Definitions Deconstructed

Mean Girls, Highway Medians, and Pie à la Mode

S. G. Lacey

Definition:
The mean of a data set is found by adding all numbers, then dividing by the number of values in the set. The median is the middle value, when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set. [REF]

Deconstruction:

Mean:

If you’re not familiar with the 2004 movie “Mean Girls”, you’re forgiven.

While this flick has turned into a cult classic, for a specific segment of impressionable teenagers at the time, it’s generally disappeared from public consciousness. The first screenplay ever written by Tina Fey, the story is an adaptation of the book “Queen Bees and Wannabees”, a self-help offering penned by author Rosalind Wiseman, in 2002.

For the purposes of this mathematical analysis, there are a few important details which can be extracted from the movie’s plot. Based on clues in the film, the main character, and the trio of antagonists, are in 11th grade, at a private school in the suburbs of Chicago. This information provides a nice proxy demographic: American females roughly 16.5 years old.

Fortunately, growth for girls tends to level off around age 16. All 5 of the main actresses who played students at the fictional North Shore High School were older than this waypoint at the time of filming, so we can use their stated adult height as a proxy. Here is the breakdown.

Lindsey Lohan (1986) = 5’5”
Rachel McAdams (1978) = 5’4”
Lacey Chabert (1982) = 5’2”
Amanda Seyfreid (1985) = 5’3”
Lizzy Caplan (1982) = 5’4”

Despite the small data set, converting this data to inches, summing up these height values, then dividing by 5, yields an average, or arithmetic mean, height of 63.6 inches.

One important part of statistical analysis is making sure to take into account all the variables. Ignoring the obviously skewed whiteness in the movie, we can use a more representative sample. The ethnicity breakdown reported at Evanston Public High School, in the suburbs of Chicago, the largest, and closest, proxy to the fictional venue, is as follows.

White = 46%
African American = 26%
Hispanic = 19%
Asian = 6%

Checking a few private schools in the region reveals a similar racial distribution. Per the recent 2020 U.S. Census, aside from a roughly 10% higher African America contingent, and subsequent reduction in Whites, the other ratios are very close. Since this distribution isn’t drastically different than the national race breakdown, that should provide a reasonable starting point for this analysis.

A multitude of online sources report the height and weight of girls throughout the United States at various ages. We will use the most recent Centers for Disease Control survey, which was last conducted in 2002, just 2 years before the release of “Mean Girls”.

This data finds an identical 64.2” average height for women, at ages 17 through 19, so height seems to be pretty stable during this period. Weigh is a different story, increasing annually, but that’s a risky subject to broach. As it turns out, the selection of actresses to play the students is fairly representative, at just 0.6” too short, all other demographic anomalies aside.

Ironically, Tina Fey portrays math teacher Ms. Norbury in “Mean Girls”, and is instrumental in the Mathletes Club, which plays a key role in the plot. Maybe she sensed her story would be used to provide context on statistical terms in the future.

Median:
If you’re not familiar with a highway median, you must be good at dozing off on road trips.

This term refers to the open space between traffic traveling in each direction on a road. This feature, be it a tall retaining wall, grassy expanse, or raised concrete platform, helps provide spacing between heavy, sometimes uncontrollable, objects, moving at 150 mph relative velocity.

Highway median technology, like nearly every element of society, has seen significant engineering advancements over the past half century. The primary goal of a median is to prevent cars, which lose control, from crossing over into oncoming traffic.

The Federal Highway Administration reports that 8% of deaths on divided highways are due to head-on collisions. Median barriers are critical in this regard, as they offer a 97% reduction in such crashes.

You’re likely familiar with the various types of structures installed on medians to prevent such vehicle cross-over. Traditional metal guard rails, commonly used on the edge of roadways, as well as the middle. Heavy grey concrete, or water-filled, orange plastic, Jersey barriers, named after the state where they are most common. Flexible beam walls, which restrict vision when stagnant, but are easy to see through while moving. And the newest technology, tightly strung cables, whose flexibility imparts less damaging to vehicles, and makes them easier to repair.

