(and why your locker combination is actually a permutation)
Welcome to the third installment of my Cheat Sheet for Stats. Be sure to check out Part 1 and Part 2.
Permutations and combinations are useful to someone interested in determining the total number of items from a set or group. This is especially helpful in probability when calculating a denominator and/or numerator.
The difference between a permutation and a combination is simple to understand – if you pay close attention to how the items/objects/people are chosen (and ignore semantics). In this post I’ll give you definitions, formulas, and examples of both permutations and combinations. But first, I’ll discuss the Fundamental Counting Principle and factorials.
The Fundamental Counting Principle
Also known as the multiplication counting rule, this principle says to multiply all possible events together to find the total number of outcomes.
A simple example starts with packing for a vacation. Say you pack 4 shirts, 3 pairs of pants, and 2 pairs of shoes. How many possible outfits can you make? (Assume they all match, or you are 5 years old and don’t give a flip.)
The fundamental counting principle says you now have:
4 * 3 * 2 = 24 possible outfits
Here’s another example. Let’s say your company requires a 5-value verification code consisting of 3 numerical values and 2 alphabetic values (in that order and case sensitive). How many possible verification codes can be produced?
Sometimes it helps to see what is going on:
And visualize the values in each position:
There are 10 total digits to consider (0 – 9) and 26 letters in the alphabet – 52 if case-sensitive. The trick is to multiply to find the total possible outcomes:
= 2,704,000 different verification codes
And what if the requirement changed to 3 numbers and 2 letters (same order), but no repeats? We’d have to take away the number of options for each digit/letter as they are used:
= 1,909,440 different verification codes
There is a little more math involved if you can put these values in any order and I won’t cover that in this post.
At first glance, a factorial looks like a very excited number. For instance, 5! might appear to be yelling, “FIVE!” (Silly teacher joke – works better in person.) The exclamation point is actually an operator telling us to multiply that number by all integers less than that number down to 1.
Permutations apply the Fundamental Counting Principle to determine the number of ways you can arrange members of a group. The permutation formula calculates the number of arrangements of n objects taken r at a time:
For example, let’s say you and 29 other people are in the running for 3 distinct prizes. Your names are in a hat and prizes are only given to the first, second, and third names drawn (the best prize being first). The number of ways 30 people can take first, second, and third prize is called a permutation. In a permutation, the order in which the items or people are arranged “matters”. (And by matters, you could say the order is noted, or apparent.)
For the prize example, you can calculate this using the formula for permutations:
And this goes back to the fundamental counting principle since a portion of the numerator cancels with the denominator:
Simplifying the expression to 30*29*28 = 24,360 ways 3 individuals can be awarded first, second, and third prize from a group of 30 in a random drawing. If we were merely drawing 3 names all at once with no difference in prizes, it would NOT be considered a permutation.
Luckily you really don’t need to know the formula to calculate a permutation. The Excel function for permutations is PERMUT:
Note: There is another Excel function for permutations with repetitions – that one is PERMUTATIONA. For this example, you would use that if we drew names for the three prizes and each time the name was returned to the hat, making it possible for the same person to win all 3 times.
Now suppose you and 29 other individuals are in the running for 3 prizes, all with the same value. Your names are in a hat and all three names are drawn at once. Because no order or arrangement is involved, this type of counting technique is called a combination. The combination formula also calculates n objects taken r at a time:
For the newest version of our prize example, we are taking 3 names from the hat at one time and there is no difference between prizes. Here is that calculation:
Once the 27! in the numerator and denominator cancel, we are left with the 24,360 in the numerator, but still divide by 3! (which is 3*2 = 6):
Which results in only 4060 possible combinations.
The Excel function for combinations is COMBIN:
A Locker Combination is Actually a Permutation
Now consider locker combinations. Let’s assume a typical dial lock (Right, left, right) in which there are 39 numbers on the dial and your code has 3 numbers. Does order matter? Absolutely! If you try to open the lock using your 3 number code but in a different order, the locker will not open. So how many possible codes does this locker have?
