Business Statistics Quiz: Risk And Volatility Measures
20 questions · exam conditions
0:00
Risk And Volatility MeasuresQuestion 1 of 20

When presenting volatility metrics to an executive team with a limited statistical background, a financial analyst chooses to report standard deviation rather than variance. What is the primary reason for this choice?

The standard deviation is always a smaller, more manageable number than the variance.
The standard deviation is expressed in the same units as the original data, making it more intuitive to interpret.
The standard deviation is a more accurate measure of risk because it cannot be negative.
The calculation for standard deviation is simpler and less prone to computational errors than that for variance.
← Back to quizzes

Business Statistics Quiz

Business Statistics Quiz: Risk And Volatility Measures

Practice Risk And Volatility Measures in Business Statistics with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Risk And Volatility Measures, giving you a quick way to practice the rules, question types, and explanations that matter most for Business Statistics.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

When presenting volatility metrics to an executive team with a limited statistical background, a financial analyst chooses to report standard deviation rather than variance. What is the primary reason for this choice?

  1. The standard deviation is always a smaller, more manageable number than the variance.
  2. The standard deviation is expressed in the same units as the original data, making it more intuitive to interpret. (correct answer)
  3. The standard deviation is a more accurate measure of risk because it cannot be negative.
  4. The calculation for standard deviation is simpler and less prone to computational errors than that for variance.
Explanation: The main advantage of the standard deviation over the variance is its interpretability. Standard deviation is expressed in the same units as the original data (e.g., dollars, units sold, etc.), whereas variance is in squared units (e.g., dollars squared). This makes standard deviation directly comparable to the mean and easier to understand in a practical business context.

Question 2

A market research firm conducts a survey with a sample of n=10 respondents and calculates a sample variance of 18. They later decide to merge this data with a second independent sample of n=10 from the same population, which, by chance, has the exact same data values as the first. What is the sample variance of the combined sample of n=20?

  1. 9.00
  2. 18.00
  3. 18.95 (correct answer)
  4. 36.00
Explanation: This requires careful application of the sample variance formula. \n1. For the first sample, the sum of squared deviations (SSD) is SSD1=s12×(n1−1)=18×(10−1)=18×9=162SSD_1 = s_1^2 \times (n_1-1) = 18 \times (10-1) = 18 \times 9 = 162. \n2. Since the second sample is identical, its mean and SSD are also the same: SSD2=162SSD_2 = 162. \n3. When combining the samples, the new mean is the same as the old mean. The total SSD is SSDtotal=SSD1+SSD2=162+162=324SSD_{total} = SSD_1 + SSD_2 = 162 + 162 = 324. \n4. The new sample size is nnew=10+10=20n_{new} = 10 + 10 = 20. \n5. The new sample variance is snew2=SSDtotal/(nnew−1)=324/(20−1)=324/19≈18.95s_{new}^2 = SSD_{total} / (n_{new}-1) = 324 / (20-1) = 324 / 19 \approx 18.95.

Question 3

A sample of 20 shipping orders has a sample variance of 38 (days²). A 21st order is added to the sample, and its shipping time is exactly equal to the original sample mean. What is the variance of the new sample of 21 orders?

  1. 36.10 (correct answer)
  2. 38.00
  3. 39.90
  4. 40.00
Explanation: This problem requires understanding the formula for sample variance. \n1. The original sum of squared deviations (SSD) can be found from the sample variance formula: s2=SSD/(n−1)s^2 = SSD / (n-1). So, SSD=s2×(n−1)=38×(20−1)=38×19=722SSD = s^2 \times (n-1) = 38 \times (20-1) = 38 \times 19 = 722.
  1. When a new data point equal to the mean is added, the mean of the new, larger sample remains unchanged. The deviation for this new point is (mean - mean) = 0. Therefore, the SSD does not change.
  2. The new sample variance is calculated with the same SSD but with the new sample size, n_new = 21. The denominator becomes (n_new - 1) = 20.
  3. New Variance = SSD / (n_new - 1) = 722 / 20 = 36.1.

