Unit 5 - Practice Quiz

The range is the simplest measure of dispersion, calculated as the difference between the maximum and minimum values in a dataset (Range = Highest Value - Lowest Value).

To find the range, subtract the lowest value from the highest value. Highest value = 25, Lowest value = 8. So, Range = 25 - 8 = 17.

The range only considers the two most extreme data points (maximum and minimum), making it very sensitive to outliers, which can give a misleading idea of the data's dispersion.

Quartile Deviation is calculated as half of the interquartile range (IQR), which is the difference between the third quartile () and the first quartile (). Hence, it is called the Semi-Interquartile Range.

The Quartile Deviation is half the distance between the third and first quartiles. The formula is correctly expressed as .

Since it is based on and , the Quartile Deviation considers the spread of the central half of the dataset, making it less sensitive to extreme values than the range.

Mean Deviation measures the average distance of each data point from a central point, which is typically the mean, median, or mode.

A key property of the arithmetic mean is that the sum of the deviations of all observations from it is always zero (). Using absolute values ensures that deviations do not cancel each other out.

This formula correctly represents the mean deviation, which is the sum of the absolute differences between each value () and the mean (), divided by the total number of observations (n).

Standard deviation () is defined as the positive square root of the variance (). This brings the measure back to the original units of the data.

If all values are identical, there is no variation or spread in the data. Each value is equal to the mean, so all deviations are zero, resulting in a standard deviation of 0.

A small standard deviation signifies that the data values tend to be very close to the mean, indicating low variability. A large standard deviation indicates high variability.

The Greek letter sigma () represents the standard deviation of an entire population, whereas 's' is typically used for the standard deviation of a sample.

The CV is a relative measure because it expresses the standard deviation as a percentage of the mean, making it a unit-less value that can be used to compare datasets with different units or means.

The formula for the Coefficient of Variation is the ratio of the standard deviation to the mean, usually multiplied by 100 to express it as a percentage.

A smaller CV indicates less variability relative to the mean. In finance, this translates to lower volatility and greater consistency in the stock's returns or price.

Because the CV is a unit-less, relative measure, its main advantage is in comparing the dispersion of two or more series that are measured in different units (e.g., comparing height in cm vs. weight in kg) or have significantly different means.

Skewness is a measure of the asymmetry of the probability distribution of a real-valued random variable about its mean. It indicates the extent to which a distribution deviates from being symmetrical.

A symmetrical distribution has its left and right sides as mirror images of each other. This results in a skewness value of zero.

In a positively skewed distribution, the long tail is on the right. The extreme high values in the tail pull the mean to the right, making it the largest value, followed by the median, and then the mode.

Original data: {3, 8, 10, 12, 15, 18, 25}. Original Range = Max - Min = 25 - 3 = 22. New data with outlier: {3, 8, 10, 12, 15, 18, 25, 50}. New Range = 50 - 3 = 47. The change in range is 47 - 22 = 25. The range is highly sensitive to extreme values (outliers).

First, sort the data: {10, 20, 25, 30, 40, 45, 50}. There are n=7 observations. \ First Quartile () = Value of item = Value of item = 2nd item = 20. \ Third Quartile () = Value of item = Value of item = 6th item = 45. \ Quartile Deviation = .

Original variance () = 16. Therefore, the original standard deviation (σ) = . When each observation is multiplied by a constant 'k', the new standard deviation becomes k (original standard deviation). Here, k=2. So, the new standard deviation = 2 4 = 8. Note that the new variance would be , and .

To compare relative volatility (or consistency), we use the Coefficient of Variation (CV). CV = (Standard Deviation / Mean) 100. \ For the first stock: CV = (12 / 150) 100 = 8%. \ For the second stock: CV = (4.5 / 30) * 100 = 15%. \ Since the CV of the second stock (15%) is higher than the first (8%), the second stock is considered more volatile.

For a moderately skewed distribution, the empirical relationship between mean, median, and mode is: Mean - Mode ≈ 3(Mean - Median). \ Substituting the given values: 45 - 48 = 3(45 - Median). \ -3 = 135 - 3 Median. \ 3 Median = 138. \ Median = 138 / 3 = 46.

First, calculate the mean (). . \ Next, find the absolute deviations from the mean: |4-8|, |6-8|, |8-8|, |10-8|, |12-8|, which are {4, 2, 0, 2, 4}. \ Sum of absolute deviations = 4 + 2 + 0 + 2 + 4 = 12. \ Mean Deviation = .

The formula for population variance is . \ First, calculate the mean: . \ Now, calculate the variance: . \ The standard deviation () is the square root of the variance, so .

