[Article] Can AI Boost Cracker Barrel’s Stagnant Sales?

Cracker Barrel (NASDAQ: CBRL), the restaurant and gift shop chain, is one of many successful businesses whose sales in constant dollars have stagnated after a period of rapid, even exponential growth. Historically, Cracker Barrel relied almost exclusively on roadside billboards and word of mouth for its dramatic growth from a small business in Lebanon, TN in 1969 to sales of about $2.4 billion in 2004. Adjusted by the US Consumer Price Index (CPI), Cracker Barrel’s 2022 sales were only $2.1 billion in 2004 dollars.

This decline in real sales has occurred despite or even because of a sustained attempt to diversify away from the historically successful billboard advertising into other media.

Is it possible to use modern Artificial Intelligence (AI) technologies such as ChatGPT or other less well known methods to boost Cracker Barrel or other businesses now stagnant sales?

This article examines Cracker Barrel’s publicly reported expenditures on billboards and other media using Artificial Intelligence based Math Recognition to evaluate the effectiveness of Cracker Barrel’s strategies since 2004, generally supporting the company’s focus on other media, now about two thirds of ad expenditures, although also indicating the effectiveness of current advertising methods whether billboards or “other media” is small compared to the spectacular results in the 1980’s and 1990s.

Indeed the public financial data suggests Cracker Barrel may be fighting a negative effect from growing US Gross Domestic Product (GDP), needing to advertise more and more simply to sustain, never mind boost, company revenues.

Cracker Barrel’s publicly available information on the company’s marketing and advertising expenditures is very limited, falling into two aggregate categories: billboards and “other media” which includes radio, TV, Internet, and other non-billboard methods. The data is annual for the entire United States. There is no geographical breakdown, association with specific marketing or advertising campaigns in time, location, or other important factors.

It is almost certain our Math Recognition would construct more complex models from such details and likely make more reliable predictions from more detailed data which Cracker Barrel undoubtedly has in the internal accounting systems than the relatively simple model constructed from the public data.

What is Math Recognition?

Math recognition is a key part of scientific and engineering practice. It is identifying and recognizing the mathematical formulas and objects describing the data. The Bell Curve often encountered in grading in high school or college is a well known and relatively easy to identify mathematical formula. In addition to grades it describes the frequency of heights in adults of the same sex and many other common measured quantities.

However, scientists, engineers, and financial analysts often struggle to find a viable mathematical formula or formulas for real world data. Without such a formula it is impossible to make predictions or optimize the output of systems.

Our Math Recognizer has a large and growing database of known mathematics including obscure and difficult to identify mathematical objects: special functions, differential equations, etc.. The Math Recognizer uses AI methods to recognize these mathematical formulas in data.

What is the mathematical relationship if any between a company’s advertising expenditures on competing methods and actual sales?

Photo of John Wanamaker who owned large department store in Philadelphia.  Famous quote on advertising is attributed to him.
John Wanamaker (1838-1922)

Half the money I spend on advertising is wasted; the trouble is, I don’t know which half.

John Wanamaker (1838-1922)

This is an OLD problem in business.

The Math Recognizer is able to identify a relatively simple mathematical model for the Cracker Barrel data combined with the US Gross Domestic Product (GDP) from the St Louis Federal reserve. The GDP is used as a proxy for the overall state of the economy, something the company cannot control. For example, in the revenues reported by Cracker Barrel we can easily see a sharp drop attributable to the 2020 COVID pandemic lockdowns, something clearly beyond the control of the company.

The model roughly “explains” about 86 percent of the variation in the data. This is a roughly correct interpretation of the R**2 or coefficient of determination Goodness of Fit (GoF) metric used in the analysis.

Once we have identified a model with good agreement with the data, we can optimize the output, meaning maximize the sales in this case, given a projected budget. This is actually somewhat disappointing in that the program recommends to spend the entire budget on the “other media,” non-billboard, category with a slight dip in sales due to the negative effect of GDP according to the model.

Budget in Units of $1K ($1,000), about $89.5 Million in 2022

Let’s consider a larger future budget of $150 million, an abrupt increase of about $60 million over 2022. In this case we see the expected sales in current dollars to jump from about $3.2 billion in 2022 to almost $4 billion, an increase of about $800 million per year.