Question 4

An operations manager is analyzing two suppliers based on delivery times. Supplier A has a mean delivery time of 5 days with a standard deviation of 1 day. Supplier B has a mean delivery time of 5 days with a standard deviation of 2 days. Assuming the manager is risk-averse regarding delivery times, which statement is most accurate?

  1. Supplier B is preferred because higher volatility indicates a chance for much faster delivery.
  2. Supplier A is preferred because, for the same expected delivery time, it offers less uncertainty. (correct answer)
  3. Both suppliers are equally acceptable because their mean delivery times are identical.
  4. It is impossible to decide without knowing the shape of the delivery time distributions.
Explanation: For a risk-averse decision-maker, lower volatility (risk) is preferred when expected outcomes are equal. Both suppliers have the same mean delivery time (5 days), but Supplier B has a higher standard deviation (2 days vs. 1 day). This means Supplier B's delivery times are more spread out and less predictable. A risk-averse manager would prefer the lower uncertainty offered by Supplier A.

Question 5

A dataset of 100 customer satisfaction scores has a standard deviation of 5 points. A data entry error is discovered: one score was entered as 100, but it should have been 50. The mean score for the original dataset was 60. How will correcting this single outlier affect the standard deviation?

  1. It will increase because the range of the data has decreased.
  2. It will remain unchanged because only one data point out of 100 was adjusted.
  3. It will decrease because a value far from the mean has been moved closer to the mean. (correct answer)
  4. It is impossible to determine the effect without the complete dataset.
Explanation: Standard deviation measures the average distance of data points from the mean. The erroneous score of 100 was 40 points away from the original mean of 60. The corrected score of 50 is only 10 points away from the original mean. By moving an extreme value significantly closer to the center of the distribution, the overall spread of the data is reduced. Since variance and standard deviation are sensitive to outliers, this correction will cause the standard deviation to decrease.

Question 6

A financial analyst is comparing the risk profiles of two mutual funds using their monthly return data. Fund Alpha has monthly returns with a standard deviation of 2.1% and a mean return of 1.4%. Fund Beta has monthly returns with a variance of 0.0009 and a mean return of 0.9%. If an investor prioritizes lower relative risk, which fund should they choose and why?

  1. Fund Beta, because its coefficient of variation of 3.33 indicates lower relative risk than Fund Alpha's coefficient of variation of 1.50
  2. Fund Beta, because its coefficient of variation of 3.33 is lower than Fund Alpha's coefficient of variation of 1.50
  3. Fund Alpha, because its absolute standard deviation of 2.1% is lower than Fund Beta's absolute standard deviation of 3.0%
  4. Fund Alpha, because its coefficient of variation of 1.50 is lower than Fund Beta's coefficient of variation of 3.33 (correct answer)
Explanation: When comparing investments with different return levels, you need to measure relative risk rather than absolute risk. The coefficient of variation (CV) standardizes risk by dividing standard deviation by the mean return, allowing fair comparison between investments with different expected returns. First, let's calculate each fund's coefficient of variation. Fund Alpha has a standard deviation of 2.1% and mean return of 1.4%, so its CV = 2.1% ÷ 1.4% = 1.50. Fund Beta has a variance of 0.0009, which means its standard deviation is 0.0009=0.03=3.0%\sqrt{0.0009} = 0.03 = 3.0\%. With a mean return of 0.9%, Fund Beta's CV = 3.0% ÷ 0.9% = 3.33. Fund Alpha's lower coefficient of variation (1.50 vs. 3.33) indicates lower relative risk, making it the better choice for risk-averse investors. Answer D correctly identifies this relationship. Answer A incorrectly states that Fund Beta has lower relative risk when it actually has higher relative risk. Answer B makes the same error, claiming 3.33 is lower than 1.50. Answer C focuses on absolute standard deviation rather than relative risk—while Fund Beta does have higher absolute risk (3.0% vs. 2.1%), this misses the point about relative risk comparison. Remember: when comparing investments with different return levels, always use the coefficient of variation rather than standard deviation alone. The CV tells you how much risk you're taking per unit of expected return, which is the fair way to compare different investment opportunities.

Question 7

A quality control engineer measures the diameter of manufactured bolts and finds that the process has a variance of 0.25 mm². After implementing a new manufacturing technique, the variance decreases to 0.16 mm². What percentage reduction in risk, as measured by standard deviation, has been achieved?

