Provided Formulas

Key Statistical Formulas

Understanding and applying statistical formulas is essential for analyzing data in business statistics. Below are the main formulas you need to know for Test 1:

• Sample Variance: Measures the average squared deviation from the mean in a sample.

• Expected Value (Discrete Random Variable): The mean value expected from a probability distribution.

• Variance (Discrete Random Variable): Measures the spread of a probability distribution.

Chapter 1: Statistics, Data, and Statistical Thinking

Descriptive vs. Inferential Statistics

Statistics is divided into two main branches: descriptive and inferential statistics.

• Descriptive Statistics: Uses numerical and graphical methods to summarize and present information from a data set.

• Inferential Statistics: Uses sample data to make estimates, predictions, or generalizations about a population.

Key Definitions

• Experimental Unit: The object or entity being studied.

• Population: The complete set of units under investigation.

• Variable: A characteristic or property measured or observed.

• Sample: A subset of the population selected for analysis.

Types of Data

• Quantitative Data: Numerical values (e.g., income, age).

• Qualitative Data: Non-numerical categories (e.g., political party, breed).

Chapter 2: Methods for Describing Sets of Data

Frequency Distributions and Histograms

Frequency distributions and histograms are used to organize and visualize data.

• Class: Categories for qualitative data or intervals for quantitative data.

• Class Frequency: Number of observations in a class.

• Class Relative Frequency: Class frequency divided by total observations.

• Class Percentage: Relative frequency multiplied by 100.

Measures of Central Tendency

• Mean: Arithmetic average of a data set.

• Median: Middle value when data is ordered.

Skewness and Symmetry

• Skewed Right: Mean > Median

• Skewed Left: Mean < Median

• Symmetrical: Mean = Median

Range, Variance, and Standard Deviation

• Range: Difference between largest and smallest values.

• Sample Variance: Measures variability in a sample.

• Sample Standard Deviation: Positive square root of sample variance.

The larger the variance or standard deviation, the more variable the data are. Sample statistics (s, s²) are used to estimate population parameters (σ, σ²).

Empirical Rule

• 68% of data falls within 1 standard deviation of the mean.

• 95% within 2 standard deviations.

• 99.7% within 3 standard deviations.

Z-Scores

Z-scores measure how many standard deviations a value is from the mean.

• Sample z-score:

• Population z-score:

Percentiles

• The pth percentile is the value below which p% of the data falls.

• Example: 90th percentile means 90% of data is below that value.

Chapter 3: Probability

Basic Probability Concepts

Probability quantifies the likelihood of events occurring in an experiment.

• Probability: A number between 0 and 1 representing the chance of an event.

• Experiment: A process that leads to a single, unpredictable outcome.

• Sample Point: The most basic outcome of an experiment.

• Sample Space: The set of all possible sample points.

• Event: A specific collection of sample points.

Probability Rules

• All sample point probabilities must be between 0 and 1.

• The sum of all sample point probabilities in a sample space must equal 1.

Finding Probabilities

• Probability of an event is its relative frequency in repeated experiments.

• Example: Flipping a fair coin, probability of heads = 0.5.

Set Operations and Complements

• The probability of complementary events sums to 1.

Chapter 4: Random Variables and Probability Distributions

Random Variables

A random variable assigns numerical values to outcomes of an experiment.

• Discrete Random Variable: Takes countable values.

• Continuous Random Variable: Takes values in intervals (uncountable).

Probability Distribution Requirements

• for all values of x

• (sum over all possible values of x)

Expected Value and Variance

• Expected Value (Mean):

• Variance:

Binomial Distribution

The binomial distribution models the number of successes in a fixed number of independent trials, each with the same probability of success.

• Characteristics:

  • n identical trials

  • Two possible outcomes per trial: success (S) or failure (F)

  • Probability of success (p) remains constant

  • Trials are independent

• Probability Distribution Formula: where , = number of trials, = number of successes,

Mean, Variance, and Standard Deviation of Binomial Distribution

• Mean:

• Variance:

• Standard Deviation:

Excel Functions for Binomial Distribution

BINOM.DIST Function

• BINOM.DIST(a, n, p, TRUE): Computes cumulative probability of up to a successes in n trials.

• BINOM.DIST(a, n, p, FALSE): Computes probability of exactly a successes in n trials.

• Arguments:

  • a: Number of successes

  • n: Number of trials

  • p: Probability of success