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

Statistical analysis is widely used in all business disciplines. For example, marketing researchers analyze consumer spending patterns 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 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.

In this chapter, I provide a brief introduction to the concepts that are covered throughout the book. I introduce several important techniques that help you to measure and analyze the statistical properties of real-world variables, such as stock prices, interest rates, corporate profits, and so on.

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 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).

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 August 1, 2023 and September 29, 2023.

With a line graph, it's easy to see trends or patterns in a data set. 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.

FIGURE 1-2: A line graph of gold prices.

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.

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 relationship between the two variables.

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.

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 halves; half of the elements in the data set are less than or equal to the median, and the remaining half are greater than or equal to 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 may be used to determine the most common number of years of education among the employees of a firm. It may 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 (because 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...