Question 1

A researcher collects the following data: 5, 8, 12, 15, 20. What is the range of this dataset?

Accepted Answer

$$20 - 5 = 15$$

Question 2

Given a frequency table with values $$x_i$$ and frequencies $$f_i$$, which formula correctly calculates the mean for the frequency table?

Accepted Answer

$$\frac{\sum (x_i f_i)}{\sum f_i}$$

Question 3

A class interval has a lower limit of 10 and an upper limit of 20. What is the class midpoint?

Accepted Answer

$$\frac{10 + 20}{2} = 15$$

Question 4

Which formula is used to calculate the sample standard deviation?

Accepted Answer

$$s = \sqrt{\frac{\sum (x - \bar{x})^2$}{n - 1}}$$

Question 5

A student scores 85 on a test where the mean is 75 and the standard deviation is 5. What is the student's z-score?

Accepted Answer

$$z = \frac{85 - 75}{5} = 2$$

Question 6

If the highest value in a dataset is 50 and the lowest value is 10, and you want to create 8 classes, what is the class width?

Accepted Answer

$$\frac{50 - 10}{8} = 5 ($$increase to next integer if needed)

Question 7

Which formula correctly calculates the population variance?

Accepted Answer

$$\$$sigma^2$$ = \frac{\sum (x - \mu)^2$}{N}$$

Question 8

A hospital records the weights of 5 patients: 60, 70, 80, 90, and 100 kg. What is the sample mean?

Accepted Answer

$$\bar{x} = \frac{60 + 70 + 80 + 90 + 100}{5} = 80$$

Question 9

Which formula is used to calculate the standard error of the mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$$

Question 10

A teacher wants to find the interquartile range (IQR) for a dataset. Which formula should be used?

Accepted Answer

$$IQR = Q_3$ - Q_1$

Question 11

A weighted mean is needed for a set of exam scores with different weights. Which formula should be used?

Accepted Answer

$$\bar{x} = \frac{\sum w x}{\sum w}$$

Question 12

If a dataset has a highest value of 90 and a lowest value of 30, what is the midrange?

Accepted Answer

$$\frac{90 + 30}{2} = 60$$

Question 13

A population consists of the values 2, 4, 6, 8. What is the population mean?

Accepted Answer

$$\mu = \frac{2 + 4 + 6 + 8}{4} = 5$$

Question 14

A sample has values 3, 7, 7, 10. What is the sample variance?

Accepted Answer

$$s^2$$ = \frac{$$(3-6.75)^2$$ + $$(7-6.75)^2$$ + $$(7-6.75)^2$$ + $$(10-6.75)^2$$}{4-1} = 8.25$$

Question 15

A population has values 5, 10, 15. What is the population standard deviation?

Accepted Answer

$$\sigma = \sqrt{\frac{(5-10)^2$ + (10-10)^2$ + (15-10)^2$}{3}} = \sqrt{\frac{25 + 0 + 25}{3}} = \sqrt{16.67} \approx 4.08$$

Question 16

A sample of 25 students has a standard deviation of 4. What is the standard error of the mean?

Accepted Answer

$$\sigma_{\bar{x}} = \frac{4}{\sqrt{25}} = 0.8$$

Question 17

A dataset has quartiles $$Q_1 = 20$ and Q_3$ = 50. $$What is the interquartile range?

Accepted Answer

$$IQR = 50 - 20 = 30$$

Question 18

A sample has values 2, 4, 6, 8, 10. What is the midrange?

Accepted Answer

$$\frac{2 + 10}{2} = 6$$

Fundamental Statistical Measures and Formulas

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