  1. 44% reduction because the standard deviation decreased from 0.5 mm to 0.28 mm
  2. 36% reduction because the variance decreased from 0.25 to 0.16, a reduction of 0.09
  3. 20% reduction because the standard deviation decreased from 0.5 mm to 0.4 mm (correct answer)
  4. 64% reduction because this matches the proportional change in variance from 0.25 to 0.16
Explanation: When you encounter quality control problems involving variance and standard deviation, remember that risk is typically measured by standard deviation (not variance), since it's in the same units as the original measurements. To find the percentage reduction in risk, you need to work with standard deviations. First, convert the variances to standard deviations by taking the square root: 0.25=0.5\sqrt{0.25} = 0.5 mm and 0.16=0.4\sqrt{0.16} = 0.4 mm. The reduction is 0.5−0.4=0.10.5 - 0.4 = 0.1 mm, which represents a 0.10.5=0.20=20%\frac{0.1}{0.5} = 0.20 = 20\% reduction. Choice C correctly identifies this 20% reduction in standard deviation from 0.5 mm to 0.4 mm. Choice A makes a calculation error when computing the new standard deviation, stating it's 0.28 mm instead of 0.4 mm. This leads to an incorrect 44% reduction figure. Choice B confuses the concept by calculating the percentage based on variance reduction rather than standard deviation reduction. While the variance did decrease by 0.09 (from 0.25 to 0.16), risk reduction should be measured using standard deviation since it represents the actual spread in the original units. Choice D incorrectly applies the variance reduction percentage (64%) to standard deviation. Since standard deviation is the square root of variance, the percentage changes are different for these two measures. Remember: when measuring risk reduction in quality control, always work with standard deviation rather than variance, since standard deviation is expressed in the same units as your measurements and provides a more intuitive sense of variability.

Question 8

A manufacturing company tracks daily production output. Over the past 50 days, the variance in daily output was 144 units². If the company implements a new quality control system that reduces variability by 25%, what will be the new standard deviation of daily output?

  1. 9 units because reducing variability by 25% means the standard deviation becomes 75% of the original value
  2. 10.39 units because reducing variance by 25% results in a new variance of 108 units², and standard deviation equals the square root of variance (correct answer)
  3. 8.66 units because the standard deviation reduction should be calculated as 12 × √0.75 to account for the square root relationship
  4. 108 units because this represents the new variance value after the 25% reduction in variability
Explanation: Original variance = 144 units², so original standard deviation = √144 = 12 units. A 25% reduction in variance means new variance = 144 × (1 - 0.25) = 144 × 0.75 = 108 units². The new standard deviation = √108 ≈ 10.39 units. Choice A incorrectly assumes the standard deviation reduces by 25% rather than the variance. Choice C applies an incorrect square root adjustment. Choice D confuses variance with standard deviation.

Question 9

The variance of monthly sales revenue for a retail store is $16,000,000. To boost sales, the company introduces a new policy that adds a fixed $2,000 bonus to every salesperson's monthly commission, which is a component of sales revenue. Assuming this is the only change, what will be the new variance of the monthly sales revenue?

  1. $12,000,000
  2. $16,000,000 (correct answer)
  3. $16,002,000
  4. $20,000,000
Explanation: Variance and standard deviation are measures of dispersion or spread. Adding a constant value to every data point in a set shifts the entire set but does not change its spread. The differences between each data point and the mean remain the same. Therefore, the variance is unaffected. The new variance will remain $16,000,000.

Question 10

A quality control process has a variance of 0.04 cm² in the diameter of ball bearings. If a new polishing step is implemented that uniformly reduces the diameter of every ball bearing by 0.01 cm, what will be the standard deviation of the diameters after the new step is applied?

  1. 0.19 cm
  2. 0.20 cm (correct answer)
  3. 0.03 cm²
  4. 0.04 cm²
Explanation: This question involves two concepts. First, subtracting a constant from every value in a dataset shifts the mean but does not change the spread. Therefore, the variance and standard deviation remain unchanged. Second, the question provides the variance and asks for the standard deviation. \nOriginal Variance = 0.04 cm². \nThe new polishing step does not change the variance. \nNew Variance = 0.04 cm². \nStandard Deviation = Variance=0.04=0.20\sqrt{\text{Variance}} = \sqrt{0.04} = 0.20 cm.

