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13.7 The R Statistical Analysis Tool
R is an open-source statistical analysis tool that is widely used in the finance industry. It provides an integrated
suite of functions for data analysis, graphing, and statistical programming. R is increasingly being used as a
data analysis and statistical tool as it is an open-source language, and additional features are constantly being
added by the user community. This tool can be used on many different computing platforms and can be
downloaded at The R Project for Statistical Computing (https://openstax.org/r/The-R-Project).
Key Terms
arithmetic mean a measure of center of a data set, calculated by adding up the data values and dividing the
sum by the number of data values
bar graph a chart that presents categorical data in a summarized form based on frequency or relative
frequency
bivariate data paired data in which each value of one variable is paired with a value of a second variable
data visualization the use of graphical displays, such as bar charts, histograms, and scatter plots, to help
interpret patterns and trends in a data set
empirical rule a rule that provides the percentages of data values falling within one, two, and three standard
deviations from the mean for a bell-shaped (normal) distribution
expected value a weighted average of the values of a variable where the weights are the associated
probabilities
exponential distribution a continuous probability distribution that is useful for calculating probabilities
within the time between events
frequency distribution a method of organizing and summarizing a data set that provides the frequency
with which each value in the data set occurs
geometric mean a measure of center of a data set, calculated by multiplying the data values and then
raising the product to the exponent , where n is the number of data values
histogram a graphical display of continuous data showing class intervals on the horizontal axis and
frequency or relative frequency on the vertical axis
interquartile range (IQR) a number that indicates the spread of the middle half, or middle 50%, of the data;
the difference between the third quartile (Q3) and the first quartile (Q1)
median the middle value in an ordered data set
mode the most frequently occurring data value in a data set
normal distribution a bell-shaped distribution curve that is used to model many measurements, including
IQ scores, salaries, heights, weights, blood pressures, etc.
outliers data values that are significantly different from the other data values in a data set
percentiles numbers that divide an ordered data set into hundredths; often used to indicate position of a
data value in a data set
population data data representing all the outcomes or measurements that are of interest
portfolio a collection of financial investments, such as stocks, bonds, mutual funds, certificates of deposit,
etc.
probability distribution a mathematical function that assigns probabilities to various outcomes
quartiles numbers that divide an ordered data set into quarters; the second quartile is the same as the
median
sample data data representing outcomes or measurements collected from a subset or part of a population
scatter plot (or scatter diagram) a graphical display that shows the relationship between a dependent
variable and an independent variable
standard deviation a measure of the spread of a data set that indicates how far a typical data value is from
the mean
time series graph a graphical display used to show measurement data plotted versus time, where time is
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displayed on the horizontal axis
variance the measure of the spread of data values calculated as the square of the standard deviation
weighted mean a measure of center of a data set in which each data value has a corresponding weighting
x-axis the horizontal axis in a rectangular coordinate system
y-axis the vertical axis in a rectangular coordinate system
z-score (or z-value) a measure of the position of a data value in the data set, calculated by subtracting the
mean from the data value and then dividing the difference by the standard deviation
CFA Institute
This chapter supports some of the Learning Outcome Statements (LOS) in this CFA® Level I Study Session
(https://openstax.org/r/cfa-level1-study-session). Reference with permission of CFA Institute.
Multiple Choice
1. A data set of salaries contains an outlier salary. The best measure of center to use for this data set is the
________.
a. mean
b. median
c. mode
d. standard deviation
2. A portfolio includes shares of United Airlines stock that were purchased at different times and different
prices. Which measure is best to determine the average cost of the shares of the stock?
a. mean
b. median
c. weighted mean
d. standard deviation
3. Standard deviation is a measure of the ________.
a. center of a data set
b. position of a data value in a data set
c. area under a normal curve
d. spread of a distribution
4. How are standard deviation and variance related?
a. The two measures are equal to one another.
b. Variance is the square root of the standard deviation.
c. Standard deviation is the square root of the variance.
d. The two squared measures are equal to one another.
5. Which of the following is the best definition of a z-score?
a. the distance of a data value from the mean
b. the number of standard deviations that a data value is from the mean
c. the distance of a data value from the mean divided by the sample size
d. the number of quartiles that a data value is from the mean
6. The results of a standardized test indicate that you are in the 85th percentile. What is the best
interpretation of this result?
a. You scored in the top 85% of all students taking the test.