Budget in Units of $1K ($1,000), about $89.5 Million in 2022, new budget of $150 Million
Budget in Units of $1K ($1,000), about $89.5 Million in 2022, new budget of $150 Million

For this increase to pay for itself, the additional $800 million in sales must cost no more than $740 million dollars — a profit margin of about 7.5% (seven point five percent). That is a good profit margin for a restaurant. Restaurants often average only 3-5 percent profit margins.

Conclusion

A simple model based on broad financial numbers like these should not be taken very seriously although it may give some insights into a company. Indeed this simple analysis appears to support Cracker Barrel’s existing policies of diversifying advertising out of billboards, despite the disappointing results with inflation adjusted sales stagnant.

An in depth analysis of finely grained internal accounting data may be able to yield more detailed, reliable, and actionable results including a predictive model.

(C) 2023 by John F. McGowan, Ph.D.

About Me

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech).

[Video] Is the “conspiracy theory” label stopping your from reaching your desired audience?

Is the “conspiracy theory” label stopping you from reaching your desired audience?

Has the thought-stopping pejorative phrase “conspiracy theory” ever caused serious problems discussing certain ideas or even objective facts with your audience, friends, family, or colleagues? Today even the simple word “conspiracy” is increasingly used this way. How can you overcome the thought stopping effect of “conspiracy theory” and expand your audience?

“Conspiracy theory” labelers frequently use superficially plausible arguments backed up by no data or a single or few examples. For example: “conspiracies will always or almost always fail because someone would have talked,” citing for example the exposure of the Watergate burglary failure and the downfall of Richard Nixon. This would for example suggest unsolved murders by conspiracies, such as “gang,” “Mafia” or “organized crime” killings are exceptionally rare or nonexistent.

What does the data actually tell us about the frequency and success rate of conspiracies?

For data and more details see:

Text Article URL: http://wordpress.jmcgowan.com/wp/conspiracies-what-does-the-data-show/

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(C) 2023 by John F. McGowan, Ph.D.

About Me

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech).

[Video] How to Sell a Product That Does Not Work

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About Me

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech).

How to Tell Scientifically if Advertising Works Explainer Video

AdEvaluator Sales Projections
How to Tell Scientifically if Advertising Works Explainer Video

[Slide 1]

“Half the money I spend on advertising is wasted; the trouble is I don’t know which half.”

This popular quote sums up the problem with advertising.

[Slide 2]

There are many advertising choices today including not advertising, relying on word of mouth and other “organic” growth. Is the advertising working?

[Slide 3]

Proxy measures such as link clicks can be highly misleading. A bad advertisement can get many clicks, even likes but reduce sales by making the product look bad in an entertaining way.

[Animation Enter]

[Wait 2 seconds]

[Slide 4]

Did the advertising increase sales and profits? This requires analysis of the product sales and advertising expenses from your accounting program such as QuickBooks. Raw sales reports are often difficult to interpret unless the boost in sales is extremely large such as doubling sales. Sales are random like flipping a coin. This means a small but profitable increase such as twenty percent is often difficult to distinguish from chance alone.

[Slide 5]

Statistical analysis and computer simulation of a business can give a quantitative, PREDICTIVE answer. We can measure the fraction of days with zero, one, two, or more unit sales with advertising — the green bars in the plot shown — and without advertising, the blue bars.

[Slide 6]

With these fractions, we can simulate the business with and without advertising.

The bar chart shows the results for one thousand simulations of a year of business operations. Because sales are random like flipping a coin, there will be variations in profit from simulation to simulation due to chance alone.

The horizontal axis shows the change in profits in the simulation compared to the actual sales without advertising. The height of the bars shows the FRACTION of the simulations with the change in profits on the horizontal axis.

The blue bars are the fractions for one-thousand simulations without advertising.

[Animation Enter]

The green bars are the fractions for one-thousand simulations with advertising.

[Animation Enter]

The vertical red bar shows the average change in profits over ALL the simulations WITH THE ADVERTISING.

There is ALWAYS an increased risk from the fixed cost of the advertising — $500 per month, $6,000 per year in this example. The green bars in the lower left corner show the increased risk with advertising compared to the blue bars without advertising.

If the advertising campaign increases profits on average and we can afford the increased risk, we should continue the advertising.

[Slide 7]

This analysis was performed with Mathematical Software’s AdEvaluator Free Open Source Software. AdEvaluator works for sales data where there is a SINGLE change in the business, a new advertising campaign.

Our AdEvaluator Pro software for which we will charge money will evaluate cases with multiple changes such as a price change and a new advertising campaign overlapping.

