Geometric mean (AQA GCSE Statistics): Revision Notes

Worked Example: Find the geometric mean of 5, 7, and 12

Step 1: Count the number of values

• We have 3 values, so $n = 3$

Step 2: Multiply all values together

• $5 \times 7 \times 12 = 420$

Step 3: Take the nth root (in this case, the cube root)

• Geometric mean = $\sqrt[3]{420} = 7.49$ (to 2 decimal places)

Calculator tip: You can use the $\sqrt[3]{}$ function on your calculator, or raise 420 to the power of $\frac{1}{3}$ (since $\sqrt[3]{x} = x^{1/3}$).

Worked Example: Finding Unknown Values

Problem: The geometric mean of four numbers is 6. Two of the numbers are 4.5 and 8. The third and fourth numbers are equal. Calculate the values of the third and fourth numbers.

Solution: Since we know the geometric mean is 6, we can work backwards:

Step 1: Set up the equation

• Geometric mean = $\sqrt[4]{4.5 \times 8 \times x \times x} = 6$
• Where $x$ represents both the third and fourth numbers (since they're equal)

Step 2: Rearrange the equation

• $6 = \sqrt[4]{4.5 \times 8 \times x^2}$
• $6^4 = 4.5 \times 8 \times x^2$
• $1296 = 36x^2$

Step 3: Solve for x

• $x^2 = 1296 \div 36 = 36$
• $x = \sqrt{36} = 6$

Therefore, the third and fourth numbers are both 6.

Worked Example: Using Percentage Multipliers

Problem: The number of cars sold by a car manufacturer increased by 4% in year 1 and decreased by 5% in year 2. Calculate the geometric mean of these two percentage changes.

Solution: When dealing with percentage changes, we use multipliers rather than the percentage values themselves:

Step 1: Convert percentages to multipliers

• An increase of 4% means multiplying by 1.04
• A decrease of 5% means multiplying by 0.95

Step 2: Apply the geometric mean formula

• Geometric mean = $\sqrt{1.04 \times 0.95}$
• Geometric mean = $\sqrt{0.988} = 0.994$ (to 3 decimal places)

This tells us that the overall effect is a slight decrease, with an average multiplier of 0.994.