Ch. 4

OPAN 2101: Business Statistics

Instructor: Professor Amrita Kundu - Georgetown University

Topics Covered

• Covariance and Correlation (Sections 4.1-4.3)

Key Statistical Terms

• Statistic: A characteristic or measure obtained by using the data values from a sample. For example, the sample mean ($\bar{x}$) or sample standard deviation (s) are statistics that describe the characteristics of a subset of data.

• Sample: A subset of the population used to represent the entire group. E.g., if you want to know the average height of students in a university, you might measure the height of a random selection of 100 students as a sample.

• Parameter: A characteristic or measure obtained by using all the data values from a population. E.g., the true population mean ($\mu$) or population standard deviation ($\sigma$) are parameters. These are often unknown and estimated by statistics.

• Population: The entire group of individuals or instances about whom we hope to learn. E.g., all registered voters in a country, all products produced by a factory in a month, or the expected value ($E[X]$) of a variable in the entire population.

• Z-Score: The number of standard deviations a data point is from the mean, calculated as $Z = \frac{X - \bar{X}}{s}$, where $X$ is the individual data value, $\bar{X}$ is the mean of the sample (or population), and $s$ is the standard deviation of the sample (or population). A positive Z-score indicates the data point is above the mean, while a negative Z-score indicates it's below the mean.

Scatterplots

• Definition: A scatterplot is a graph that depicts the relationship between two quantitative variables by plotting individual data points along two axes. Each point represents a pair of values for the two variables.

• Purpose: Effective for identifying trends, patterns, and relationships between variables, helping to visualize if there's an association.

• Association: The relationship between variables observed in the scatterplot. This describes how changes in one variable correspond to changes in the other.

Types of Associations

• Negative Association: A pattern running from the upper left to the lower right indicates that as one variable increases, the other generally decreases. Example: The more hours a student spends watching TV, the lower their test scores tend to be.

• Positive Association: A pattern running from the lower left to the upper right indicates that as one variable increases, the other also increases. Example: As the number of hours studied increases, exam scores tend to increase.

Relationships

• Linear Relationships: If the points in the scatterplot form a straight line or close to it, there exists a linear relationship. This suggests a constant rate of change between the variables.

• Non-Linear Relationships: If the relationship does not follow a clear linear pattern (e.g., it forms a curve), it is categorized as non-linear. Example: The relationship between the amount of fertilizer used and crop yield might be curved, increasing up to a point and then declining.

• Outliers: Points that fall far away from the overall pattern of the scatter plot and may indicate anomalies in the data, measurement errors, or unique cases that warrant further investigation.

Covariance

• Definition: Covariance measures how much two random variables vary together. Specifically, it indicates the strength and direction of the linear relationship between the two variables.

• Formula (Sample Covariance): $\text{Cov}(X,Y) = \frac{\sum<em>{i=1}^{n}[(X</em>i - \bar{X})(Y_i - \bar{Y})]}{n-1}$

  • $X<em>i$ and $Y</em>i$ are individual sample points.

  • $\bar{X}$ and $\bar{Y}$ are the sample means of variables X and Y, respectively.

  • $n$ is the number of data points (pairs).

• Properties:

  1. $\text{Cov}(X,Y) = \text{Cov}(Y,X)$ (Symmetry)

  2. $\text{Cov}(X,X) = \text{Var}(X)$ (The covariance of a variable with itself is its variance)

• Excel Functions:

  • Sample covariance: =COVAR.S(array1, array2)

  • Population covariance: =COVAR.P(array1, array2)

• Population Parameter (Population Covariance): Denoted by $\text{Cov}(X, Y)$, it is calculated with the population means and total population size, represented as: $\text{Cov}(X,Y) = \frac{\sum<em>{i=1}^{N}[(X</em>i - \mu<em>X)(Y</em>i - \mu_Y)]}{N}$

  • Where $\mu<em>X$ and $\mu</em>Y$ are the population means, and $N$ is the total number of data points in the population.

Correlation

• Definition: Correlation is a standardized measure of the relationship between two variables, providing both the direction and strength of that relationship. Unlike covariance, its value is always between -1 and 1, making it easier to interpret.

• Formula (Sample Correlation Coefficient): $r<em>{XY} = \frac{\text{Cov}(X,Y)}{s</em>X s_Y}$

  • Where $\text{Cov}(X,Y)$ is the sample covariance between X and Y.

  • $s<em>X$ and $$s