[Slide 8]

AdEvaluator is available on our web site:

mathematical-software.com

[Web Site Animation Entrance]

Click on the Downloads TAB for our Downloads page.

[Web Site Animation Exit]

[Download Links Animation Entrance]

AdEvaluator can be downloaded from GitHub or as a ZIP file directly from the downloads page on our web site.

[Download Links Animation Exit]

Or scan this QR code to go to the Downloads page.

This is John F. McGowan, Ph.D., CEO of Mathematical Software. I have many years experience solving problems using mathematics and mathematical software including work for Apple, HP Labs, and NASA. I can be reached at ceo@mathematical-software.com

###

Technical Article: http://wordpress.jmcgowan.com/wp/how-to-tell-scientifically-if-advertising-boosts-sales-and-profits/

(C) 2019 by John F. McGowan, Ph.D.

About Me

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech).

How to Tell Scientifically if Advertising Boosts Sales and Profits

Annual Profit Projections

“Half the money I spend on advertising is wasted; the trouble is I don’t know which half.”

John Wanamaker, (attributed)
US department store merchant (1838 – 1922)

Between $190 billion and $270 billion is spent on advertising in the United States each year (depending on source). It is often hard to tell whether the advertising boosts sales and profits. This is caused by the unpredictability of individual sales and in many cases the other changes in the business and business environment occurring in addition to the advertising. In technical terms, the evaluation of the effect of advertising on sales and profits is often a multidimensional problem.

Many common metrics such as the number of views, click through rates (CTR), and others do not directly measure the change in sales or profits. For example, an embarrassing or controversial video can generate large numbers of views, shares, and even likes on a social media site and yet cause a sizable fall in sales and profits.

Because individual sales are unpredictable, it is often difficult or impossible to tell whether a change in sales is caused by advertising, simply due to chance alone or some combination of advertising and luck.

The plot below shows the simulated daily sales for a product or service with a price of $90.00 per unit. Initially, the business has no advertising, relying on word of mouth and other methods to acquire and retain customers. During this “no advertising” period, an average of three units are sold per day. The business then contracts with an advertising service such as Facebook, Google AdWords, Yelp, etc. During this “advertising” period, an average of three and one half units are sold per day.

Daily Sales
Daily Sales

The raw daily sales data is impossible to interpret. Even looking at the thirty day moving average of daily sales (the black line), it is far from clear that the advertising campaign is boosting sales.

Taking the average daily sales over the “no advertising” period, the first six months, and over the “advertising” period (the blue line), the average daily sales was higher during the advertising period.

Is the increase in sales due to the advertising or random chance or some combination of the two causes? There is always a possibility that the sales increase is simply due to chance. How much confidence can we have that the increase in sales is due to the advertising and not chance?

This is where statistical methods such as Student’s T test, Welch’s T test, mathematical modeling and computer simulations are needed. These methods compute the effectiveness of the advertising in quantitative terms. These quantitative measures can be converted to estimates of future sales and profits, risks and potential rewards, in dollar terms.

Measuring the Difference Between Two Random Data Sets

In most cases, individual sales are random events like the outcome of flipping a coin. Telling whether sales data with and without advertising is the same is similar to evaluating whether two coins have the same chances of heads and tails. A “fair” coin is a coin with an equal chance of giving a head or a tail when flipped. An “unfair” coin might have a three fourths chance of giving a head and only a one quarter chance of giving a tail when flipped.

If I flip each coin once, I cannot tell the difference between the fair coin and the unfair coin. If I flip the two coins ten times, on average I will get five heads from the fair coin and seven and one half (seven or eight) heads from the unfair coin. It is still hard to tell the difference. With one hundred times, the fair coin will average fifty heads and the unfair coin seventy-five heads. There is still a small chance that the seventy five heads came from a fair coin.

The T statistics used in Student’s T test (Student was a pseudonym used by statistician William Sealy Gossett) and Welch’s T test, a more advanced T test, are measures of the difference in a statistical sense between two random data sets, such as the outcome of flipping coins one hundred times. The larger the T statistic the more different the two random data sets in a statistical sense.

William Sealy Gossett (Student)
William Sealy Gossett (Student)

Student’s T test and Welch’s T test convert the T statistics into probabilities that the difference between the two data sets (the “no advertising” and “advertising” sales data in our case) is due to chance. Student’s T test and Welch’s T test are included in Excel and many other financial and statistical programs.

