Module 3

Question 1

Population vs sample mean

Answer 1

Symbols:
Σ: Sum
μ: Population mean
N: Population size
Sample variance uses n−1 in the denominator to ensure it is an unbiased estimator.

Question 2

Measures of central location

Answer 2

Mean, median, mode

Question 3

Median

Answer 3

Definition: The middle value of a dataset that divides it into two equal halves.
Key Point:
The median is useful when outliers are present.
If mean and median differ, outliers likely exist.

Question 4

Mode

Answer 4

Definition: The value(s) occurring most frequently in a dataset.

Types:
Unimodal: One mode
Bimodal: Two modes
Multimodal: Three or more modes (less useful)

Examples:
Acetech salaries: $40,000 is the mode (most common).
Women’s sweatshirt sizes: “L” is the mode.

Question 5

Differences between mean, median, and mode

Answer 5

• Use mean when data is symmetrical and free of outliers.
• Use median when data contains outliers or is skewed.
• Use mode for categorical variables or to identify most frequent values.

Question 6

Weighted mean

Answer 6

A mean where different data points are assigned specific weights.
Useful when observations contribute unequally to the average.

Question 7

Histograms

Answer 7

Purpose:
Visualizes data distribution, clustering, spread, and shape.

Key Characteristics:
Symmetric: Mirror image on both sides of the center.
Skewed:
Positive: Long right tail.
Negative: Long left tail.
Symmetric and Unimodal

Distribution:
Mean = Median = Mode.

Question 8

Percentiles

Answer 8

Definition:

Divide data into 100 equal parts.

Specific percentiles:
25th Percentile: Q1
50th Percentile: Q2 (Median)
75th Percentile: Q3

Application:
Ideal for large datasets.
Used in five-number summaries (Minimum, Q1, Median, Q3, Maximum)

Question 9

What is the five number summary?

Answer 9

Minimum, Q1, Median (Q2), Q3, Maximum.

Purpose:
Summarizes the spread and relative position of data.
Example: Growth and Value variables.

Question 10

Measures of dispersion

Answer 10

Variance and standard deviation

Variance:
The average of squared differences between observations and the mean.
Units are squared.

Standard Deviation:
Square root of variance, returning to the original units.
Represents typical spread around the mean.

Question 11

Importance of variance and standard deviation

Answer 11

Variance: Highlights how data points differ from the mean.

Standard Deviation: Provides a clear measure of spread in the same units as the data.

Use Cases: Evaluate consistency, risk, or variability in data (e.g., financial analysis).

Question 12

The Empirical Rule

Answer 12

Bell-shaped distribution:
68% of data within 1 standard deviation of the mean.
95% within 2 standard deviations.
99.7% within 3 standard deviations.

Question 13

Z-scores for Outlier Detection

Answer 13

Measures how many standard deviations a value is from the mean.

For a symmetric, bell-shaped distribution, outliers have Z-scores less than -3 or greater than +3.

Question 14

What is covariance?

Answer 14

Measures the degree to which two variables change together.

Question 15

What is correlation?

Answer 15

Standardized measure of covariance, ranging from -1 to 1.

Question 16

Z-score equation and what does it stand for?

Answer 16

Z-score = X−μ/σ

X is the value,
μ is the mean
σ is the standard deviation.

Question 17

Standard deviation

Answer 17

Square root of variance