Question 1

Which of the following best describes the sample mean $$\ x\bar{} \  in $$statistics?

Accepted Answer

It is the balancing point or center of mass of the sample values.

Question 2

Given the sample values: 4, 4, 5, 6, 8, 14, 22, what is the sample mean $$\ x\bar{} \ $$?

Accepted Answer

$$\ x\bar{} = \frac{4+4+5+6+8+14+22}{7} = 9 \ $$

Question 3

Which formula correctly calculates the sample standard deviation $$\ s \ $$ for a sample of $$\ n \ $$ values $$\ x_1$, x_2$, ..., x_n \ $$?

Accepted Answer

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

Question 4

Why do we divide by $$\ n-1 \ $$ instead of $$\ n \ $$ when calculating the sample variance?

Accepted Answer

To correct for the bias in estimating the population variance from a sample.

Question 5

A sample has values: 7, 9, 9, 16, 19. What is the sample mean $$\ x\bar{} \ $$?

Accepted Answer

$$\ x\bar{} = \frac{7+9+9+16+19}{5} = 12 \ $$

Question 6

Which of the following statements about the sample standard deviation $$\ s \  is $$TRUE?

Accepted Answer

It measures how much sample values differ from the sample mean.

Question 7

If a sample contains extreme values (outliers), how does this affect the sample standard deviation $$\ s \ $$?

Accepted Answer

The sample standard deviation increases, reflecting greater variability.

Question 8

Given the sample values: 15, 1, 357, 1225, what is the sum of squared deviations if the mean is $$\ x\bar{} = 299.4 \ $$ and the deviations are 4479, 1, 49, 1, 1?

Accepted Answer

$$\ 4479 + 1 + 49 + 1 + 1 = 4531 \ $$

Question 9

A sample has $$\ n = 7 \ $$ values. After calculating the sum of squared deviations as $$\ 278 \ $$, what is the sample variance $$\ s^2$ \ $$?

Accepted Answer

$$\ s^2$ = \frac{278}{6} = 46.33 \ $$

Question 10

Which step comes immediately before taking the square root when calculating the sample standard deviation?

Accepted Answer

Divide the sum of squared deviations by $$\ n-1 \  to $$get the sample variance.

Question 11

In a sample of $$\ n = 11 \ $$ values, the sum of squared deviations is $$\ 2994 \ . $$What is the sample variance $$\ s^2$ \ $$?

Accepted Answer

$$\ s^2$ = \frac{2994}{10} = 299.4 \ $$

Question 12

If the sample variance $$\ s^2$ \  is$$ $$\ 299.4 \ $$, what is the sample standard deviation $$\ s \ $$?

Accepted Answer

$$\ s = \sqrt{299.4} = 17.3031 \ $$

Question 13

Which of the following best interprets the sample standard deviation $$\ s \ $$?

Accepted Answer

It is roughly the average distance of sample points to the sample mean.

Question 14

A sample has values: 4, 4, 5, 6, 8, 14, 22. After calculating the sample mean $$\ x\bar{} = 9 \ $$, what is the deviation for the value 14?

Accepted Answer

$$\ 14 - 9 = 5 \ $$

Question 15

Which of the following is NOT a property of the sample mean $$\ x\bar{} \ $$?

Accepted Answer

It is always equal to the sample median.

Question 16

If you want to systematically calculate the sample standard deviation, which sequence of steps should you follow?

Accepted Answer

Find the mean, calculate deviations, square deviations, sum squared deviations, divide by $$\ n-1 \ $$, take the square root.

Question 17

Which of the following real-world scenarios best illustrates the use of sample standard deviation?

Accepted Answer

A medical researcher measures blood pressure in a group of patients to assess variability.

Question 18

If the sample mean $$\ x\bar{} \  is $$the balancing point of the sample values, what happens if you replace one value with a much larger number?

Accepted Answer

The sample mean increases, shifting the center toward the larger value.

Question 19

Which formula would you use to calculate the sample mean for $$\ n \ $$ values $$\ x_1$, x_2$, ..., x_n \ $$?

Accepted Answer

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

Sample Mean and Sample Standard Deviation: Concepts and Calculation

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