Question 11

A fund manager evaluates two stocks. Stock X has a mean monthly return of 2.0% and a variance of 0.0016. Stock Y has a mean monthly return of 1.0% and a variance of 0.0009. Based on the coefficient of variation, which statement is most accurate?

  1. Stock X is twice as risky as Stock Y.
  2. Both stocks have the same level of relative risk.
  3. Stock X is riskier because its variance is higher.
  4. Stock Y is 50% riskier than Stock X. (correct answer)
Explanation: When comparing investment risks, you need the coefficient of variation (CV), which measures relative risk by standardizing volatility against returns. The CV equals the standard deviation divided by the mean, allowing you to compare assets with different return levels. For Stock X: Standard deviation = 0.0016=0.04\sqrt{0.0016} = 0.04 or 4%. CV = 0.040.02=2.0\frac{0.04}{0.02} = 2.0 For Stock Y: Standard deviation = 0.0009=0.03\sqrt{0.0009} = 0.03 or 3%. CV = 0.030.01=3.0\frac{0.03}{0.01} = 3.0 Stock Y's CV is 50% higher than Stock X's (3.0 vs 2.0), meaning Stock Y is 50% riskier on a relative basis. This makes answer D correct. Answer A incorrectly suggests Stock X is riskier, when it's actually Stock Y. The calculation would need Stock X's CV to be twice Stock Y's CV, but it's the reverse relationship. Answer B is wrong because the coefficients of variation are clearly different (2.0 vs 3.0), so relative risk levels aren't equal. Answer C falls into the trap of using absolute variance rather than relative risk. While Stock X does have higher variance (0.0016 vs 0.0009), this ignores that Stock X also has twice the expected return. Raw variance doesn't account for the return differential. Remember: When comparing investments with different return levels, always use the coefficient of variation rather than standard deviation or variance alone. CV reveals which investment gives you more "bang for your risk buck."

Question 12

The number of daily customers at a store is approximately normally distributed with a mean of 450 and a standard deviation of 30. According to the Empirical Rule, approximately 68% of days will have a customer count between which two values?

  1. 420 and 480 (correct answer)
  2. 390 and 510
  3. 440 and 460
  4. 360 and 540
Explanation: The Empirical Rule states that for a normal distribution, approximately 68% of the data falls within one standard deviation of the mean (μ±1σ\mu \pm 1\sigma). \nGiven μ=450\mu = 450 and σ=30\sigma = 30. \nThe lower bound is 450−1×30=420450 - 1 \times 30 = 420. \nThe upper bound is 450+1×30=480450 + 1 \times 30 = 480. \nTherefore, approximately 68% of days will see between 420 and 480 customers.

Question 13

The total variance of a firm's profits is composed of systematic variance (related to the overall economy) and unsystematic variance (specific to the firm). The firm undertakes a major strategic initiative to diversify its operations by acquiring a company in a completely unrelated industry. What is the most likely effect on the components of its profit variance?

  1. Both systematic and unsystematic variance will decrease significantly.
  2. Systematic variance will decrease, while unsystematic variance will remain unchanged.
  3. Unsystematic variance will decrease, while systematic variance will remain largely unchanged. (correct answer)
  4. Both systematic and unsystematic variance will remain unchanged as diversification does not affect volatility.
Explanation: This question applies a core principle of diversification from finance. Systematic risk is market-wide and cannot be eliminated through diversification. It affects all industries. Unsystematic risk is firm-specific or industry-specific. By diversifying into an unrelated industry, the firm reduces its exposure to shocks that affect only its original industry. This action directly targets and reduces the unsystematic component of variance. The systematic component, tied to the broader economy, is not reduced by this type of diversification.

Question 14

A quality control engineer needs to compare the consistency of two manufacturing machines. Machine 1 produces bolts with a mean length of 50 mm and a standard deviation of 0.1 mm. Machine 2 produces pistons with a mean weight of 800 g and a standard deviation of 2 g. Which machine is relatively more consistent in its output?

