Q: If the values $$x_1 = 2$, x_2$ = 4$$, $$x_3 = 8$, what is the geometric mean G$?

$$G = (2 \times 4 \times 8$$)^{1/3}$$

Q: A medical researcher collects the following data on the number of times a gene appears in samples: $$x_i = 1, 2, 3$$ with frequencies $$f_1 = 5$$, $$f_2 = 3$$, $$f_3 = 2$. What is the arithmetic mean?

$$\frac{5 \times 1 + 3 \times 2 + 2 \times 3}{5 + 3 + 2}$$

Q: A chemist measures concentrations: $$x_1 = 1$, x_2$ = 3$$, $$x_3 = 9$. What is the geometric mean?

$$(1 \times 3 \times 9$$)^{1/3}$$

Q: If the frequencies in a distribution are $$f_1 = 2$, f_2$ = 3$$, $$f_3 = 5$ and the values are x_1$ = 10$$, $$x_2 = 20$, x_3$ = 30$$, what is the arithmetic mean?

$$\frac{2 \times 10 + 3 \times 20 + 5 \times 30}{2 + 3 + 5}$$

Question 1

Which formula correctly represents the arithmetic mean of $$n$$ values $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

$$\frac{1}{n} (x_1$ + x_2$ + ... + x_n)$$

Question 2

Given a frequency distribution with values $$x_i$$ and frequencies $$f_i$$, what is the formula for the arithmetic mean?

Accepted Answer

$$\frac{\sum_{i=1}^{n} f_i x_i}{\sum_{i=1}^{n} f_i}$$

Question 3

If the values $$x_1 = 2$, x_2$ = 4$$, $$x_3 = 8$, what is the geometric mean G$?

Accepted Answer

$$G = (2 	imes 4 	imes 8$$)^{1/3}$$

Question 4

Which of the following best describes the harmonic mean of $$n$$ non-zero observations $$x_1$$, $$x_2$$, ..., x_n$$?

Accepted Answer

$$H = \frac{n}{\sum_{i=1}^{n} \frac{1}{x_i}}$$

Question 5

A medical researcher collects the following data on the number of times a gene appears in samples: $$x_i = 1, 2, 3$$ with frequencies $$f_1 = 5$$, $$f_2 = 3$$, $$f_3 = 2$. What is the arithmetic mean?

Accepted Answer

$$\frac{5 	imes 1 + 3 	imes 2 + 2 	imes 3}{5 + 3 + 2}$$

Question 6

Which mean is most appropriate for averaging rates, such as speed or efficiency?

Accepted Answer

Harmonic mean

Question 7

If the arithmetic mean of a set of numbers is 10, what is the sum of 5 numbers in the set?

Accepted Answer

$$10 	imes 5 = 50$$

Question 8

A chemist measures concentrations: $$x_1 = 1$, x_2$ = 3$$, $$x_3 = 9$. What is the geometric mean?

Accepted Answer

$$(1 	imes 3 	imes 9$$)^{1/3}$$

Question 9

Which mean is most affected by extremely large or small values in a data set?

Accepted Answer

Arithmetic mean

Question 10

A student records the following times to complete a task: 2 min, 4 min, 8 min. What is the harmonic mean?

Accepted Answer

$$\frac{3}{\frac{1}{2} + \frac{1}{4} + \frac{1}{8}}$$

Question 11

In a frequency distribution, what does $$N$$ represent?

Accepted Answer

The total number of observations, $$N = f_1$ + f_2$ + ... + f_n$$

Question 12

Which mean is best used for calculating average growth rates over time?

Accepted Answer

Geometric mean

Question 13

If $$x_1 = 5$, x_2$ = 10$$, $$x_3 = 20$, what is the harmonic mean?

Accepted Answer

$$\frac{3}{\frac{1}{5} + \frac{1}{10} + \frac{1}{20}}$$

Question 14

A company wants to average the rates at which its machines produce widgets per hour. Which mean should be used?

Accepted Answer

Harmonic mean

Question 15

If the frequencies in a distribution are $$f_1 = 2$, f_2$ = 3$$, $$f_3 = 5$ and the values are x_1$ = 10$$, $$x_2 = 20$, x_3$ = 30$$, what is the arithmetic mean?

Accepted Answer

$$\frac{2 	imes 10 + 3 	imes 20 + 5 	imes 30}{2 + 3 + 5}$$

Question 16

Which mean is calculated by taking the reciprocal of the arithmetic mean of the reciprocals of the data values?

Accepted Answer

Harmonic mean

Question 17

A biologist measures the growth rates of bacteria: 1.2, 1.5, and 1.8 per hour. What is the geometric mean?

Accepted Answer

$$(1.2 	imes 1.5 	imes 1.8$$)^{1/3}$$

Question 18

Which mean is most appropriate for averaging multiplicative processes, such as investment returns?

Accepted Answer

Geometric mean

Question 19

If a frequency distribution has $$x_1 = 2$, x_2$ = 5$$, $$x_3 = 7$ and f_1$ = 4$$, $$f_2 = 2$, f_3$ = 4$$, what is the arithmetic mean?

Accepted Answer

$$\frac{4 	imes 2 + 2 	imes 5 + 4 	imes 7}{4 + 2 + 4}$$

Measures of Central Tendency: Arithmetic, Geometric, and Harmonic Means

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