*exact*value of the coupons in today's paper?"

Furthermore, I wondered about which coupons the image refers to? I assumed that it refers to the contents of special inserts such as SmartSource™ Magazine, RedPlum.com™, and P&G BrandSaver. I call these inserts *coupon circulars*.

There are other interesting statistical questions that you can ask about Sunday coupons, such as "What is the distribution of the values of coupons? Are $0.50 coupons the most prevalent or are larger values the norm?"

##### Data Collection

To research these questions, I collected data about the coupons in coupon circulars for 12 Sundays in early 2011. I recorded three kinds of coupons:

- General coupons that can be redeemed at any store and that have a certain value (such as $0.50 or $1.00) when you buy one or more specified items.
- Coupons that reward you with a free item when you buy a certain number of like items at the regular price. These "buy one, get one" (BOGO) coupons do not have a fixed value, but depend on the store price of the item. The coupon is valid up to some specified maximum price.
- The coupon circulars also include a small number of store-specific coupons such as for Red Lobster, Boston Market, or Baskin Robbins. Because these coupons are part of the coupon circulars, I counted them as well.

You can download the data and a SAS program that analyzes the data.

##### The Distribution of Coupon Values

As I recorded the values of coupons each week, it soon became clear that $1.00 coupons appear most frequently. The following graph (click to enlarge) shows the distribution of denominations of coupons in my Sunday newspaper:

A few statistics are apparent:

- $1.00 coupons are the most prevalent, followed by $0.50 and $2.00 coupons.
- There are 24 unique denominations of coupons in the data, but the six most common denominations account for more than 80% of the coupons.
- Denominations increase by $0.05 between $0.25 and $0.75, although it is relatively rare to find a $0.70 coupon! Between $0.75 and $1.50, denominations increase by $0.25. Between $1.50 and $4.00, denominations increase by $0.50. Higher denominations are a multiple of $1.00.

##### Claim versus Reality: How Much Can You Really Save?

The following table sums up the total values of coupons in three categories: general coupons, BOGO coupons, and the store coupons that are intermingled with the other two categories.

A few facts are apparent:

- Week 9 is an outlier in that the claimed value ($208) is much, much, less than the actual value of the coupons. Did the paper make a typographical error? Did they intend to print "Up to $280 in coupons" or "Up to $308 in coupons"? Another alternative: there were two coupon circulars that week, one with $204 worth of coupons and the other with $115. Did someone fail to tabulate the coupons in the second circular?
- Week 8 is the only week in which the total value of the coupons did not meet or exceed the advertised claim.
- For two weeks (Weeks 3, and 10), the "claimed value" is about the same as the values of coupons in the General category.
- For most weeks (Weeks 1, 2, 4, 6, 7, and 12), the "claimed value" is exceeded by the sum of the values of coupons in the General and BOGO categories. That is, you do not need to redeem store-specific coupons in order to meet or exceed the claimed value.

##### Predicting the Actual Coupons Values from the Advertised Claim

The previous graph indicates a linear relationship between the actual and the claimed values of the coupons. It is interesting to try to use linear regression to predict the total value of the coupons based on the claimed (starburst) values. However, the graph also shows two observations that have high-leverage values for the regression (Weeks 1 and 10) and one value that is an outlier (Week 9). Consequently, you might decide to use the ROBUSTREG procedure to fit a robust regression line to the data. (This is carried out in the downloadable program for least trimmed squares (LTS) regression.)

The LTS regression model (shown in the following graph) states that the expected value of the Sunday coupons is about $11 more than the advertised claim: predicted = $10.87 + claim. In other words, if the newspaper advertises "up to $181 in coupons," the model predicts $192 in coupons.

##### Summary

In summary:- The value of coupons in my local Sunday newspaper over a 12-week period varied from $43 in one week to over $400 another week.
- The most frequent coupon denomination is $1.00.
- Although the newspaper claims a specific value for the coupons each Sunday, it is not clear how that number is obtained.
- The claimed value usually differs from the actual value by less than $20, but the claimed value twice underestimated the actual value by $70 or more.
- According to a robust regression model, the expected value of the actual coupon value is $11 more than the claimed value.