Chapter 1
The Art and Science of Business Statistics
In This Chapter
Looking at the key properties of data
Understanding probability’s role in business
Sampling distributions
Drawing conclusions based on results
This chapter provides a brief introduction to the concepts that are covered throughout the book. I introduce several important techniques that allow you to measure and analyze the statistical properties of real-world variables, such as stock prices, interest rates, corporate profits, and so on.
Statistical analysis is widely used in all business disciplines. For example, marketing researchers analyze consumer spending patterns in order to properly plan new advertising campaigns. Organizations use management consulting to determine how efficiently resources are being used. Manufacturers use quality control methods to ensure the consistency of the products they are producing. These types of business applications and many others are heavily based on statistical analysis.
Financial institutions use statistics for a wide variety of applications. For example, a pension fund may use statistics to identify the types of securities that it should hold in its investment portfolio. A hedge fund may use statistics to identify profitable trading opportunities. An investment bank may forecast the future state of the economy in order to determine which new assets it should hold in its own portfolio.
Whereas statistics is a quantitative discipline, the ultimate objective of statistical analysis is to explain real-world events. This means that in addition to the rigorous application of statistical methods, there is always a great deal of room for judgment. As a result, you can think of statistical analysis as both a science and an art; the art comes from choosing the appropriate statistical technique for a given situation and correctly interpreting the results.
Representing the Key Properties of Data
The word data refers to a collection of quantitative (numerical) or qualitative (non-numerical) values. Quantitative data may consist of prices, profits, sales, or any variable that can be measured on a numerical scale. Qualitative data may consist of colors, brand names, geographic locations, and so on. Most of the data encountered in business applications are quantitative.
The word data is actually the plural of datum; datum refers to a single value, while data refers to a collection of values.
You can analyze data with graphical techniques or numerical measures. I explore both options in the following sections.
Analyzing data with graphs
Graphs are a visual representation of a data set, making it easy to see patterns and other details. Deciding which type of graph to use depends on the type of data you’re trying to analyze. Here are some of the more common types of graphs used in business statistics:
Histograms: A histogram shows the distribution of data among different intervals or categories, using a series of vertical bars.
Line graphs: A line graph shows how a variable changes over time.
Pie charts: A pie chart shows how data is distributed between different categories, illustrated as a series of slices taken from a pie.
Scatter plots (scatter diagrams): A scatter plot shows the relationship between two variables as a series of points. The pattern of the points indicates how closely related the two variables are.
Histograms
You can use a histogram with either quantitative or qualitative data. It’s designed to show how a variable is distributed among different categories. For example, suppose that a marketing firm surveys 100 consumers to determine their favorite color. The responses are
|
Red:
|
23
|
|
Blue:
|
44
|
|
Yellow:
|
12
|
|
Green:
|
21
|
The results can be illustrated with a histogram, with each color in a single category. The heights of the bars indicate the number of responses for each color, making it easy to see which colors are the most popular (see Figure 1-1).
Illustration by Wiley, Composition Services Graphics
Figure 1-1: A histogram for preferred colors.
Based on the histogram, you can see at a glance that blue is the most popular choice, while yellow is the least popular choice.
Line graphs
You can use a line graph with quantitative data. It shows the values of a variable over a given interval of time. For example, Figure 1-2 shows the daily price of gold between April 14, 2013 and June 2, 2013:
Illustration by Wiley, Composition Services Graphics
Figure 1-2: A line graph of gold prices.
With a line graph, it’s easy to see trends or patterns in a data set. In this example, the price of gold rose steadily throughout late April into mid-May before falling back in late May and then recovering somewhat at the end of the month. These types of graphs may be used by investors to identify which assets are likely to rise in the future based on their past performance.
Pie charts
Use a pie chart with quantitative or qualitative data to show the distribution of the data among different categories. For example, suppose that a chain of coffee shops wants to analyze its sales by coffee style. The styles that the chain sells are French Roast, Breakfast Blend, Brazilian Rainforest, Jamaica Blue Mountain, and Espresso. Figure 1-3 shows the proportion of sales for each style.
