The Mean, Median, and Mode

 

The Mean, Median, and Mode
Given the observations from a sample: 8, 10, 9, 12, 12

To calculate the mean, we need to sum up all the values and divide by the number of observations: Mean = (8 + 10 + 9 + 12 + 12) / 5 = 51 / 5 = 10.2

To calculate the median, we need to arrange the observations in ascending order and find the middle value. Since we have an odd number of observations, the median is the middle value: Median = 10

To calculate the mode, we need to find the value that appears most frequently in the observations: Mode = 12

Therefore, for the given observations, the mean is 10.2, the median is 10, and the mode is 12.

The Five-Point Summary and Outliers
Using 500 observations, the following five-point summary was obtained for a variable:

Min = 125 Q1 = 200 Median = 300 Q3 = 550 Max = 1300

a. Q1 represents the first quartile, which is the value separating the lower 25% of the data from the upper 75% of the data. In this case, Q1 is 200.

b. The interquartile range (IQR) is calculated by subtracting Q1 from Q3: IQR = Q3 – Q1 = 550 – 200 = 350

To determine if any outliers exist, we can use the rule that considers values outside the range of Q1 – (1.5 * IQR) to Q3 + (1.5 * IQR) as potential outliers. In this case: Lower bound = Q1 – (1.5 * IQR) = 200 – (1.5 * 350) = -275 Upper bound = Q3 + (1.5 * IQR) = 550 + (1.5 * 350) = 1075

Any values below -275 or above 1075 would be considered potential outliers.

c. To determine if the distribution is symmetric or skewed, we can compare the location of the median and mean. If they are approximately equal, the distribution is symmetric. If the mean is greater than the median, it indicates right-skewness, and if the mean is less than the median, it indicates left-skewness.

Without information about the mean, we cannot determine if the distribution is symmetric or skewed.

Arithmetic and Geometric Mean Returns
Suppose at the beginning of Year 1, you decide to invest $1,000 in a mutual fund. The following table shows the returns (in %) for the past four years:

Year Annual Return
1 17.3
2 19.6
3 6.8
4 8.2
a. To calculate the arithmetic mean return, we sum up all the annual returns and divide by the number of years: Mean Return = (17.3 + 19.6 + 6.8 + 8.2) / 4 = 51.9 / 4 = 12.975%

The arithmetic mean return represents the average return over the four years.

b. To calculate the geometric mean return, we multiply all the annual returns and take the fourth root: Geometric Mean Return = ((1 + 17.3/100) * (1 + 19.6/100) * (1 + 6.8/100) * (1 + 8.2/100))^(1/4) – 1 Geometric Mean Return ≈ ((1.173) * (1.196) * (1.068) * (1.082))^(1/4) – 1 Geometric Mean Return ≈ (1.537)^0.25 – 1 ≈ 0.1236 or 12.36%

The geometric mean return represents the compounded annual return over the four years.

c. To calculate how much money would be accumulated by the end of Year 4, we can use the formula for compound interest: Accumulated Amount = Principal * (1 + Rate)^Time

Principal = $1,000 Rate = Geometric Mean Return = 0.1236 or 12.36% Time = 4 years

Accumulated Amount = $1,000 * (1 + 0.1236)^4 ≈ $1,000 * (1.1236)^4 ≈ $1,000 * 1.6152 ≈ $1,615.20

Therefore, by the end of Year 4, you would have accumulated approximately $1,615.20.

Population Data Analysis
Consider the following population data: 34, 42, 12, 10, and 22.

a. To calculate the range, we subtract the lowest value from the highest value: Range = Max – Min = 42 – 10 = 32

b. To calculate MAD (Mean Absolute Deviation), we need to find the mean of absolute deviations from the mean: Mean = (34 + 42 + 12 + 10 +22) / 5 =120 /5 =24

Deviation from Mean: |34-24| =10 |42-24| =18 |12-24| =12 |10-24| =14 |22-24| =2

MAD= (10+18+12+14+2)/5=56/5=11.2

c. To calculate the population variance, we need to find the squared deviations from the mean and take their average: Squared Deviation from Mean: (34-24)^2=100 (42-24)^2=324 (12-24)^2=144 (10-24)^2=196 (22-24)^2=4

Variance=(100+324+144+196+4)/5=768/5=153.6

d. To calculate the population standard deviation, we take the square root of the variance: Standard Deviation=√153.6≈12.403

Stock Returns Comparison
Consider the following summary measures for the annual returns for Stock 1 and Stock 2 over the past 13 years:

Stock 1: x̄ = 9.62% and s = 23.58% Stock 2: x̄ = 12.38% and s = 15.45%

a. To determine which stock had a higher average return, we compare their means: Stock 2 has a higher average return with x̄ = 12.38%.

b. To determine which stock was riskier over this time period, we can compare their standard deviations: Stock 1 has a higher standard deviation with s = 23.58%.

Given that Stock 2 has a higher average return and Stock 1 has a higher standard deviation, it is not surprising that Stock 2 was riskier over this time period.

c. To compare their Sharpe ratios, we need to consider their excess returns over a risk-free rate and divide by their standard deviations. Given a risk-free rate of 3%, let’s calculate their Sharpe ratios:

Sharpe Ratio for Stock 1: (x̄ – Risk-Free Rate) / s Sharpe Ratio for Stock 2: (x̄ – Risk-Free Rate) / s

Assuming a Risk-Free Rate of 3%: Sharpe Ratio for Stock 1: (9.62 -3)/23.58≈0.268 Sharpe Ratio for Stock 2: (12.38-3)/15.45≈0.584

Therefore, Stock 2 has a higher Sharpe ratio than Stock 1. The Sharpe ratio measures how much excess return an investment generates per unit of risk taken. In this case, Stock 2 has a higher Sharpe ratio, implying that it provides better risk-adjusted returns compared to Stock 1.

Bell-Shaped Distribution Observations
Observations are drawn from a bell-shaped distribution with a mean of 20 and a standard deviation of 2.

a. To approximate what percentage of observations fall between 18 and 22, we can use the empirical rule or normal distribution properties. According to the empirical rule, approximately68% of observations fall within one standard deviation of the mean. Therefore, approximately68%of observations fall between (20-2)=18and(20+2)=22.

b. To approximate what percentage of observations fall between16and24, Since one standard deviation covers approximately68%of observations on both sides of the mean, we can approximate that approximately95%(68%+27% within two standard deviations) of observations fall between16and24.

c.Approximately what percentage of observations are less than16, Using properties of a normal distribution, we can say that approximately2%of observations are less than16 since it falls beyond two standard deviations below themean(20).