  1. Machine 1 is more consistent because its standard deviation of 0.1 is smaller than Machine 2's of 2.
  2. Machine 2 is more consistent because its coefficient of variation is 0.0025, which is smaller than Machine 1's of 0.0020.
  3. Machine 1 is more consistent because its coefficient of variation is 0.0020, which is smaller than Machine 2's of 0.0025. (correct answer)
  4. Their consistencies cannot be compared because the products and units of measurement are different.
Explanation: To compare relative consistency or volatility when units and means are different, the coefficient of variation (CV = Standard Deviation / Mean) must be used. A lower CV indicates higher relative consistency. \nFor Machine 1: CV = 0.1 mm / 50 mm = 0.0020. \nFor Machine 2: CV = 2 g / 800 g = 0.0025. \nSince Machine 1 has a smaller coefficient of variation (0.0020 < 0.0025), it is relatively more consistent.

Question 15

A U.S.-based multinational corporation finds that the standard deviation of its European division's annual profit is €3 million. If the average exchange rate during the period was $1.10 per euro, what is the variance of the annual profit expressed in U.S. dollars?

  1. $3.30 million
  2. $3.63 million
  3. $9.90 million
  4. $10.89 million (correct answer)
Explanation: This is a multi-step problem involving unit conversion and the properties of variance. First, convert the standard deviation from euros to dollars. If you multiply a dataset by a constant 'k', the new standard deviation is 'k' times the old standard deviation. Here, k = 1.10. \nNew Standard Deviation (in dollars) = €3 million * 1.10 $/€ = 3.3million.\nSecond,thequestionasksforthevariance,whichisthesquareofthestandarddeviation.\nNewVariance(indollarssquared)=(3.3 million. \nSecond, the question asks for the variance, which is the square of the standard deviation. \nNew Variance (in dollars squared) = (3.3 million)² = $10.89 million.

Question 16

An investor creates a portfolio with two assets, A and B, which have a perfect negative correlation (ρ=−1\rho = -1). Both assets have a standard deviation of 12%. The investor allocates 50% of the funds to Asset A and 50% to Asset B. What is the standard deviation of this portfolio?

  1. 0% (correct answer)
  2. 6%
  3. 12%
  4. 24%
Explanation: The formula for the variance of a two-asset portfolio is σp2=wA2σA2+wB2σB2+2wAwBρABσAσB\sigma_p^2 = w_A^2\sigma_A^2 + w_B^2\sigma_B^2 + 2w_Aw_B\rho_{AB}\sigma_A\sigma_B. \nGiven: wA=0.5w_A = 0.5, wB=0.5w_B = 0.5, σA=0.12\sigma_A = 0.12, σB=0.12\sigma_B = 0.12, and ρAB=−1\rho_{AB} = -1. \nσp2=(0.5)2(0.12)2+(0.5)2(0.12)2+2(0.5)(0.5)(−1)(0.12)(0.12)\sigma_p^2 = (0.5)^2(0.12)^2 + (0.5)^2(0.12)^2 + 2(0.5)(0.5)(-1)(0.12)(0.12) \nσp2=(0.25)(0.0144)+(0.25)(0.0144)−(0.5)(0.0144)\sigma_p^2 = (0.25)(0.0144) + (0.25)(0.0144) - (0.5)(0.0144) \nσp2=0.0036+0.0036−0.0072=0\sigma_p^2 = 0.0036 + 0.0036 - 0.0072 = 0. \nThe standard deviation is the square root of the variance, so σp=0=0\sigma_p = \sqrt{0} = 0. A perfectly negatively correlated portfolio can, with the right weights, eliminate all risk.

Question 17

An analyst examines the profits of a small business for the last four quarters: $30k, $50k, $20k, and $40k. The analyst's goal is to measure the volatility of only these specific four quarters to report to the owner. Which of the following is the correct volatility measure and value?