Karl Pearson's coefficient of skewness is given by the formula: . \ We are given , Mean = 60, and SD = 8. \ -0.75 = . \ -0.75 * 8 = 60 - Mode. \ -6 = 60 - Mode. \ Mode = 60 + 6 = 66.

The formula for the Coefficient of Variation (CV) is CV = (Standard Deviation / Mean) 100. \ We are given CV = 40% and Mean = 70. \ 40 = (Standard Deviation / 70) 100. \ 0.40 = Standard Deviation / 70. \ Standard Deviation = 0.40 * 70 = 28. \ The question asks for the variance, which is the square of the standard deviation. \ Variance = .

In a symmetric distribution, the median () is equidistant from the first quartile () and the third quartile (). Therefore, the median is the average of and . \ Median = .

The range is calculated as the difference between the maximum and minimum values in a dataset. Its main weakness is that it completely ignores the distribution of the data between these two extremes and is extremely sensitive to outliers, which can give a misleading picture of the overall dispersion.

A key property of mean deviation is that the sum of the absolute values of the deviations, , is at its minimum when A is the median of the distribution. It is smaller when calculated from the median than from the mean or mode.

The second set of numbers {7, 8, 10, 11} is obtained by adding a constant, 5, to each number in the first set {2, 3, 5, 6}. Standard deviation is a measure of dispersion and is unaffected by a change of origin (i.e., adding or subtracting a constant from all observations). Therefore, the standard deviation remains the same.

In a positively skewed (or right-skewed) distribution, the tail on the right side of the distribution is longer or fatter than the left side. This pulls the mean to the right of the median. The characteristic relationship is Mean > Median > Mode. Since 30 > 28 > 24, the distribution is positively skewed.

Standard deviation is an absolute measure of dispersion, expressed in the same units as the data. The Coefficient of Variation (CV) is a relative measure (a percentage) that is unit-free. This allows for a meaningful comparison of the dispersion of datasets that have different units (e.g., height in cm vs. weight in kg) or whose means are drastically different.

The Interquartile Range (IQR) is defined as the difference between the third quartile () and the first quartile (). The formula is IQR = . \ Given IQR = 30 and . \ 30 = 75 - . \ = 75 - 30 = 45.

The sum of the squared deviations from the mean is given as . The number of items is n=5. \ The formula for variance () is . \ . \ The standard deviation () is the square root of the variance. \ .

First, sort the data: {3, 3, 4, 5, 9, 10, 12, 18}. There are n=8 observations. \ The median is the average of the 4th and 5th terms: Median = . \ Now, find the absolute deviations from the median (7): |3-7|, |3-7|, |4-7|, |5-7|, |9-7|, |10-7|, |12-7|, |18-7|. \ The deviations are: {4, 4, 3, 2, 2, 3, 5, 11}. \ Sum of absolute deviations = 4+4+3+2+2+3+5+11 = 34. \ Mean Deviation = .

A negatively skewed (or left-skewed) distribution is characterized by a long tail on the left side. The presence of lower-value outliers pulls the mean to the left of the median. The general relationship for such a distribution is Mean < Median < Mode.

Standard deviation is an absolute measure of dispersion. A standard deviation of 20 might be very large for a dataset with a mean of 10 (CV=200%), but very small for a dataset with a mean of 1000 (CV=2%). To judge whether the dispersion is high or low in a relative sense, one must compare it to the mean, which is precisely what the Coefficient of Variation (CV) does.

This problem requires using the property that relates the sum of squared deviations from an arbitrary point 'A' to the sum of squared deviations from the mean ().

The formula is:
Given: , , , and .

1. Substitute the given values into the formula:
   
   
   
2. Solve for the sum of squared deviations from the mean:
   
3. Calculate the variance ():
   
4. Calculate the standard deviation ():
   
5. Calculate the Coefficient of Variation (CV):
   

This question requires a multi-step synthesis of skewness formulas and the empirical relationship between mean, median, and mode.

1. Use Karl Pearson's primary formula for skewness:
   
   We are given , Mean = 27, and SD = 10. We can find the Mode directly from this.
   
   
   

The information about the empirical relationship Mean - Mode ≈ 3(Mean - Median) is extra information designed to test whether the student knows which formula to use. While one could use it to find the median, the question asks for the mode, which can be found directly. If we were to proceed with the other formula:

1. Use Pearson's second formula:
   
   
   
   Median = 30.

2. Now use the empirical relation: Mean - Mode = 3(Mean - Median)
   
   
   Mode = 27 + 9 = 36.

Both methods yield the same result, but the first one is more direct.

This is a data correction problem requiring recalculation of the standard deviation.

1. Find original sums:
   Given