4.2 Proceso de desarrollo
4.2.2 Implementación
1. The branch of statistics that summarizes the important characteristics of large data sets is known as:
A. Inferential statistics.
B. Descriptive statistics.
C. Quantitative statistics.
2. A parameter is defined as:
A. All members of a special group.
B. A descriptive measure of a population characteristic.
C. A subset of a population.
3. Which of the following measurement scales should most likely be used to measure cash dividends per share?
A. Nominal scale.
B. Ordinal scale.
C. Ratio scale.
4. The actual number of observations in a given interval is known as:
A. Absolute frequency.
B. Relative frequency.
C. Cumulative frequency.
5. An analyst gathered the following information about the required rate of return of different stocks held in a portfolio:
Interval Req. ROR Absolute Frequency
1 5%–6% 1
2 6%–7% 3
3 7%–8% 4
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4 8%–9% 2
5 9%–10% 3
The relative frequency & cumulative relative frequency of Interval 4 are closest to:
Relative Frequency Cumulative Relative Frequency
A. 15% 23%
B. 20% 77%
C. 15% 77%
6. If a sample consists of the historical monthly stock returns of Trukmaker Inc. for the past 5 years, then the mean computed from these observations will be a:
A. Time-series mean B. Cross-sectional mean C. Weighted mean
7. A portfolio consists of 40% common stock, 20% preferred stock, 15%
bonds, and 25% real estate. The returns on these asset classes are 12%
on common stock, 10% on preferred stock, and 8% on both real estate and bonds. The mean return on the portfolio is closest to:
A. 8%
B. 10%
C. 12%
8. An individual invests $1,000 in a particular security on two different dates ($2,000 total). The price of the security is $10 on the first day and $12 on the second. The investor’s average purchase price is closest to:
A. 11 B. 10.84 C. 10.91
9. The 4th quintile for the following distribution of returns is closest to:
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8%, 6%, 12%, 18%, 25%, 8%, 9%, 17%, 14%, 10%
A. 17%
B. 17.8%
C. 18.5%
10. Which of the following portfolios has the smaller range and which one most likely has greater risk?
Portfolio A Portfolio B
16.2% 9.2%
20.5% 8.4%
−11.1% −1.6%
7.2% 13.2%
Smaller Range Higher Risk
A. A B
B. B A
C. B B
11. What is the mean absolute deviation and variance of the sample investment returns given below:
Year Return 2004 8%
2005 9%
2006 −7%
2007 5%
MAD Variance A. 6.375% 40.69 B. 5.375% 54.25 C. 6.375% 54.25
12. What is the minimum percentage of a distribution that will lie within 3
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standard deviations of the mean?
A. 75%
B. 82%
C. 89%
13. An analyst gathered the following information about a common stock portfolio:
Arithmetic mean 12%
Geometric mean 12%
Harmonic mean 11.5%
Variance of returns 270 Portfolio Beta 1.25 Risk-free ROR 5%
Compute the coefficient of variation and Sharpe ratio.
Coefficient of Variation Sharpe Ratio
A. 1.26 0.43
B. 1.26 2.05
C. 1.37 0.43
14. Compared to the standard deviation, the mean absolute deviation is always:
A. Less than or equal to the standard deviation B. Greater than or equal to the standard deviation C. Less than the standard deviation
15. A distribution that has a kurtosis of +3 is known as:
A. Kurtosis B. Leptokurtic C. Mesokurtic
16. Excess kurtosis of normal distribution equals:
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A. 3 B. 2 C. 0
Use the following frequency distribution for Questions 17 to 19:
Annual Return (R) Frequency
−30% ≤ R < −20% 1
−20% ≤ R < −10% 4
−10% ≤ R < 0% 15 0% ≤ R < 10% 10 10% ≤ R < 20% 14 20% ≤ R < 30% 2 17. The modal interval is:
A. −10% ≤ R < 0%
B. −5
C. −30% ≤ R < −20%
18. The sample size is:
A. 7 B. 46 C. 45
19. The cumulative relative frequency of the fifth interval is:
A. 44 B. 0.956 C. 0.304
Use the following information for Questions 20 to 23:
The annual returns of the various stocks for 2008 are given in the table below:
Stock ABC DEF GHI JKL MNO PQR
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Return 21% −10% 5% 13% −19% 5%
20. The arithmetic mean of the returns on these stocks for the year 2008 is:
A. 12.17%
B. 2.5%
C. 5%
21. The median return of these stocks for the year 2008 is:
A. 9%
B. −19%
C. 5%
22. The range and standard deviation of the stock returns, assuming that they represent a population, are:
Range Std Dev
A. 40 32.9
B. 40 13.43 C. 28.16 40
23. The geometric and harmonic means for the returns data are:
G-Mean H-Mean A. 10.4% 2.5%
B. 1.6% 16.1%
C. 1.02% 8.86%
24. Which of the following statements is most accurate?
A. Descriptive statistics refer to methods used to make forecasts, estimates, or draw conclusions about a larger data set based on a smaller representative set.