  1. Sample standard deviation of $12.91k
  2. Population standard deviation of $11.18k (correct answer)
  3. Sample variance of $166.67k²
  4. Population variance of $125.00k²
Explanation: The key phrase is 'only these specific four quarters,' which implies the data set is a population, not a sample of a larger set of possibilities. Therefore, population variance/standard deviation formulas (dividing by N) should be used. 1. Calculate the mean: (30+30+50+20+20+40)/4 = $35k. 2. Calculate sum of squared deviations: (30-35)² + (50-35)² + (20-35)² + (40-35)² = 25 + 225 + 225 + 25 = 500. 3. Population variance: 500/4 = 125k². 4. Population standard deviation: √125 ≈ $11.18k. The sample formulas (dividing by n-1=3) would give variance of 500/3 = 166.67k² and standard deviation of √166.67 = 12.91k, but these are incorrect since we're treating this as the complete population.

Question 18

The variance of a set of project completion times is 36 days². The project manager needs to report this volatility metric on an international dashboard where the standard unit of time is weeks. What is the standard deviation of the completion times in weeks?

  1. 0.86 weeks (correct answer)
  2. 1.17 weeks
  3. 5.14 weeks
  4. 0.73 weeks²
Explanation: This is a two-step transformation problem. \n1. First, find the standard deviation in the original units (days). The standard deviation is the square root of the variance: σdays=36 days2=6 days\sigma_{days} = \sqrt{36 \text{ days}^2} = 6 \text{ days}. \n2. Second, convert the standard deviation from days to weeks. Since there are 7 days in a week, we divide by 7. This is equivalent to multiplying by a constant k = 1/7. The standard deviation scales directly with this constant. \nσweeks=σdays×(1/7)=6/7≈0.857\sigma_{weeks} = \sigma_{days} \times (1/7) = 6 / 7 \approx 0.857 weeks.

Question 19

An investment fund manager is comparing two portfolios. Portfolio A has an expected return of 12% with a standard deviation of 8%, while Portfolio B has an expected return of 15% with a standard deviation of 18%. If an investor's primary concern is risk-adjusted return measured by the coefficient of variation, which statement best describes the comparison?

  1. Portfolio A is less risky per unit of return because its coefficient of variation is approximately 0.67 compared to Portfolio B's 1.20 (correct answer)
  2. Portfolio B is less risky per unit of return because its coefficient of variation is approximately 0.83 compared to Portfolio A's 1.50
  3. Portfolio A is less risky per unit of return because its coefficient of variation is approximately 1.50 compared to Portfolio B's 0.83
  4. Portfolio B is less risky per unit of return because its coefficient of variation is approximately 1.20 compared to Portfolio A's 0.67
Explanation: The coefficient of variation (CV) = standard deviation ÷ expected return. For Portfolio A: CV = 8% ÷ 12% = 0.67. For Portfolio B: CV = 18% ÷ 15% = 1.20. A lower CV indicates less risk per unit of return, so Portfolio A is preferable. Choice B incorrectly calculates CVs as 0.83 and 1.50. Choice C has the correct CVs but assigns them to wrong portfolios. Choice D reverses the interpretation of which portfolio is better.

Question 20

A retail chain analyzes monthly sales data for two product categories. Category X has a mean monthly sales of $50,000 with a variance of $900,000,000. Category Y has a mean monthly sales of $75,000 with a standard deviation of $25,000. Which category exhibits greater relative volatility, and what is the difference in their coefficients of variation?

  1. Category X has greater relative volatility, with coefficients of variation differing by approximately 0.27 (correct answer)
  2. Category Y has greater relative volatility, with coefficients of variation differing by approximately 0.33
  3. Category X has greater relative volatility, with coefficients of variation differing by approximately 0.60
  4. Both categories have identical relative volatility, with coefficients of variation differing by approximately 0.00
Explanation: For Category X: Standard deviation = √900,000,000 = $30,000. Coefficient of variation = 30,000/50,000 = 0.60. For Category Y: Coefficient of variation = 25,000/75,000 = 0.333. Category X has greater relative volatility (0.60 > 0.333). The difference in coefficients of variation is 0.60 - 0.333 = 0.267 ≈ 0.27. Choice B incorrectly identifies Category Y as having greater volatility. Choice C gives the CV for Category X rather than the difference. Choice D incorrectly suggests equal volatility.