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b. You scored in the top 15% of all students taking the test.
c. Your score on the test is 85 when measured on a scale from 0 to 100.
d. You scored in the bottom 15% of all students taking the test.
7. The interquartile range is ________.
a. the middle 50% of a data set
b. the upper 50% of a data set
c. the lower 50% of a data set
d. equal to the median
8. In a frequency distribution table, the sum of the relative frequencies must be equal to ________.
a. the sample size
b. 1, or 100%
c. zero
d. the standard deviation of the distribution
9. A change in the standard deviation of a normal distribution will result in ________.
a. a change in the location of the peak of the curve
b. a change in the area under the curve
c. a change in the shape of the curve
d. a change that shifts the graph to the left or the right
10. When calculating an expected value, ________.
a. the result should always be 1
b. the result should always be a positive value
c. the result should always be a negative value
d. the result can be a positive or negative value
11. The area under a normal curve between a z-score of -2 and a z-score of +2 is ________.
a. 0.68
b. 0.95
c. 0.997
d. dependent on the mean and standard deviation
12. A scatter plot is a visualization for ________.
a. univariate data only
b. bivariate data only
c. either univariate or bivariate data
d. test scores
13. Which of the following is NOT a benefit of using the R statistical analysis tool?
a. Additional features are constantly being added by the user community.
b. It can be used on many computer platforms, including Mac, Windows, and Linux.
c. It is free to download.
d. Users pay an annual subscription fee.
Review Questions
1. Explain the considerations that determine whether the mean or the median is the best measure of central
tendency for a data set.
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2. Explain the difference between a mean and a weighted mean.
3. Explain why the standard deviation of a data set cannot be a negative value.
4. Explain what a negative z-score, a positive z-score,and a z-score of zero imply.
5. Explain how quartiles can be used to detect outliers in a data set.
Problems
1. You purchased 1,000 shares of a stock for $12 per share. Then, two months later, you purchased an
additional 500 shares of the same stock at $9 per share. Calculate the weighted mean of the purchase
price for the total of 1,500 shares.
2. You score a 60 on a biology test. The mean test grade is 70, and the standard deviation is 5. Calculate and
interpret your corresponding z-score.
3. You score a 60 on a biology test. The mean test grade is 70, and the standard deviation is 5. What
percentile does your grade correspond to?
4. A fast food restaurant has measured service time for customers waiting in line, and the service time
follows an exponential distribution with a mean waiting time of 1.9 minutes. The restaurant has a
guarantee that if customers wait in line for more than 5 minutes, their meal is free. What is the probability
that a customer will receive a free meal?
5. The total value of your portfolio consists of approximately 65% stock assets, 25% bonds, and 10% cash
equivalents. Historical returns have shown that stocks provide a return of 12%, bonds provide a return of
3.5%, and cash savings provide a return of 1.5%. What is the expected value of the return on this portfolio?
6. The distribution of the average annual return of the S&P 500 over a 50-year time period follows a normal
distribution with a mean rate of return of 10.5% and a standard deviation of 14.3%. What is the probability
that an average annual return will fall between -3.8% and 24.8%?
7. Write a short R program to find the expected return for the data set in Table 13.16.
Historical Return on
United Airlines Stock
Associated
Probability
12% 15%
5% 35%
2% 25%
-5% 14%
-10% 11%
Table 13.16
Video Activity
Normal Distribution Stock Return Calculations
Click to view content (https://openstax.org/r/stock-return)
1. Assume the return on stocks follows a normal distribution. Is it more likely that a stock will return between
-1 and +1 standard deviations from the mean or between -2 and +2 standard deviations from the mean?
Why?
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https://openstax.org/r/stock-return
2. Would an investor be likely to prefer a stock that has a smaller standard deviation for annual stock returns
or one with a larger standard deviation for annual stock returns? Why?
Portfolio Weights
Click to view content (https://openstax.org/r/port-weights)
3. What are the reasons for calculating portfolio weights? What useful information does this provide to the
investor?
4. What are the advantages and disadvantages of the equal weighting approach and the market cap
weighting approach for portfolio allocation strategy?
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	Chapter 13 Statistical Analysis in Finance
	Key Terms
	CFA Institute
	Multiple Choice
	Review Questions
	Problems
	Video Activity