Mean, Median, Mode & Sharpe

Overview

Statistics are an important tool for evaluating the investment worthiness of a product or security. This chapter focuses on four common descriptive statistics:

• Mean
• Median
• Mode
• Range

It then reviews alpha, beta, and the Sharpe ratio.

🔑 Definitions at a glance

Mean

The mean is the average value in a set.

🔑 To calculate the mean:

1. Add up all the values.
2. Divide by the number of values.

Mean = sum of all values ÷ number of values

Worked example

A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the mean of the annual returns?

Answer = 5.75%

Add the annual returns (10%, 15%, 5%, -7%) to get 23%. Then divide by the number of returns (4). 23% ÷ 4 = 5.75%

Median

The median is the middle value in a set after the values are ordered from lowest to highest.

🔑 To calculate the median:

1. Put the values in order.
2. If there’s an odd number of values, the median is the middle one.
3. If there’s an even number of values, the median is the average of the two middle values.

Worked example — even number of values

A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the median of the annual returns?

Answer = 7.5%

First, order the returns from lowest to highest: -7%, 5%, 10%, 15% There are four values (an even number), so average the two middle values (5% and 10%). (5% + 10%) ÷ 2 = 7.5%

Worked example — odd number of values

A security obtains annual returns of 10%, 15%, 5%, -7%, and 3% over the past five years. What is the median of the annual returns?

Answer = 5%

First, order the returns from lowest to highest: -7%, 3%, 5%, 10%, 15% 5% is the middle value, so it’s the median.

Mode

The mode is the value that occurs most often.

• ⚠️ If no value repeats, there is no mode.
• If multiple values repeat, the mode is the one that repeats the most.

Worked example — no mode

A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the mode of the annual returns?

Answer = There is no mode

None of the returns repeat, so there is no mode.

Worked example — mode exists

A security obtains annual returns of 10%, 15%, 5%, and -7%, 10%, 8%, and 12% over the past seven years. What is the mode of the annual returns?

Answer = 10%

10% is the only value that repeats, so it’s the mode.

Range

The range is the difference between the highest value and the lowest value.

Range = highest value − lowest value

Worked example

A security obtains annual returns of 10%, 15%, 5%, and -7% over the past four years. What is the range of the annual returns?

Answer = 22%

First, order the returns from lowest to highest: -7%, 5%, 10%, 15% The lowest return is -7% and the highest return is 15%. The range is the difference between them: -7% - 15% = 22%

⚠️ The page’s final arithmetic line prints as “-7% - 15% = 22%”, which is the two endpoints written in the order they were just named rather than a correctly ordered subtraction. The arithmetic that produces the page’s own stated answer of 22% is highest − lowest = 15% − (-7%) = 22%. Use that orientation.

Alpha & beta

Alpha

A common way to evaluate the effectiveness of a fund manager is by using alpha. Alpha measures whether a fund overperformed or underperformed its expected return.

If a question gives you the expected return, the calculation is straightforward.

🔑 Alpha formula (simple form)

Alpha = actual return − expected return

Worked example

An investor determines the expected return of a large-cap stock mutual fund over a year to be +14%. At the end of the year, the actual return was +17%. What is the fund’s alpha?

Alpha = 17% − 14% = 3

Beta

More math-based alpha questions typically introduce another figure: beta.

The page’s summary table:

Math-based question type 1 — inferring the market return

An investor is comparing two different funds in an investment analysis. BCD stock fund maintains a beta of 1.0, while TUV stock fund maintains a beta of 1.5. Last year, BCD stock fund’s performance was +14%, while TUV stock fund’s performance was +19%. What was TUV stock fund’s alpha last year?

Because alpha measures overperformance or underperformance, we need TUV’s actual return and its expected return.

• TUV’s actual return is given: +19%.
• TUV’s expected return is not stated directly.

TUV has a beta of 1.5, meaning it has historically moved 1.5 times as much as the market (in the same direction, since beta is positive). So its expected return is:

Expected return = 1.5 × 14% = 21%

Now apply the alpha formula:

Alpha = actual return − expected return Alpha = 19% − 21% Alpha = -2

An alpha of -2 means TUV underperformed expectations by 2%.

🔑 Alpha formula (expanded form)

Alpha = (PR − RF) − (Beta × (MR − RF))

Where:

• The portfolio return and market return are usually given in the question.
• The risk-free rate of return is the return on a relatively risk-free security. The most commonly cited risk-free security is the 3-month Treasury bill. It’s considered close to risk-free because of its short maturity and U.S. government backing, although all securities carry at least some risk.

Math-based question type 2 — expanded formula

An investor is analyzing the market and the returns of a small-cap stock fund held in their portfolio. The fund was up 28% while maintaining a beta of 2.5 last year. During the same year, the S&P 500 was up 10%, the Russell 2000 was up 14%, and the 3-month Treasury bill gained 2%. What is the small-cap stock fund’s alpha?

Answer: -4

Alpha = (PR − RF) − (Beta × (MR − RF)) Alpha = (28% − 2%) − (2.5 × (14% − 2%)) Alpha = 26% − (2.5 × 12%) Alpha = 26% − 30% Alpha = -4

This fund manager underperformed expectations by 4%, leading to an alpha of -4.

Active vs. passive management

• 📌 Alpha is most relevant when evaluating an actively managed fund because active managers aim to outperform a benchmark (a relevant market index). For example, if a small-cap stock fund manager tries to beat the Russell 2000 by selecting small-cap stocks, alpha helps measure whether those choices added value.

Sharpe ratio

This ratio measures risk-adjusted returns for a security or portfolio. In plain terms, it measures “bang for the buck,” or investment efficiency: how much return you’re getting for the amount of risk you’re taking.

🔑 Formula

Sharpe ratio = (Actual return − Risk free rate) ÷ Standard deviation

⚠️ Reconstructed fraction. The page’s text extraction flattens fractions and prints the denominator first — it renders as “Sharpe ratio = Standard deviation / Actual return - Risk free rate”. The correct orientation is (actual return − risk-free rate) over standard deviation, confirmed by the page’s own interpretation that a higher Sharpe ratio means more efficiency (more return per unit of risk) — which only holds when standard deviation is the denominator.

Components:

• The higher the Sharpe ratio, the more efficient the security or portfolio (more return per unit of risk).
• 📌 It’s unlikely you’ll be asked to calculate the Sharpe ratio on the exam, but you may see questions about what the ratio measures or what each part of the formula represents.

Key points

Mean

• Average of relevant factors

Median

• Middle number in a set of factors

Mode

• Most frequently recurring factor

Range

• Difference between lowest and highest factor

Alpha

• Measures over or underperformance of a portfolio or security
• Positive alpha = overperformance
• Zero alpha = meeting expectations
• Negative alpha = underperformance

Beta

• Volatility measure as compared to the market (benchmark index)

Sharpe ratio

• Measures risk-adjusted return of security portfolio
• 🔑 Sharpe ratio = (Actual return − Risk free rate) ÷ Standard deviation ⚠️ (reconstructed from the page’s flattened “Sharpe ratio = Standard deviation / Actual return - Risk free rate”)

Sources

Primary/official references for the material in this chapter. Every link was fetched and returned HTTP 200 on 2026-08-15.