B. Inferential statistics refer to how large volumes of data are converted into useful and readily understood information by summarizing their important characteristics.
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C. The arithmetic mean of a particular data set is an example of a sample statistic.
25. Which of the following types of measurement scales sorts data into categories that are ranked according to a certain characteristic, but does not tell us anything about the differences between categories?
A. Nominal scales B. Ordinal scales C. Interval scales
26. Which of the following statements regarding frequency distributions is least accurate?
A. Any particular observation could fall into one or more intervals.
B. The set of intervals should include the entire range of values in the data set.
C. The interval with the highest frequency is called the modal class.
27. Which of the following statements is most likely?
A. The cumulative frequency for an interval is the proportion of total observations that lies in that particular interval.
B. The cumulative absolute frequency for a particular interval is the number of observations that are less than the upper bound for the interval.
C. The cumulative relative frequency for a particular interval is the number of observations that are less than the midpoint of the interval.
Use the following information to answer Questions 28 to 35:
Cool Pop Soda Corp’s annual profit margins since its incorporation are presented below:
Year 1 11%
Year 2 9%
Year 3 4%
Year 4 −2%
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Year 5 1%
Year 6 4%
28. The arithmetic mean for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 4.5%
B. 6%
C. 11%
29. The median for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 11%
B. 4.5%
C. 4%
30. The mode for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 4%
B. 4.5%
C. 11%
31. The mean absolute deviation for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 3 B. 3.67 C. 19.56
32. The variance for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 3.67 B. 4.42 C. 19.58
33. The standard deviation for Cool Pop Soda Corp’s profit margins over
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the years is closest to:
A. 1.91%
B. 4.43%
C. 19.58%
34. Given that the data set is only a representative sample of profit
margins over the company’s entire existence, the variance for Cool Pop Soda Corp’s profit margins over the 6 years is closest to:
A. 23.5 B. 19.56 C. 13.47
35. Given that the data set is only a representative sample of profit margins over the company’s entire existence, the standard deviation for Cool Pop Soda Corp’s profit margins over the 6 years is closest to:
A. 3.67%
B. 4.42%
C. 4.85%
36. Junglelala’s returns over its first 4 years of operations are −1.1%, 1.1%, 1.9%, and 2.8% respectively. The geometric mean of its returns is closest to:
A. 1.165%
B. 1.175%
C. 1.185%
37. The maximum percentage of observations in a data set that lies further than 1.85 standard deviations away from the mean is closest to:
A. 29.2%
B. 45.9%
C. 70.8%
38. A distribution of returns that has a median that is greater than its mean is most likely:
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A. Positively skewed B. Negatively skewed
C. A symmetric distribution
39. Stock A has an average return of 11% and a standard deviation of 4%.
Stock B has an average return of 33% and a standard deviation of 14%.
Stock C has an average return of 7% and a standard deviation of 0.33%. Given a risk-free rate of 5%, which investment is most attractive based on the coefficient of variation?
A. A B. C C. B
40. Portfolio A has a mean return of 11% and a standard deviation of 3%.
Portfolio B has a mean return of 17% and a standard deviation of 6%.
The risk-free rate of interest is 3%. The respective Sharpe ratios of the two portfolios are closest to:
Portfolio A Portfolio B
A. 2.33 2.67
B. 2.67 2.33
C. 2.33 2.83
41. A leptokurtic distribution most likely:
A. Has a negative excess kurtosis and is more peaked and has thinner tails than a normal distribution
B. Has a positive excess kurtosis and is more peaked and has fatter tails than a normal distribution
C. Has a zero excess kurtosis and is less peaked and has fatter tails than a normal distribution
42. The Sharpe ratio most likely measures:
A. Total risk per unit of return B. Downside risk
C. Excess return per unit of risk
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43. Which of the following computations uses the sum of cubed deviations from the mean?
