7.2: Means and Variance

Question 1

What is measures of central tendency? What can it be used to represent?

Answer 1

Measures of central tendency identify the centre, or average, of a data set.

It can be used to represent the typical, or expected, value in the data set.

Question 2

How is population mean computed?

Answer 2

Population mean = sum of all observed values in the population / number of observations in the population.

Question 3

What is sample mean and how is it computed?

Answer 3

Sample mean = sum of all the values in a sample of a population / number of observations in the sample.

Sample mean is used to make inferences about the population mean.

Question 4

Define arithmetic mean and provide examples. What is it used to measure and what properties does it have (4) ?

Answer 4

Arithmetic mean is the sum of the observation values divided by the number of observations.

Examples of arithmetic means are population mean and sample mean.

It is the most widely used measure of central tendency and has the following properties:

1. All interval and ratio data sets have an arithmetic mean.
2. All data values are considered and included in the arithmetic mean computation.
3. A data set has only one arithmetic mean such that the arithmetic mean is unique.
4. The sum of the deviations of each observation in the data set from the mean is always zero.

Question 5

What is the sum of mean deviations?

Answer 5

Sum of mean deviations = zero

Question 6

What is weighted mean and how is it computed?

Answer 6

Weighted mean recognizes that different observations may have a disproportionate influence on the mean.

The weighted mean of a set of numbers is computed as the sum of observed value multiplied by its corresponding weight.

Question 7

Define median and why it is important.

Answer 7

Median is the midpoint of a data set when the data is arranged in ascending or descending order.

Median is important because the arithmetic mean can be affected by outliers and when this happens, median is a better measure of central tendency than the mean because it is not affected by extreme values that may actually be the result of errors in the data.

Question 8

Define mode, unimodal, bimodal, and trimodal.

Answer 8

Mode is the value that occurs the most frequently in a data set.

A data set may have more than one mode or no mode.

Unimodal: one value that appears frequently
Bimodal: two values that appear frequently
Trimodal: three values that appear frequently.

Question 9

Define geometric mean and how it is computed. How is the computation of the geometric mean for a returns data set different?

Answer 9

Geometric mean is often used when calculated investment returns over multiple periods or when measuring compound growth rates.

Geometric mean is computed as follows:
G = (X1 x X2 x …. x Xn)^(1/n)

Geometric mean for a returns data set is computed as follows:

1 + Rg = [(1 + R1) x (1 + R2) x … x (1 + Rn)]^(1/n)

where Rg is the geometric mean return and Rn is the return for period t.

Question 10

What is the relationship between geometric mean and arithmetic mean?

Answer 10

Equal when there is no variability in the observations (observations are equal). Geometric mean is always less than or equal to the arithmetic mean and the difference increases as the dispersion of the observations increases.

Question 11

When is harmonic mean used and how is it computed?

Answer 11

Harmonic mean is used for certain computations, such as the average cost of shares purchased over time.

Harmonic mean = number of observations / sum of 1 over value of observation

Question 12

What is the relationship between harmonic mean, arithmetic mean, and geometric mean?

Answer 12

When values are not equal: harmonic mean < geometric mean < arithmetic mean

Question 13

Define quantile.

Answer 13

Quantile is the general term for a value at or below which a stated proportion of the data in a distribution lies.

Examples of quantiles include:

• Quartiles (the distribution is divided into quarters)
• Quintiles (the distribution is divided into fifths)
• Decile (the distribution is divided into tenths)
• Percentiles (the distribution is divided into hundredths)

Question 14

How is the position of the observation at a given percentile?

Answer 14

Ly = (n +1)(y/100)

where y is the given percentile and n is the data points sorted in ascending order.

Question 15

What are the two measures of location?

Answer 15

Measures of location are quantiles and measures of central tendency.

Question 16

What is dispersion?

Answer 16

Dispersion is defined as the variability around the central tendency.

Question 17

What is range and how is it computed?

Answer 17

Range is a measure of variability between the largest and the smallest value in the data set.

Range = maximum value - minimum value

Question 18

Define mean absolute deviation (MAD) and how it is computed.

Answer 18

Mean absolute deviation is the average of the absolute values of the deviations of individual observations from the arithmetic mean.

MAD = sum of absolute value of X

Question 19

What is population variance? What is the disadvantage to using variance? How is this problem mitigated?

Answer 19

Population variance is defined as the average of the squared deviations from the mean.

Disadvantage with using variance is the difficulty of interpreting it.

The problem is mitigated through the use of standard deviation, which is the square root of the population variance.

Question 20

What is sample variance and how is it calculated?

Answer 20

Sample variance is the measure of dispersion that applies when evaluating a sample of n observations from a population.

The sample variance is calculated using:
sum of observed value minus mean squared / n - 1

Question 21

What is the most noteworthy difference between the population variance and sample variance? Why?

Answer 21

Population variance uses whole population size N and sample variance uses n - 1.

N will underestimate the population parameter, especially for small samples. The underestimation is known as biased estimator.

n - 1 improves the statistical properties.

Question 22

What is biased estimator?

Answer 22

N will underestimate the population parameter, especially for small samples. The underestimation is known as biased estimator.

Question 23

How is sample standard deviation computed?

Answer 23

Sample standard deviation is the square root of the sample variance.

Sample standard deviation = sum of observed value less mean squared / n - 1