Illustration by Wiley, Composition Services Graphics
Figure 1-3: A pie chart for coffee sales.
The chart shows that Espresso is the chain’s best-selling style, while Jamaica Blue Mountain accounts for the smallest percentage of the chain’s sales.
Scatter plots
A scatter plot is designed to show the relationship between two quantitative variables. For example, Figure 1-4 shows the relationship between a corporation’s sales and profits over the past 20 years.
Each point on the scatter plot represents profit and sales for a single year. The pattern of the points shows that higher levels of sales tend to be matched by higher levels of profits, and vice versa. This is called a positive trend in the data.
Illustration by Wiley, Composition Services Graphics
Figure 1-4: A scatter plot showing sales and profits.
Defining properties and relationships with numerical measures
A numerical measure is a value that describes a key property of a data set. For example, to determine whether the residents of one city tend to be older than the residents in another city, you can compute and compare the average or mean age of the residents of each city.
Some of the most important properties of interest in a data set are the center of the data and the spread among the observations. I describe these properties in the following sections.
Finding the center of the data
To identify the center of a data set, you use measures that are known as measures of central tendency; the most important of these are the mean, median, and mode.
The mean represents the average value in a data set, while the median represents the midpoint. The median is a value that separates the data into two equal halves; half of the elements in the data set are less than the median, and the remaining half are greater than the median. The mode is the most commonly occurring value in the data set.
The mean is the most widely used measure of central tendency, but it can give deceptive results if the data contain any unusually large or small values, known as outliers. In this case, the median provides a more representative measure of the center of the data. For example, median household income is usually reported by government agencies instead of mean household income. This is because mean household income is inflated by the presence of a small number of extremely wealthy households. As a result, median household income is thought to be a better measure of how standards of living are changing over time.
The mode can be used for either quantitative or qualitative data. For example, it could be used to determine the most common number of years of education among the employees of a firm. It could also be used to determine the most popular flavor sold by a soft drink manufacturer.
Measuring the spread of the data
Measures of dispersion identify how spread out a data set is, relative to the center. This provides a way of determining if the members of a data set tend to be very close to each other or if they tend to be widely scattered. Some of the most important measures of dispersion are
Variance
Standard deviation
Percentiles
Quartiles
Interquartile range (IQR)
The variance is a measure of the average squared difference between the elements of a data set and the mean. The larger the variance, the more “spread out” the data is. Variance is often used as a measure of risk in business applications; for example, it can be used to show how much uncertainty there is over the returns on a stock.
The standard deviation is the square root of the variance, and is more commonly used than the variance (since the variance is expressed in squared units). For example, the variance of a series of gas prices is measured in squared dollars, which is difficult to interpret. The corresponding standard deviation is measured in dollars, which is much more intuitively clear.
Percentiles divide a data set into 100 equal parts, each consisting of 1 percent of the total. For example, if a student’s score on a standardized exam is in the 80th percentile, then the student outscored 80 percent of the other students who took the exam. A quartile is a special type of percentile; it divides a data set into four equal parts, each consisting of 25 percent of the total. The first quartile is the 25th percentile of a data set, the second quartile is the 50th percentile, and the third quartile is the 75th percentile. The interquartile range identifies the middle 50 percent of the observations in a data set; it equals the difference between the third and the first quartiles.
Determining the relationship between two variables
For some applications, you need to understand the relationship between two variables. For example, if an investor wants to understand the risk of a portfolio of stocks, it’s essential to properly measure how closely the returns on the stocks track each other. You can determine the relationship between two variables with two measures of association: covariance and correlation.
Covariance is used to measure the tendency for two variables to rise above their means or fall below their means at the same time. For example, suppose that a bioengineering company finds that increasing research and development expenditures typically leads to an increase in the development of new patents. In this case, R&D spending and new patents would have a positive covariance. If the same company finds that rising labor costs typically reduce corporate profits, then labor costs and profits would have a negative covariance. If the company finds that profits are not related to the average daily temperature, then these two variables will have a covariance that is very close to zero.