A. Sample standard deviation B. Sample skewness
C. Sample kurtosis
44. The geometric mean return approximately equals:
A. The arithmetic mean return minus 2 times the standard deviation of returns.
B. The arithmetic mean return minus half the variance of returns.
C. The arithmetic mean return plus half the variance of returns.
45. Which of the following data sets suffers from the lowest level of skewness?
A. One with a sample skewness of −1.5 B. One with a sample skewness of zero C. One with a sample skewness of +1
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Reading 7: Statistical Concepts and Market Returns
1. The branch of statistics that summarizes the important characteristics of large data sets is known as:
A. Inferential statistics.
B. Descriptive statistics.
C. Quantitative statistics.
Answer: B
Descriptive statistics is the study of how data can be summarized effectively to describe the important aspects of large data sets.
2. A parameter is defined as:
A. All members of a special group.
B. A descriptive measure of a population characteristic.
C. A subset of a population.
Answer: B
A parameter is a descriptive measure of a population characteristic.
3. Which of the following measurement scales should most likely be used to measure cash dividends per share?
A. Nominal scale.
B. Ordinal scale.
C. Ratio scale.
Answer: C
Cash dividends per share can be best measured on a ratio scale. Ratio scales have a true zero point as the origin. For the variable in question, zero represents the absence of dividends.
4. The actual number of observations in a given interval is known as:
A. Absolute frequency.
B. Relative frequency.
C. Cumulative frequency.
Answer: A
The actual number of observations in a given interval is called frequency or absolute frequency.
5. An analyst gathered the following information about the required rate of return of different stocks held in a portfolio:
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Interval Req. ROR Absolute Frequency
1 5%–6% 1
2 6%–7% 3
3 7%–8% 4
4 8%–9% 2
5 9%–10% 3
The relative frequency & cumulative relative frequency of Interval 4 are closest to:
Relative Frequency Cumulative Relative Frequency
A. 15% 23%
B. 20% 77%
C. 15% 77%
Answer: C
Cumulative relative frequency is the cumulative frequency of 4th interval divided by the total number of observations:
6. If a sample consists of the historical monthly stock returns of Trukmaker Inc. for the past 5 years, then the mean computed from these observations will be a:
A. Time-series mean B. Cross-sectional mean C. Weighted mean Answer: A
A firm’s returns over a period of time are an example of time-series data. The mean of time-series data is called a time-series mean.
7. A portfolio consists of 40% common stock, 20% preferred stock, 15% bonds, and 25% real estate. The returns on these asset classes are 12% on common stock, 10% on preferred stock, and 8% on both real estate and bonds. The mean return on the portfolio is closest to:
A. 8%
B. 10%
C. 12%
Answer: B
Relative frequency = Absolute frequency of each interval × 100 = = 15.38 % or 15 % Total no. of observations 2
13
CRF = 1 + 3 + 4 + 2 = = 0.7962 or 77 %
13 10
13
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Portfolio return = (0.4)(0.12)+(0.2)(0.1)+(0.15)(0.08)+(0.25)(0.08) = 0.1 or 10%
8. An individual invests $1,000 in a particular security on two different dates ($2,000 total).
The price of the security is $10 on the first day and $12 on the second. The investor’s average purchase price is closest to:
A. 11 B. 10.84 C. 10.91 Answer: C
9. The 4th quintile for the following distribution of returns is closest to:
8%, 6%, 12%, 18%, 25%, 8%, 9%, 17%, 14%, 10%
A. 17%
B. 17.8%
C. 18.5%
Answer: B
First rearranging the data in ascending order:
6%, 8%, 8%, 9%, 10%, 12%, 14%, 17%, 18%, 25%
Quintiles are fifths (1/5) so the 4th quintile is 80% of the observations.
This means that when the data is arranged in ascending order, the 4th quintile is 8th data point from the left plus 0.8 times the distance between the 8th and 9th values:
= 17 + 0.8(18 − 17)
= 17.8%
10. Which of the following portfolios has the smaller range and which one most likely has greater risk?
Harmonic mean = 2 = 10.909
1 +
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Smaller Range Higher Risk
A. A B
B. B A
C. B B
Answer: B
Range of Portfolio A = 20.5% − (−11.1%) = 31.6%
Range of Portfolio B = 13.2% − (−1.6%) = 14.8%
Portfolio B has the smaller range and Portfolio A is riskier as it has a higher range.
