The arithmetic mean is what most people call the "average." It is found by adding all the values together and dividing by the number of values.

Formula

For n values: $x_1, x_2, x_3, ..., x_n$

$\text{Arithmetic Mean} = \frac{x_1 + x_2 + x_3 + ... + x_n}{n}$

Worked Example

The masses of four parcels at a courier shop in Dar es Salaam are: 5 kg, 8 kg, 20 kg, and 15 kg. Find the mean mass.

Solution:

Total mass = 5 + 8 + 20 + 15 = 48 kg

Number of parcels = 4

Mean mass = $\frac{48}{4}$ = 12 kg

When data is grouped with frequencies, we use a different approach:

1. Multiply each value (x) by its frequency (f) to get fx
2. Add all fx values to get the total
3. Add all frequencies to get the total number of items
4. Divide total fx by total frequency

Worked Example

Mean = $\frac{38}{10}$ = 3.8

The geometric mean is useful when dealing with rates of growth, ratios, or numbers that multiply together. It is the nth root of the product of n values.

Formula

For n positive values: $x_1, x_2, x_3, ..., x_n$

$\text{Geometric Mean} = \sqrt[n]{x_1 \times x_2 \times x_3 \times ... \times x_n}$

Or equivalently: $\text{GM} = (x_1 \times x_2 \times ... \times x_n)^{\frac{1}{n}}$

Worked Example

A shopkeeper recorded the price increase ratios of a commodity over four years: 1.2, 1.5, 0.8, and 1.1. Find the average growth ratio.

Solution:

Geometric Mean = $\sqrt[4]{1.2 \times 1.5 \times 0.8 \times 1.1}$

First, multiply: 1.2 × 1.5 × 0.8 × 1.1 = 1.584

Now find the fourth root: $\sqrt[4]{1.584}$

Using a calculator: $(1.584)^{0.25}$ ≈ 1.123

The average annual growth ratio is approximately 1.123, meaning prices grew by about 12.3% on average per year.

Note

The geometric mean is always less than or or equal to the arithmetic mean. It is preferred when averaging ratios, percentages, or growth rates because it accounts for compounding effects.

Compound interest is interest calculated on both the initial principal and the accumulated interest from previous periods. This causes money to grow exponentially, unlike simple interest which is calculated only on the principal.

Formula

$A = P\left(1 + \frac{r}{n}\right)^{nt}$

Where:

• A = amount accumulated (principal + interest)
• P = principal (initial amount)
• r = annual interest rate (as a decimal)
• n = number of times interest is compounded per year
• t = time in years

If interest is compounded annually, n = 1, and the formula simplifies to:

$A = P(1 + r)^t$

Worked Example 1

Fatuma deposited 200,000 TZS in a savings account that pays 6% compound interest per year. How much will she have after 5 years?

Given:

• P = 200,000 TZS
• r = 6% = 0.06
• n = 1 (compounded annually)
• t = 5 years

Solution:

$A = 200,000 \times (1 + 0.06)^5$

$A = 200,000 \times (1.06)^5$

$A = 200,000 \times 1.338225$

$A =$ 267,645 TZS

Fatuma will have approximately 267,645 TZS after 5 years.

Worked Example 2

A businessman borrowed 1,000,000 TZS at 12% per year compound interest. How much will he owe after 3 years?

Solution:

$A = 1,000,000 \times (1 + 0.12)^3$

$A = 1,000,000 \times 1.12^3$

$A = 1,000,000 \times 1.404928$

$A =$ 1,404,928 TZS

He will owe approximately 1,404,928 TZS after 3 years.

• Arithmetic Mean: Add all values, divide by count. Used for typical everyday averaging.
• Geometric Mean: Take the nth root of the product of n values. Used for growth rates, ratios, and multiplicative data.
• Compound Interest: Amount grows according to $(1 + r)^t$ when compounded annually. The interest earns interest over time.

In Tanzania, these concepts are used regularly. For example, when a farmer in Mbeya takes a loan of 500,000 TZS from a microfinance institution at 15% compound interest to buy fertilizer, understanding compound interest helps the farmer calculate exactly how much will be owed after 2 or 3 years, enabling better financial planning and avoiding debt traps. Similarly, shop owners in Arusha use geometric mean to calculate average monthly growth rates of sales, while arithmetic mean helps schools compute the average scores of students in examinations.