Correlation is a closely related measure. It’s defined as a value between –1 and 1, so interpreting the correlation is easier than the covariance. For example, a correlation of 0.9 between two variables would indicate a very strong positive relationship, whereas a correlation of 0.2 would indicate a fairly weak but positive relationship. A correlation of –0.8 would indicate a very strong negative relationship; a correlation of –0.3 would indicate a weak negative relationship. A correlation of 0 would show that two variables are independent (that is, unrelated).
Probability: The Foundation of All Statistical Analysis
Probability theory provides a mathematical framework for measuring uncertainty. This area is important for business applications since all results from the field of statistics are ultimately based on probability theory. Understanding probability theory provides fundamental insights into all the statistical methods used in this book.
Probability is heavily based on the notion of sets. A set is a collection of objects. These objects may be numbers, colors, flavors, and so on. This chapter focuses on sets of numbers that may represent prices, rates of return, and so forth. Several mathematical operations may be applied to sets — union, intersection, and complement, for example.
The union of two sets is a new set that contains all the elements in the original two sets. The intersection of two sets is a set that contains only the elements contained in both of the two original sets (if any.) The complement of a set is a set containing elements that are not in the original set. For example, the complement of the set of black cards in a standard deck is the set containing all red cards.
Probability theory is based on a model of how random outcomes are generated, known as a random experiment. Outcomes are generated in such a way that all possible outcomes are known in advance, but the actual outcome isn’t known.
The following rules help you determine the probability of specific outcomes occurring:
The addition rule
The multiplication rule
The complement rule
You use the addition rule to determine the probability of a union of two sets. The multiplication rule is used to determine the probability of an intersection of two sets. The complement rule is used to identify the probability that the outcome of a random experiment will not be an element in a specified set.
Random variables
A random variable assigns numerical values to the outcomes of a random experiment. For example, when you flip a coin twice, you’re performing a random experiment, since:
All possible outcomes are known in advance
The actual outcome isn’t known in advance
The experiment consists of two trials. On each trial, the outcome must be a “head” or a “tail.”
Assume that a random variable X is defined as the number of “heads” that turn up during the course of this experiment. X assigns values to the outcomes of this experiment as follows:
|
Outcome
|
X
|
|
{TT}
|
0
|
|
{HT, TH}
|
1
|
|
{HH}
|
2
|
T represents a tail on a single flip
H represents a head on a single flip
TT represents two consecutive tails
HT represents a head followed by a tail
TH represents a tail followed by a head
HH represents two consecutive heads
X assigns a value of 0 to the outcome TT because no heads turned up. X assigns a value of 1 to both HT and TH because one head turned up in each case. Similarly, X assigns a value of 2 to HH because two heads turned up.
Probability distributions
A probability distribution is a formula or a table used to assign probabilities to each possible value of a random variable X. A probability distribution may be discrete, which means that X can assume one of a finite (countable) number of values, or continuous, in which case X can assume one of an infinite (uncountable) number of different values.
For the coin-flipping experiment from the previous section, the probability distribution of X could be a simple table that shows the probability of each possible value of X, written as P(X):
|
X
|
P(X)
|
|
0
|
0.25
|
|
1
|
0.50
|
|
2
|
0.25
|
The probability that X = 0 (that no heads turn up) equals 0.25 because this experiment has four equally likely outcomes: HH, HT, TH, and TT and in only one of those cases will there be no heads. You compute the other probabilities in a similar manner.
Discrete probability distributions
Several specialized discrete probability distributions are useful for specific applications. For business applications, three frequently used discrete distributions are:
Binomial
Geometric
Poisson
You use the binomial distribution to compute probabilities for a process where only one of two possible outcomes may occur on each trial. The geometric distribution is related to the binomial distribution; you use the geometric distribution to determine the probability that a specified number of trials will take place before the first success occurs. You can use the Poisson distribution to measure the probability that a given number of events will occur during a given time frame.
Continuous probability distributions
Many continuous distributions may be used for business applications; two of the most widely used are:
Uniform
Normal
The uniform distribution is useful because it represents variables that are evenly distributed over a given interval. For example, if the length of time until the next defective part arrives on an assembly line is equally likely to be any value between one and ten minutes, then you may use the uniform distribution to compute probabilities for the time until the next defective part arrives.