11. What is the mean absolute deviation and variance of the sample investment returns given below:
Year Return 2004 8%
2005 9%
2006 −7%
2007 5%
MAD Variance A. 6.375% 40.69 B. 5.375% 54.25 C. 6.375% 54.25 Answer: B
12. What is the minimum percentage of a distribution that will lie within 3 standard deviations of the mean?
A. 75%
B. 82%
C. 89%
Answer: C
Applying Chebyshev’s inequality 1 − 1/k2: 1 − (1/32) = 89%
Mean = x− = (8 % + 9 % + ( − 7 % ) + 5)/4 = 3.75 %
MAD = ( 8 − 3.75 + 9 − 3.75 + − 7 − 3.75 + 5 + 3.75 )/4 = 5.375 %∣∣ ∣∣ ∣∣ ∣∣ ∣∣ ∣∣ ∣∣ ∣∣
Variance = (8 − 3.75)2 +(9 − 3.75)2 +( − 7 − 3.75)2 +(5 − 3.75)2 = 54.25 4 − 1
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13. An analyst gathered the following information about a common stock portfolio:
Arithmetic mean 12%
Compute the coefficient of variation and Sharpe ratio.
Coefficient of Variation Sharpe Ratio
A. 1.26 0.43
B. 1.26 2.05
C. 1.37 0.43
Answer: C
14. Compared to the standard deviation, the mean absolute deviation is always:
A. Less than or equal to the standard deviation B. Greater than or equal to the standard deviation C. Less than the standard deviation
Answer: A
The MAD is always less than or equal to the standard deviation as the standard deviation attaches a heavier weight to larger deviations.
15. A distribution that has a kurtosis of +3 is known as:
A. Kurtosis B. Leptokurtic C. Mesokurtic Answer: C
A mesokurtic distribution has a kurtosis of 3 and excess kurtosis of zero.
16. Excess kurtosis of normal distribution equals:
A. 3
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C. 0 Answer: C
While kurtosis for the normal distribution is 3, the excess kurtosis of a normal distribution equals 0.
Use the following frequency distribution for Questions 17 to 19:
Annual Return (R) Frequency 17. The modal interval is:
A. −10% ≤ R < 0%
B. −5
C. −30% ≤ R < −20%
Answer: A
The modal interval is the interval with the highest frequency, which in this case is (−10%
≤ R < 0%).
The sample size equals the total frequency (1+4+15+10+14+2 = 46).
19. The cumulative relative frequency of the fifth interval is:
A. 44 B. 0.956 C. 0.304 Answer: B
Cumulative relative frequency is the cumulative frequency divided by the total frequency (44/46 = 0.956)
Use the following information for Questions 20 to 23:
The annual returns of the various stocks for 2008 are given in the table below:
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Stock ABC DEF GHI JKL MNO PQR Return 21% −10% 5% 13% −19% 5%
20. The arithmetic mean of the returns on these stocks for the year 2008 is:
A. 12.17%
B. 2.5%
C. 5%
Answer: B
The arithmetic mean is the sum of the returns divided by the total number of stocks:
(21−10+5+13−19+5)/6 = 2.5%
21. The median return of these stocks for the year 2008 is:
A. 9%
B. −19%
C. 5%
Answer: C
The median is the ½(n+1)th element in the dataset. In this case, it is the 3.5th element, which lies between the 3rd and the 4th items. The average of 5% and 5% equals 5%.
22. The range and standard deviation of the stock returns, assuming that they represent a population, are:
Range Std Dev A. 40 32.9 B. 40 13.43 C. 28.16 40 Answer: B
The range is the difference between the highest and lowest value in the dataset [21 − (−19)
= 40]. The standard deviation is the average of the square root of the sum of squared deviations from the mean:
23. The geometric and harmonic means for the returns data are:
Geometric Mean Harmonic Mean
A. 10.4% 2.5%
B. 1.6% 16.1%
C. 1.02% 8.86%
Std Dv. = σ = √(21 − 2.5)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾2 +( − 10 − 2.5)2 +(5 − 2.5)2 +( − 19 − 2.5)2 +(5 − 2.5)2‾ = 13.43 6
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Answer: B
The geometric mean is calculated as:
The harmonic mean is calculated as follows:
24. Which of the following statements is most accurate?
A. Descriptive statistics refer to methods used to make forecasts, estimates, or draw conclusions about a larger data set based on a smaller representative set.
B. Inferential statistics refer to how large volumes of data are converted into useful and readily understood information by summarizing their important characteristics.
C. The arithmetic mean of a particular data set is an example of a sample statistic.
Answer: C
Descriptive statistics refer to how large volumes of data are converted into useful and readily understood information by summarizing their important characteristics.
Inferential statistics refer to methods used to make forecasts, estimates, or draw conclusions about a larger data set based on a smaller representative set.
25. Which of the following types of measurement scales sorts data into categories that are ranked according to a certain characteristic, but does not tell us anything about the differences between categories?
