum value, quartile 1, quartile 2 (the median), quartile 3, and the maximum value.

Box Plot- A way to graphically represent a five number summary.

Interquartile Range- The IQR is found within a box plot, it represents the difference between the first and third quartile.

Standard Score (Z-Score)- This is how many standard deviations a certain value is from the mean.

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4.1) Introduction to Probability
Key Concepts:

Probability Experiment- A probability experiment is anything in which the result/outcome is determined by chance.

Sample Space- This is all of the collective results possible in a probability experiment.

Event- This is a subsection of events that can occur within a probability experiment.

Tree Diagram- A tree diagram organizes outcomes in a systematic manner.

Subjective Probability- An educated guess regarding the probability of an event.

Experimental Probability (Empirical Probability)- This is found by performing an experiment.

Classical Probability (Theoretical Probability)- This is when all outcomes are equally likely to occur.

4.2) Addition Rules for Probability
Key Concepts:

Compliment- A compliment of an event (E) are all of the other possible outcomes outside of E. The sum of E and its compliment is equal to 1.

Mutually Exclusive- An event that shares no outcomes is mutually exclusive.

4.3) Multiplication Rules for Probability
Key Concepts:

Multistage Experiment- An experiment with multiple steps is called a multistage experiment.

"With Replacement"- This refers to maintaining the same number of variables when performing a probability experiment.

Independent- An independent event is not influenced by the probability of another event.

"Without Replacement"- This is when the number of variables is lessened throughout the process of the experiment.

Dependent- An event that cannot happen without certain conditions.

Conditional Probability- The probability of one event given that another occurs first.

Fundamental Counting Principle- This is when you multiply each possible outcome in a multistage experiment to get the total number of possibilities.

4.4) Combinations and Permutations
Key Concepts:

Factorial- The product of all positive integers less or equal to the given value.

Combination- This is a selection of variables without regard to their order.

Permutation- A permutation is a selection of variables picked solely on their order.

Special Permutations- These involve variables that are identical.

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5.1) Discrete Random Variable
Key Concepts:

Random variable- A variable whose numeric value is depended on the outcome of a probability experiment

Probability distribution- A table or formula that gives the probabilities of occurrence of different possible outcome of an experiment

Discrete random variable-  has a countable number of possible values. The probability of each value is between 0 and 1, the sum is 1.

Discrete probability distribution- A table or formula that gives the probability for each outcome of a discrete random variable

Expected Value-  is the value that we expect the random variable to have on average. It is the mean of the probability distribution,

Properties of a Probability Distribution

1. All of the probabilities are between 0 and 1

2. The sum of the probabilities is 1.

5.2) Binomial Distribution
Key Concepts:

Binomial distribution- A discrete probability function for problems with a fixed number of independent trials, where each trial has only two possible outcomes, success or failure. Success is counted

Properties of a Binomial Distribution

1. The experiment consists of a fixed number of identical trials.

2. Each trial is independent of the others.

3. For each trial, there are only two possible outcomes, labeled a success, and the other a failure.

4. For every trial, the probability of getting a success is called p The probability of getting a failure is then 1−p

5. The binomial random variable counts the number of successes in number of trials.

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6.1) Introduction to Normal Distribution
Key Concepts:

Normal distribution- a function that represents the distribution of many random variables as a symmetrical bell-shaped graph.

Density curve- is a theoretical curve that describes the distribution of a continuous variable.

Inflection point- a point of a curve at which a change in the direction of curvature occurs.

Horizontal asymptote- are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞.

Standard normal distribution- it is the distribution that occurs when a normal random variable has a mean of zero and a standard deviation of one.

Properties of a normal distribution

The mean, mode, and median are all equal.

The curve is symmetric at the center (i.e. around the mean, μ).

Exactly half of the values are to the left of center and exactly half the values are to the right.

The total area under the curve is 1.

6.2) Finding Area Under Normal Distribution
Key Concepts:
Cumulative normal distribution table- distribution function of a random variable X, evaluated at x, is the probability that X will take a value less than or equal to x.

6.3) Finding Probability Using a Normal Distribution
Key Concepts:
Relationship between Area and Probability for a Normal Distribution- This is the area under the normal curve for the region that is equivalent to the probability of the normally distributed random variable that is in the specific area.

Procedure

Finding Probability for Any Normal Curve

Turn each value of the random variable (x-value) to a standard score (z-value).

Find the appropriate area under the standard normal curve using the standard deviation.

6.4) Finding Values of a Normally Distributed Random Variable
Key Concepts:
Finding the value of a normally distributed random variable.

1. Using the standard normal distribution tables, a calculator, or statistical software to find the z-score corresponding to the given area under the standard normal curve.

2. Transform the z-score to the corresponding value of the random variable (x-value) using the formula x=(z)*(σ+μ)

6.5) Approximating a Binomial Distribution Using a Normal Distribution
Key Concepts:
Continuity correction- Is applied when you want to use a continuous distribution to approximate a discrete distribution. Typically, it is used when you want to use a normal distribution to approximate a binomial distribution.

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7.2) Central Limit Theorem with Means
Key Concepts:

Central Limit Theorem with Mean- These equations are based on the probability found by using the sample mean rather than the probability of a specific, individual variable.