The normal distribution is useful for a wide array of applications in many disciplines. In business applications, variables such as stock returns are often assumed to follow the normal distribution. The normal distribution is characterized by a bell-shaped curve, and areas under this curve represent probabilities. The bell-shaped curve is shown in Figure 1-5.
Illustration by Wiley, Composition Services Graphics
Figure 1-5: The bell-shaped curve of the normal distribution.
The normal distribution has many convenient statistical properties that make it a popular choice for statistical modeling. One of these properties is known as symmetry, the idea that the probabilities of values below the mean are matched by the probabilities of values that are equally far above the mean.
Using Sampling Techniques and Sampling Distributions
Sampling is a branch of statistics in which the properties of a population are estimated from samples. A population is a collection of data that someone has an interest in studying. A sample is a selection of data randomly chosen from a population.
For example, if a university is interested in analyzing the distribution of grade point averages (GPAs) among its MBA students, the population of interest would be the GPAs of every MBA student at the university; a sample would consist of the GPAs of a set of randomly chosen MBA students.
Several approaches can be used for choosing samples; a sample is a subset of the underlying population.
A statistic is a summary measure of a sample, while a parameter is a summary measure of a population. The properties of a statistic can be determined with a sampling distribution — a special type of probability distribution that describes the properties of a statistic.
The central limit theorem (CLT) gives the conditions under which the mean of a sample follows the normal distribution:
The underlying population is normally distributed.
The sample size is “large” (at least 30).
A detailed discussion of the central limit theorem can be found at http://en.wikipedia.org/wiki/Central_limit_theorem.
Statistical Inference: Drawing Conclusions from Data
Statistical inference refers to the process of drawing conclusions about a population from randomly chosen samples. In the following sections, I discuss two techniques used for statistical inference: confidence intervals and hypothesis testing.
Confidence intervals
A confidence interval is a range of values that’s expected to contain the value of a population parameter with a specified level of confidence (such as 90 percent, 95 percent, 99 percent, and so on). For example, you can construct a confidence interval for the population mean by following these steps:
1. Estimate the value of the population mean by calculating the mean of a randomly chosen sample (known as the sample mean).
2. Calculate the lower limit of the confidence interval by subtracting a margin of error from the sample mean.
3. Calculate the upper limit of the confidence interval by adding the same margin of error to the sample mean.
The margin of error depends on the size of the sample used to construct the confidence interval, whether the population standard deviation is known, and the level of confidence chosen.
The resulting interval is known as a confidence interval. A confidence interval is constructed with a specified level of probability. For example, suppose you draw a sample of stocks from a portfolio, and you construct a 95 percent confidence interval for the mean return of the stocks in the entire portfolio:
(lower limit, upper limit) = (0.02, 0.08)
The returns on the entire portfolio are the population of interest. The mean return in each sample drawn is an estimate of the population mean. The sample mean will be slightly different each time a new sample is drawn, as will the confidence interval. If this process is repeated 100 times, 95 of the resulting confidence intervals will contain the true population mean.
Hypothesis testing
Hypothesis testing is a procedure for using sample data to draw conclusions about the characteristics of the underlying population.
The procedure begins with a statement, known as the null hypothesis. The null hypothesis is assumed to be true unless strong evidence against it is found. An alternative hypothesis — the result accepted if the null hypothesis is rejected — is also stated.
You construct a test statistic, and you compare it with a critical value (or values) to determine whether the null hypothesis should be rejected. The specific test statistic and critical value(s) depend on which population parameter is being tested, the size of the sample being used, and other factors.
If the test statistic is too extreme (for example, it’s too large compared with the critical value[s]) the null hypothesis is rejected in favor of the alternative hypothesis; otherwise, the null hypothesis is not rejected.
If the null hypothesis isn’t rejected, this doesn’t necessarily mean that it’s true; it simply means that there is not enough evidence to justify rejecting it.
Hypothesis testing is a general procedure and can be used to draw conclusions about many features of a population, such as its mean, variance, standard deviation, and so on.
