Chapter 3: Numerical Descriptive Measures

Scenario: Dorothy, a financial advisor, helps an inexperienced investor compare two mutual funds: Fidelity’s Growth Index (Growth) and Fidelity’s Value Index (Value), using their annual return data (e.g., -5.50% and -8.59% in 1984, 39.91% and 22.10% in 1985, 38.42% and 31.62% in 2019).
Tasks for Dorothy:
1. Calculate and interpret the typical return for each fund.
2. Calculate and interpret the investment risk for each fund.
3. Determine which fund offers a better return relative to its risk.

Purpose: Attempt to find a typical or central value that describes a variable (e.g., typical investment return, typical graduate salary).
Key Measures Discussed: Mean, Median, Mode, Weighted Mean, Percentile.

Definition: The primary measure of central location, also called the mean or average. Calculated by summing all observations and dividing by the number of observations.
Population Mean ( $\mu$ ):
- Formula: $\mu = \frac{\sum<em>{i=1}^{N} x</em>i}{N}$
- $N$ : number of observations in the population.
- $x_i$ : value of the $i$ -th observation.
Sample Mean $( \bar{x} )$ :
- Formula: $\bar{x} = \frac{\sum<em>{i=1}^{n} x</em>i}{n}$
- $n$ : number of observations in the sample.
Example (Growth and Value Funds): Over 36 years, the mean return for Growth was greater than Value (Growth: 15.755%, Value: 12.005%).
- Caution: Investing solely based on average return can be misleading, as risk is not considered.
Sensitivity to Outliers: The mean can be highly affected by extreme values (outliers).

Definition: The middle observation in a data set when arranged in ascending order. It divides the data into two equal halves.
Calculation:
1. Arrange observations in ascending order.
2. If $N$ (or $n$ ) is odd, the median is the middle observation.
3. If $N$ (or $n$ ) is even, the median is the average of the two middle values.
Usefulness: Particularly valuable when outliers are present, as it is less sensitive to them than the mean.
Practical Application: Often preferred for money-related variables (income, wealth, house prices) because these distributions tend to be skewed by high-value outliers.
Example (Acetech Salaries):
- Salaries: 40,000, 40,000, 65,000, 90,000, 100,000, 145,000, 150,000, 550,000 (arranged).
- $N=8$ (even), so median is average of 4th and 5th values:
  - $Median = \frac{90,000 + 100,000}{2} = 95,000$
- This provides a better reflection of a typical salary compared to the mean ( $147,500$ ) or mode ( $40,000$ ).

Definition: The observation that occurs most frequently in a data set.
Types:
- Unimodal: One mode.
- Multimodal: Two or more modes.
- A data set can also have no mode (if all observations occur with the same frequency).
Limitations: Its usefulness diminishes with more than three modes. It doesn't always reflect the center of the variable, especially if the most frequent observation is an extreme value.
Example (Acetech Salaries): The modal salary is $40,000$ (occurs twice).
Categorical Variables: The mode is the only meaningful measure of central location for categorical variables (e.g., modal size L for sweatshirts S, L, L, M, S, L, M, L, L, M).

Growth: Mean (15.755%), Median (15.245%), Mode (#N/A - no observation appears more than once).
Value: Mean (12.005%), Median (15.38%), Mode (#N/A).
Excel Formulas:
- Mean: =AVERAGE(range)
- Median: =MEDIAN(range)
- Mode: =MODE(range) (or =MODE.SNGL(range) for single mode, or =MODE.MULT(range) for multiple modes).
Excel Data Analysis Toolpak: Provides