Government standards require that medians should be at least 36 feet, and preferably 60 feet, wide, in rural areas, and 10 feet wide, with a mandatory physical barrier, in urban and mountainous zones. Beyond these rough guidelines, there’s a lot of freedom for local interpretation of highway median design.

The United States Interstate Highway System was born in the 1950s, the brainchild of President Eisenhower. The Federal Aid Highway Act of 1956 provided the funding and guidance for a complete, interconnected, system of high speed, divided, roadways, to improve countrywide means of automobile transport.

The original format laid out back then, which took 35 years, and $115 billion dollars, to complete, now serves at the backbone for national vehicle traffic. The foundational structure laid out in this novel doctrine, for construction methods, road on/off protocols, and signage, still dictates many elements of today’s major thoroughfares.

The Interstate Highway System has a total length of 48,440 miles. Even more impressively, over a quarter of all vehicle miles occur on this core network of linked roads. These reliable routes are critical for transport of goods via truck, and serve as a critical means of evacuation during natural disasters, with the ability for the roadway on both sides of the median to accommodate travel in the same direction.

Much can be gleaned from U.S. Interstate route numbers. Odd values denote north to south roadways, while even monikers are associated with generally east to west directionality.

70 of the 100 available two-digit numbers are used in the current system, which covers just the continental United States. Three-digit highway values are used to identify spurs, connectors, and beltways, which are commonly found around cities; we will ignore these short stretches for the purpose of this simplified analysis.

The only even 10’s road options not utilized are I50 and I60. I2, I4, I5, I8, are the lone single digits used. 5 route numbers are duplicated, though separated by at least one state, do avoid any navigational confusion.

The longest connected road, at over 3,000 miles, is I90, which spans from Boston, MA to Seattle, WA. The shortest official highway, at just under 13 miles, is located completely within North Carolina, making its interstate signature a misnomer.

Considering all these details, what is the median highway number. Since this statistical term simply identifies the middle number of a given data set, making this determination is fairly straightforward. Any guesses?

It turns out that the median number in the U.S. Interstate Highway System is I66.

Not to be confused with U.S. Route 66, one of the most infamous roads in the country, and subject of the famous song, I66 covers a 75-mile stretch from Fort Royal, VA to Washington, DC, west to east, as denoted by the even number.

Representing the only Interstate highway heading west from the nation’s capital, I66 is often quite congested, with high toll rates. Considering the location, it’s not surprising that for portions of its length, the Washington Metro line runs in the median. That should stop any cross-over accidents.

Mode:
If you’re not familiar with the term “à la mode”, that’s just unfortunate.

Everyone likes pie. And everyone likes warm pie, with cold ice cream on top, even more. This treatment is referred to as “à la mode” in the United States, taking a French phrase, which means “fashionably in style”, and butchering it, as only Americans can do.

Regardless, this dessert is an informative, and tasty, topic to explore.

Image you’re at Thanksgiving, with all manner of relatives. As a teenager, you are responsible for helping with the massive pile of dinner dishes, so are unfortunately the last to reach the dessert table. Flexing the pruned, wrinkly, skin of your fingers, you peruse the sweet spread which has been laid out.

There are 8 pies available, offering up a wide range of options, in shape, color, crust, and most importantly, filling. In the 15th year on earth, you’ve learned most of the tricks related to sweet selection. 3 of the pies are in the telltale flimsy metal tins, which identify grocery store products. Those are definitely ones to avoid, considering the wealth of homemade offerings available.

Each pie has been cut into eight, nearly identical, slices. No doubt a result of the precise work by your OCD cousin. This allows the damage done to each delectable tart to be quickly assessed.

You know from the mound of dishes you just worked through, there are 13 individuals at this gathering. Most are at least ancillary family. All are apparently still hungry.