Simple regression analysis
Regression analysis uses sample data to estimate the strength and direction of the relationship between two or more variables. Simple regression analysis estimates the relationship between a dependent variable (Y) and a single independent variable (X).
For example, suppose you’re interested in analyzing the relationship between the annual returns of the Standard & Poor’s (S&P) 500 Index and the annual returns of Apple stock. You can assume that the returns of Apple stock are related to the returns to the S&P 500 because the index is a reflection of the overall strength of the economy. Therefore, the returns of Apple stock are the dependent variable (Y) and the returns of the S&P 500 are the independent variable (X). You can use regression analysis to measure the numerical relationship between the S&P 500 and Apple stock.
Simple regression analysis is based on the assumption that a linear relationship occurs between X and Y. A linear relationship takes this form:
Y is the dependent variable, X is the independent variable, m is the slope, and b is the intercept.
The slope tells you how much Y changes due to a specific change in X; the intercept tells you what the value of Y would be if X had a value of zero.
The goal of regression analysis is to find a line that best fits or explains the data. The population regression line is written as follows:
Yi = β0 + β1Xi + εi
In this equation, Yi is the dependent variable, Xi is the independent variable, β0 is the intercept, β1 is the slope, and εi is an error term.
A sample regression line, estimated from the data, is written as follows:
Here, 
is the estimated value of Yi, 
is the estimated value of β0, and 
is the estimated value of β1 and is the independent variable.
The sample regression line shows the estimated relationship between Y and X; you can use this relationship to determine how much Y changes due to a given change in X. You can also use it to forecast future values of Y based on assumed values of X.
After estimating the sample regression line, the results are subjected to a series of tests to determine whether the equation is valid. If the equation isn’t valid, you reject the results and try a new model.
Multiple regression analysis
With multiple regression analysis, you estimate the relationship between a dependent variable (Y) and two or more independent variables (X1, X2, and so on).
For example, suppose that Y represents annual salaries (in thousands of dollars) at a corporation. A researcher has reason to believe that the salaries at this corporation depend mainly on the number of years of job experience and the number of years of graduate education for each employee. The researcher may test this idea by running a regression in which salary is the dependent variable (Y) and job experience and graduate education are the independent variables (X1 and X2, respectively.) The population regression equation in this case would be written as
The sample regression line would be
Using multiple regression analysis introduces several additional complications compared with simple regression analysis, but you can use it for a much wider range of applications than simple regression analysis.
Forecasting techniques
You can forecast the future values of a variable, using one of several types of models. One approach to forecasting is time series models. A time series is a set of data that consists of the values of a single variable observed at different points in time. For example, the daily price of Microsoft stock taken from the past ten years is a time series.
Time series forecasting involves using past values of a variable to forecast future values.
Some forecasting techniques include:
Trend models
Moving average models
Exponential moving average models
A trend model is used to estimate the value of a variable as it evolves over time. For example, suppose annual data is used to estimate a trend model that explains the behavior of gasoline prices over time. The price is currently $3.50 per gallon, and you determine that on average, gasoline prices rise by $0.10 per year. A simple trend model that expresses this information would be written as:
In this equation, Yt represents the estimated gas price at time t, where t represents a specific year. (t = 0 represents the present time.) The term 3.50 indicates the current price of gasoline; 0.10t indicates that the price of gasoline rises by $0.10 per year. The term 
is known as an “error term”; this reflects random fluctuations in the price of gasoline over time.
A moving average model shows that the value of a variable evolves over time based on its most recent values. For example, if the price of gasoline over the past three years was:
2010 $3.25
2011 $3.32
2012 $3.42
A three-period moving average model would produce an estimated value of ($3.25 + $3.32 + $3.42) / 3 = $3.33 for 2013.
An exponential weighted average model is closely related to a moving average model. The difference is that with an exponential weighted average, older observations aren’t given the same “weight” as newer observations. The calculation of an exponential weighted average is more complex, but may give more realistic results.
The appropriate choice of model depends on the properties of the particular time series being used.
to a standard normal random variable
The Remember icon points to information that’s especially important to remember for exam purposes.
The Tip icon presents information like a memory acronym or some other aid to understanding or remembering material.
Visit