A. Nominal scales B. Ordinal scales C. Interval scales Answer: B
Nominal scales categorize or count data but do not rank them.
Interval scales rank observations in a manner such that differences between scale values are equal so that values can be added and subtracted meaningfully.
26. Which of the following statements regarding frequency distributions is least accurate?
A. Any particular observation could fall into one or more intervals.
B. The set of intervals should include the entire range of values in the data set.
C. The interval with the highest frequency is called the modal class.
Answer: A
Each observation in the data set must fall into only one interval. The intervals must be Geometric mean = √6(1.21)(0.90)(1.05)(1.13)(0.81)(1.05)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾− 1 = 0.0158 or 1.6 %
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mutually exclusive and collectively exhaustive.
27. Which of the following statements is most likely?
A. The cumulative frequency for an interval is the proportion of total observations that lies in that particular interval.
B. The cumulative absolute frequency for a particular interval is the number of observations that are less than the upper bound for the interval.
C. The cumulative relative frequency for a particular interval is the number of observations that are less than the midpoint of the interval.
Answer: B
The cumulative frequency/cumulative absolute frequency for an interval equals the number of observations that are less than the upper bound of the interval.
The cumulative relative frequency for an interval is the proportion of total observations that is less than the upper bound of the interval.
Use the following information to answer Questions 28 to 35:
Cool Pop Soda Corp’s annual profit margins since its incorporation are presented below:
Year 1 11%
28. The arithmetic mean for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 4.5%
B. 6%
C. 11%
Answer: A
[11 + 9 + 4 + (−2) + 1 + 4] / 6 = 4.5%
29. The median for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 11%
B. 4.5%
C. 4%
Answer: C
There is an even number of observations in the data set so the median equals the average of the middle two numbers after the data set has been arranged in ascending order.
(4 + 4) / 2 = 4%
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30. The mode for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 4%
B. 4.5%
C. 11%
Answer: A
The mode is the value that has the highest frequency, 4% (twice).
31. The mean absolute deviation for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 3 B. 3.67 C. 19.56 Answer: B
The MAD equals the mean of the absolute values of the deviations from the mean.
(|11 − 4.5| + |9 − 4.5| + |4 − 4.5| + |−2 − 4.5| + |1 − 4.5| + |4 − 4.5|) / 6 = 3.67 32. The variance for Cool Pop Soda Corp’s profit margins over the years is closest to:
A. 3.67 B. 4.42 C. 19.58 Answer: C
The variance equals the arithmetic mean of the squared deviations from the mean.
[(11 − 4.5)2+ (9 − 4.5)2 + (4 − 4.5)2 + (−2 − 4.5)2 + (1 − 4.5)2 + (4 − 4.5)2] / 6 = 19.58 33. The standard deviation for Cool Pop Soda Corp’s profit margins over the years is closest
to:
A. 1.91%
B. 4.43%
C. 19.58%
Answer: B
The standard deviation equals the positive square root of the variance.
√19.58 = 4.425
34. Given that the data set is only a representative sample of profit margins over the
company’s entire existence, the variance for Cool Pop Soda Corp’s profit margins over the 6 years is closest to:
A. 23.5 B. 19.56
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C. 13.47 Answer: A
For sample variance, we must use (n − 1) in the denominator
[(11 − 4.5)2+ (9 − 4.5)2 + (4 − 4.5)2 + (−2 − 4.5)2 + (1 − 4.5)2 + (4 − 4.5)2] / (6 − 1) = 23.5.
35. Given that the data set is only a representative sample of profit margins over the company’s entire existence, the standard deviation for Cool Pop Soda Corp’s profit margins over the 6 years is closest to:
A. 3.67%
B. 4.42%
C. 4.85%
Answer: C
The standard deviation of the sample is the square root of the sample variance. √23.5 = 4.847%
36. Junglelala’s returns over its first 4 years of operations are −1.1%, 1.1%, 1.9%, and 2.8%
respectively. The geometric mean of its returns is closest to:
A. 1.165%
B. 1.175%
C. 1.185%
Answer: A
(0.989 × 1.011 × 1.019 × 1.028)1/4 − 1 = 1.1646%
37. The maximum percentage of observations in a data set that lies further than 1.85 standard deviations away from the mean is closest to:
A. 29.2%
B. 45.9%
C. 70.8%
Answer: A
Using Chebyshev’s inequality: 1 − [1 / (1.85)2] = 70.8%
At least 70.8% of the data lie within 1.85 standard deviations from the mean. This implies
At least 70.8% of the data lie within 1.85 standard deviations from the mean. This implies