Descriptive and Inferential Statistics

• Two types of statistics are descriptive and inferential.

Mutually Exclusive Events

• Mutually exclusive events cannot occur simultaneously.
• Example: Rolling a 3 and rolling a 5 on a single die roll are mutually exclusive.

Descriptive Statistics

• Descriptive statistics summarize and describe data set features.
• Examples include mean, median, mode, and graphs.

Inferential Statistics

• Inferential statistics involves making predictions or inferences about a population based on a sample.
• Examples include hypothesis testing and confidence intervals.

Independent Events Notation

• P(A∩B)=P(A)×P(B) represents the probability of both independent events A and B occurring.

Z-Score

• A Z-score indicates how many standard deviations a value is from the mean.
• Formula: Z= (X−μ) / σ, where X is the value, μ is the mean, and σ is the standard deviation.
• Example: For a value of 60, given a mean of 50 and a standard deviation of 10, the Z-score is 1.

Population

• Population refers to the entire group you want to study.

Sample

• Sample is a subset of the population.

Population vs Sample

• Population: all students in the U.S.
• Sample: 200 students from Oklahoma

Quantitative Data

• Quantitative data is measurable and expressed numerically.
• Examples include height and weight.

Qualitative Data

• Qualitative data describes categories or qualities.
• Examples include gender and eye color.

Data Subset

• Data may be subsetted to focus on a specific group or variable, like analyzing data for a specific year or category.

Horizontal Bar Charts

• Axes should be clearly marked with numbers and scales.
• Bars on bar charts should have the same width.
• Vertical axes should not be stretched.

Relative Frequency Display

• Histograms or box plots graphically display the relative frequency of a numerical variable.

Relative Frequency

• Relative frequency is the proportion of times a value occurs.
• It is calculated by dividing the frequency by the total number of observations.
• Example: A value occurring 5 times out of 50 observations has a relative frequency of 0.10.

Percent Frequency

• Percent frequency is the relative frequency expressed as a percentage.

Relative vs Percent Frequency

• Relative frequency represents a proportion (fraction).
• Percent frequency is expressed as a percentage.

Reading a Frequency Table

• To read one, calculate the sum of the relevant data divided by the total number of observations, then multiply by 100.

Statistic Influenced by Outliers

• The mean is most influenced by outliers because it incorporates all data points.

Probability Example

• Given: Male customers order (150 beer, 40 wine, 20 soft drinks) and Female customers order (38 beer, 20 wine, 12 soft drinks)
• The probability that a randomly selected customer orders beer is 0.67, calculated as (150+38) / (150+40+20+38+20+12).

Time Series Data

• Time series data is collected over time.
• An example is monthly sales figures.

Cross-Sectional Data

• Cross-sectional data is collected at one point in time.
• An example is survey data of peoples incomes.

Statistic

• A statistic is a numerical value that summarizes or describes a sample feature.

Empirical Rule

• 68% of data falls within 1 standard deviation of the mean.
• 95% of data falls within 2 standard deviations of the mean.
• 99.7% of data falls within 3 standard deviations of the mean.

Dispersion Statistics

• Standard deviation is most common for showing dispersion.

Variance

• Variance measures the spread of data.
• Standard deviation is the square root of variance.

Covariance

• Covariance measures the relationship between two variables.
• A positive covariance means the variables tend to increase together.

Collectively Exhaustive Events

• A set of events is collectively exhaustive if one of the events must occur.
• Example: rolling a die (one of 1,2,3,4,5, or 6)

Union in Probability

• The union is the combined set of outcomes where event A or event B occurs (unique elements from both sets).
• Notation: A∪B

Intersection in Probability

• The intersection is the occurrence of both events A and B happening simultaneously (elements present in both sets).
• Notation: A∩B

Intersection Example

• A = {The Porch, Louie's, The Library, O'Connell's} and B = {The Library, O'Connell's, Logies, Volare}.
• A ∩ B = {The Library, O'Connell's}.

Union Example

• A = {The Porch, Louie's, The Library, O'Connell's} and B = {The Library, O'Connell's, Logies, Volare}.
• A ∪ B = {The Porch, Louie's, The Library, O'Connell's, Logies, Volare}.

Sum of Probabilities

• All probabilities of an event must add up to 1.

Simple Event

• An event with one outcome is a simple event.

Sample Space

• Sample space is the set of all possible experiment outcomes.
• Example: letter grades in class (A, B, C, D, F).

Complement of A

• The complement of A consists of all outcomes in the sample space S that are not in A.

Correlation Coefficient

• Measures the strength and direction of the linear relationship between two variables.

Variance and Standard Deviation

• Standard deviation is the square root of the variance.

Z-score probability example 1

• Given an average on exam 1 of 85 with a standard deviation of 1.5, the probability of scoring less than 75 has a z-score of -6.67.
• Z(75) = 75-85 / 1.5 = -6.67

Z-score probability example 2

• Given an average on exam 1 of 85 with a standard deviation of 1.5, the probability of scoring greater than a 90 has a z-score of 3.33
• Z(90) = 90 - 85 / 1.5 = 3.33

Explore the fundamentals of descriptive and inferential statistics, including mutually exclusive events, independent events, and Z-scores. Learn how to summarize data, make predictions, and calculate probabilities. Understand key statistical concepts such as mean, median, mode, hypothesis testing, and standard deviation.