7.3) Central Limit Theorem with Proportions
Key Concepts:

Population Proportion- This is the percentage of a population that has a certain characteristic.

Sample Proportion- This is the percentage of a sample that has a certain characteristic.

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8.1) Estimating Population Means
Key Concepts:

Point estimate- It involves the use of sample data to calculate a single number which is to serve as the best estimate of an unknown population parameter

Unbiased estimator-A point estimate that does not consistently underestimate or overestimate the population parameter

Interval estimate- A range of possible values for a population parameter

Level of confidence- The probability that the interval estimate contains the population parameter

Confidence interval- An interval estimate associated with a certain level of confidence

Margin of error- Denoted by E, and also called the maximum error of estimate, it is the largest possible distance that tells you how many percentage points your result will differ from the real population

8.2) Student's T-distribution
Key Concepts:
Properties of a T-Distribution

A t-distribution curve is symmetric and bell-shaped, centered about 0.

A t-distribution curve is completely defined by its number of degrees of freedom

The total area under a t-distribution curve equals 1.

The x-axis is a horizontal asymptote for a t-distribution curve.

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10.1) Fundamentals of Hypothesis Testing
Key Concepts:
Hypothesis- theory or premise, a claim about the number value of a population parameter, such as the population mean, proportion, or variance.
Hypothesis testing- Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter.
Alternative hypothesis-  is one in which a difference (or an effect) between two or more variables is anticipated by the researchers; that is, the observed pattern of the data is not due to a chance occurrence
Null hypothesis- the hypothesis that there is no significant difference between specified populations, any observed difference being due to sampling or experimental error.
Test statistic- used in a hypothesis test when you are deciding to support or reject the null hypothesis. The test statistic takes your data from an experiment or survey and compares your results to the results you would expect from the null hypothesis.
Statistically significant- determination by an analyst that the results in the data are not explainable by chance alone.
Level of significance- is a measure of the strength of the evidence that must be present in your sample before you will reject the null hypothesis and conclude that the effect is statistically significant.
Type I error- Rejecting a true null hypothesis
Type II error- Failing to reject a false null hypothesis

Procedures

1. State the null and alternative hypotheses.

2. Determine which distribution to use for the test statistic, and state the level of significance.

3. Gather data and calculate the necessary sample statistics.

4. Draw a conclusion and interpret the decision

10.2) Hypothesis Testing for Population Means (Sigma Known)
Key Concepts:
P-value- the p-value is the probability of obtaining results at least as extreme as the observed results of a statistical hypothesis test, assuming that the null hypothesis is correct.

10.3) Hypothesis Testing for Population Means (Sigma Unknown)
Key Concepts:
Rejection Regions for Hypothesis Tests for Population Means (σ Unknown)
Reject the null hypothesis, H0H0, if:

t≤−tα for a left-tailed test

t≥tα for a right-tailed test

|t|≥tα/2 for a two-tailed test

10.4) Hypothesis Testing for Population Proportions:

Formula
z=(pˆ−p)/ √ p(1-p)/n

10.5) Hypothesis Testing for Population Variances
Procedure
Rejection Regions for Hypothesis Tests for Population Variances and Standard Deviations
Reject the null hypothesis, H0H0, if:

χ2≤χ2(1−α) for a left-tailed test
χ2≥χ2α for a right-tailed test
χ2≤χ2(1−α/2) or χ2≥χ2α/2 for a two-tailed test

10.6) Chi-Square Test for the Goodness of Fit
Key Concepts: Null and Alternative Hypotheses for Chi-Square Tests for Goodness of Fit
When the theoretical probabilities for the various outcomes are all the same:

H0: p1=p2=…=pk

Ha: There is some difference amongst the probabilities

When the theoretical probabilities for the various outcomes differ:
H0:

p1= probability of the first outcome

p2= probability of the second outcome

pk= probability of the K^th outcome

Ha: There is some difference from the stated probabilities

Properties
Reject the null hypothesis, H0, if: χ2≥χ2α

10.7) Chi-Square Test for Association
Key Concepts:
Null and Alternative Hypotheses for Chi-Square Tests for Association
H0: The two variables in the population are independent.
Ha: The two variables in the population are not independent

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12.1) Scatter Plots and Correlation
Key Concepts:

Scatter Plot- This is a type of graph with one point for a related pair of data.

Explanatory Variable- This is the variable that influences change in another variable

Response Variable- This variable is also known as the dependent variable, because it relies on the explanatory variable to determine its "response".

Linear Relationship- This is when the points on a scatterplot roughly follow a straight line.

Positive Slope- This happens when a line appears to go up positively as each variable increases.

Negative Slope- This occurs when one variable increases and the other (the response variable) decreases.

Correlation of Coefficient- This is what is used to measure the strength of the linear relationship.

Coefficient of Determination- This measures the proportion of variability in the response variables vs the variability in the explanatory variables.

12.2) Linear Regression
Key Concepts:

Least-Squares Regression Line- Also known as "the line best fit", it is a line that can be drawn based off of a data set's slope and y-intercept.

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Warren, C., Denley, K., & Atchley, E. (2021). Beginning statistics. Charleston, SC: Hawkes Learning Systems.
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