The plot below is a histogram (bar chart) of the number of simulations with a Welch’s T statistic value. In these simulations, the advertising has no effect on the daily sales (or profits). The advertising has no effect is the null hypothesis in the language of classical statistics.

Welch's T Statistics
Welch’s T Statistics

Welch was able to derive a mathematical formula for the expected distribution — shape of this histogram — using calculus. The mathematical formula could then be evaluated quickly with pencil and paper or an adding machine, the best available technology of his time (the 1940’s).

To derive his formula using calculus, Welch had to assume that the data had a Bell Curve (Normal or Gaussian) distribution. This is at best only approximately true for the sales data above. The distribution of daily sales in the simulated data is actually the Poisson distribution. The Poisson distribution is a better model of sales data and approximates the Bell Curve as the number of sales gets larger. This is why Welch’s T test is often approximately valid for sales data.

Many methods and tests in classical statistics assume a Bell Curve (Normal or Gaussian) distribution and are often approximately correct for real data that is not Bell Curve data. We can compute better, more reliable results with computer simulations using the actual or empirical probability distributions — shown below.

Welch's T Statistic has Bell Curve Shape
Welch’s T Statistic has Bell Curve Shape

More precisely, naming one data set the reference data and the other data set the test data, the T test computes the probability that the test data is due to a chance variation in the process that produced the reference data set. In the advertising example above, the “no advertising” period sales data is the reference data and the “advertising” sales data is the test data. Roughly this probability is the fraction of simulations in the Welch’s T statistic histogram that have a T statistic larger (or smaller for a negative T statistic) than the measured T statistic for the actual data. This probability is known as a p-value, a widely used statistic pioneered by Ronald Fisher.

Ronald Aylmer Fisher
Ronald Aylmer Fisher at the start of his career

The p-value has some obvious drawbacks for a business evaluating the effectiveness of advertising. At best it only tells us the probability that the advertising boosted sales or profits, not how large the boost was nor the risks. Even if on average the advertising boosts sales, what is the risk the advertising will fail or the sales increase will be too small to recover the cost of the advertising?

Fisher worked for Rothamsted Experimental Station in the United Kingdom where he wanted to know whether new breeds of crops, fertilizers, or other new agricultural methods increased yields. His friend and colleague Gossett worked for the Guinness beer company where he was working on improving yields and quality of beer. In both cases, they wanted to know whether a change in the process had a positive effect, not the size of the effect. Without modern computers — using only pencil and paper and adding machines — it was not practical to perform simulations as we can easily today.

Welch’s T statistic has a value of -3.28 for the above sales data. This is in fact lower than nearly all the simulations in the histogram. It is very unlikely the boost in sales is due to chance. The p-value from Welch’s T test for the advertising data above — computed using Welch’s mathematical formula — is only 0.001 (one tenth of one percent). Thus it is very likely the boost in sales is caused by the advertising and not random chance. Note that this does not tell us if the size of the boost, whether the advertising is cost effective, or the risk of the investment.

Sales and Profit Projections Using Computer Simulations

We can do much better than Student’s T test and Welch’s T test by using computer simulations based on the empirical probabilities of sales from the reference data — the “no advertising” period sales data. The simulations use random number generators to simulate the random nature of individual sales.

In these simulations, we simulate one year of business operations with advertising many times — one-thousand in the examples shown — using the frequency of sales from the period with advertising. We also simulate one year of business operations without the advertising, using the frequency of sales from the period without advertising in the sales data.

Frequency of Daily Sales in Both Periods
Frequency of Daily Sales in Both Periods

We compute the annual change in the profit relative to the corresponding period — with or without advertising — in the sales data for each simulated year of business operations.

Annual Profit Projections
Annual Profit Projections

The simulations show that we have an average expected increase in profit of $5,977.66 over one year (our annual advertising cost is $6,000.00). It also shows that despite this there is a risk of a decrease in profits, some greater than the possible decreases with no advertising.

A business needs to know both the risks — how much money might be lost in a worst case — and the rewards — the average and best possible returns on the advertising investment.

Since sales are a random process like flipping a coin or throwing dice, there is a risk of a decline in profits or actual losses without the advertising. The question is whether the risk with advertising is greater, smaller, or the same. This is known as differential risk.