Looking down at the various pies, it’s immediately clear that 23 slices have been extracted. Understandable. May as well fill your plate, then get a spot in front the TV, for the 2nd half of the football game.

3 of the round discs have just a single slice removed from them. Lemon meringue, the characteristic white peaks topping the yellow custard below. Chocolate, the graham cracker crust filled with dark brown pudding, then garnished with dollops of whipped cream. And, of course, the traditional mincemeat offering. This item is cooked by one person, and consumed by one person. Your widowed uncle. Most don’t even consider this swill a desert.

Next, you spot a trio of berry filled treats, each with lattice top crust, to reveal the moist fruit below. Blueberry, denoted by tiny purple orbs, cherry, a dark red slurry, and strawberry rhubarb, the blend combining to a pinkish hue.

The first two are obviously store bought, and each have two slices removed, leaving a Pac-Man-esque shape. The third is a family staple, the sweetness of the strawberries balancing the tartness of the rhubarb. As such, only a half-circle remains, in what was clearly the largest initial ring. It must have been a good year in the garden, since your aunt only uses fresh fruit. This could be a good choice.

Next comes a Thanksgiving icon, the pumpkin pie. Less than half of this staple remains, just three burnt orange slices, sitting in the white ceramic dish. You reach out and place a finger on the edge of the pan, which is cold to the touch. Not what you’re looking for.

However, there is a clear winner in this group. At the far end of the table, in a weathered Pyrex dish, is a lone remaining slice of pie. Even from several feet away, you can see the perfection contained within this wedge.

The flakey golden crust, clearly ladened with butter. The fruity center, large wedge chunks, with a perfect combination of crunch and tenderness. But the topping is the coupe-de-gras, organic balls of brown sugar and cinnamon, condensed into tasty morels.

A homemade Dutch apple pie. Your grandmother’s specialty, as it was for her grandmother. Lucky there’s any pieces left.

This is clearly the most popular pie by far, as it has been for the entirety of your memory banks at these Thanksgiving events. Understandable.

Moving ahead with excited anticipation, you scoop up one of the remaining colored-glass dessert plates, green in this case, with a fluted swirl design. Using long fingers, you shovel the last piece of Dutch apple pie on the saucer. Amazingly, it’s still warm to the touch. Perfect.

On the corner of the table is an ornate pewter urn, with hammered floral pattern on the exterior, and two braid handles of entwined vines. This glorious vessel sits in a wide glass bowl. Large frozen cubes, suspended in icy water, keep the metal tub, and precious cargo housed within, chilled.

Moving forward, you pick up the scooper, and extract a generous helping from the tub, the tiny flecks of vanilla bean visible in the ice cream. The texture is ideal, toeing the line between rock hard and fully compliant. As the cold orb hits the hot pasty, the melting starts, creating a glorious, creamy, slurry underneath.

Now that you’ve scored the most popular dessert item, with mandatory à la mode accompaniment, you can relax. Looking down, you see there’s still a little room left on the plate. May as well toss on another slice of pie. Plus, another à la mode ice cream scoop garnish, of course.

Whether you’re mulling over your next tablet movie selection, cruising down the highway in the family minivan, or eating a hand pie and milkshake recently acquired at the roadside convenient store, just remember the following.

Mean = Math, Median = Middle, & Mode = Most

Details:

All the obscure details you might want to learn about the “Mean Girls” movie. [REF]
U.S. Census 2020 nationwide ethnicity data. [REF]
CDC data for girls in America broken down by age. [REF]
Explanation of lives saved due to highway median usage. [REF]
Details on the modern median cable barrier. [REF]
Complete list of U.S. Interstate highway numbers, lengths, and locations. [REF]
Background on the word origin of “à la mode” as ice cream accompaniment. [REF]
A breakdown of American’s Thanksgiving pie preferences. [REF]
Collection of Thanksgiving dessert recipes, in case you get hungry, and motivated. [REF]