The Problem with p-values

This is a concrete example of the problem with p-values for evaluating the effectiveness of advertising. In this case, the advertising increases the average daily sales from 100 units per day to 101 units per day. Each unit costs one dollar (a candy bar for example).

P-VALUE SHOWS BOOST IN SALES
P-VALUE SHOWS BOOST IN SALES

The p-value from Welch’s T test is 0.007 (seven tenths of one percent). The advertising is almost certainly effective but the boost in sales is much less than the cost of the advertising:

Profit Projections
Profit Projections

The average expected decline in profits over the simulations is $5,128.84.

The p-value is not a good estimate of the potential risks and rewards of investing in advertising. Sales and profit projections from computer simulations based on a mathematical model derived from the reference sales data are a better (not perfect) estimate of the risks and rewards.

Multidimensional Sales Data

The above examples are simple cases where the only change is the addition of the advertising. There are no price changes, other advertising or marketing expenses, or other changes in business or economic conditions. There are no seasonal effects in the sales.

Student’s T test, Welch’s T test, and many other statistical tests are designed and valid only for simple controlled cases such as this where there is only one change between the reference and test data. These tests were well suited to data collected at the Rothamsted Experimental Station, Guinness breweries, and similar operations.

Modern businesses purchasing advertising from Facebook, other social media services, and modern media providers (e.g. the New York Times) face more complex conditions with many possible input variables (unit price, weather, unemployment rate, multiple advertising services, etc.) changing frequently or continuously.

For these, financial analysts need to extract predictive multidimensional mathematical models from the data and then perform similar simulations to evaluate the effect of advertising on sales and profits.

Example Software

The AdEvaluator™ software used in these examples is free open source software (FOSS) developed using the Anaconda Python 3 distribution. It is available under the GNU General Public License Version 3.

AdEvaluator can be downloaded here.

Disclaimer

AdEvaluator™ is designed for cases with a single product or service with a constant unit price during both periods. AdEvaluator™ needs a reference period without the new advertising and a test period with the new advertising campaign. The new advertising campaign should be the only significant change between the two periods. AdEvaluator™ also assumes that the probability of the daily sales is independent and identically distributed during each period. This is not true in all cases. Exercise your professional business judgement whether the results of the simulations are applicable to your business.

This program comes with ABSOLUTELY NO WARRANTY; for details use -license option at the command line or select Help | License… in the graphical user interface (GUI). This is free software, and you are welcome to redistribute it under certain conditions.

We are developing a professional version of AdEvaluator™ for multidimensional cases. This version uses our Math Recognition™ technology to automatically identify good multidimensional mathematical models.

The Math Recognition™ technology is applicable to many types of data, not just sales and advertising data. It can for example be applied to complex biological systems such as the blood coagulation system which causes heart attacks and strokes when it fails. According the US Centers for Disease Control (CDC) about 633,000 people died from heart attacks and 140,000 from strokes in 2016.

Conclusion

It is often difficult to evaluate whether advertising is boosting sales and profits, despite the ready availability of sales and profit data for most businesses. This is caused by the unpredictable nature of individual sales and frequently by the complex multidimensional business environment where price changes, economic downturns and upturns, the weather, and other factors combine with the advertising to produce a confusing picture.

In simple cases with a single change, the addition of the new advertising, Student’s T test, Welch’s T test and other methods from classical statistics can help evaluate the effect of the advertising on sales and profits. These statistical tests can detect an effect but provide no clear estimate of the magnitude of the effect on sales and profits and the financial risks and rewards.

Sales and profit projections based on computer simulations using the empirical probability of sales from the actual sales data can provide quantitative estimates of the effect on sales and profits, including estimates of the financial risks (chance of losing money) and the financial rewards (typical and best case profits).

(C) 2018 by John F. McGowan, Ph.D.

About Me

John F. McGowan, Ph.D. solves problems using mathematics and mathematical software, including developing gesture recognition for touch devices, video compression and speech recognition technologies. He has extensive experience developing software in C, C++, MATLAB, Python, Visual Basic and many other programming languages. He has been a Visiting Scholar at HP Labs developing computer vision algorithms and software for mobile devices. He has worked as a contractor at NASA Ames Research Center involved in the research and development of image and video processing algorithms and technology. He has published articles on the origin and evolution of life, the exploration of Mars (anticipating the discovery of methane on Mars), and cheap access to space. He has a Ph.D. in physics from the University of Illinois at Urbana-Champaign and a B.S. in physics from the California Institute of Technology (Caltech).