If numbers couldn’t repeat, we’d have P(39,3) = 54,834 different codes (or what we call “combinations”). But if numbers could repeat, there are 39*39*39 = 59,319 possible codes – to include repeatable values, apply the PERMUTATIONA function in Excel.
Based on what you just learned, can you spot the difference between a combination and permutation? Bonus points if you can calculate the result. (Answers at the end of the post.)
A board of directors consists of 13 people. In how many ways can a chief executive officer, a director, and a treasurer be selected?
How many ways can a jury of 12 people be selected from a group of 40 people?
A GM from a restaurant chain must select 8 restaurants from 14 for a promotional program. How many different ways can this selection be done?
At Waffle House hash browns can be ordered 18 different ways. How possible orders can be made by choosing only 3 of the 18?
A locker can have a 4-digit code. How many different codes can we have if there are 25 different numbers and numbers cannot repeat in any given code?
Permutation. P(13,3) = 1716
Combination. C(40,12) = 5,586,853,480 order isn’t important here
Combination. C(14,8) = 3003
Combination. C(18,3) = 816
Permutation. P(25,4) = 303,600 (Repeating numbers within a code would give 390,625 different codes.)
Welcome to Part 2 of my Cheat Sheet for Stats series, sure to make you the life of the party! In case you missed it, Part 1 covered measures of center and spread. In Part 2 we’ll dive into measures of position and location within a dataset – specifically how to calculate, apply, and interpret them.
Measures of Position
Comparing month to month sales volume or current housing prices within your neighborhood are easy to make without involving additional calculations. However, sometimes metrics aren’t exactly comparable in absolute terms. For example, sales volume from 1989 versus 2015. Or current housing prices in Atlanta versus San Francisco. So we typically “normalize” these metrics by adjusting for inflation or cost of living so we can compare relative values.
Another way of comparing considers relative position. Percentiles, quantiles, and z-scores measure the location of a value in relation to other values in a dataset. Once you know the price of a house in Atlanta relative to the rest of the Atlanta housing market, and the price of a house in San Francisco relative to the rest of the San Francisco housing market, you can see how those two housing prices compare to each other.
Percentiles describe the position of a data point relative to the rest of the dataset using a percent. That’s the percent of the rest of the dataset that falls below the particular data point.
Finding an Unknown Percentile
Example: Pretend you’re the 2nd oldest out of a group of 20 people. To find your percentile, count the number of people YOUNGER than you (18) and divide by the total number of people:
18/20 = 0.9, or the 90th percentile
(Note: By calculating percentile using the “percent less than” formula above, the computation disallows a 100th percentile (which is not a valid percentile); however, it forces the lowest value to the 0 percentile (which is also invalid). Another popular variation on this calculation computes the percent less than AND equal to: There are 19 people at your age or younger, so 19/20 is 95th percentile. This version of the calculation fixes the “0 percentile” problem but allows the possibility for an invalid 100th percentile. Both versions of the percentile calculation are acceptable and there is no universally “correct” computation.)
Finding a Given Percentile
Example: You want to determine what height marks the 58th percentile of the 20 people. You’ll multiply the 20 people by 58%, or 20*(0.58) = 11.6. This number is called the index. Round to the nearest whole number 12 – therefore, the 12th height in your group of 20 people roughly falls at the 58th percentile.
Quantiles break the dataset up into an equal number of n pieces. They signal reference points (or positions) in the dataset to which individual data points can be compared. Specific examples of quantiles are deciles (slicing the data into 10 equal pieces), terciles (3 equal pieces), and quartiles (4 equal pieces). I’ll elaborate using quartiles, but all quantiles follow the same logic:
Quartiles break the dataset up into 4 equal pieces so data points can be compared within the dataset relative to the four quarters of data.
I love watching football, which is broken up into four quarters. If I turn the TV on and the game has already started, I know how far the game has progressed in time relative to the quarter displayed on the screen.
Lower Quartile (Q1) – Roughly the 25th percentile. 25% of the data falls below this point and 75% lies above.
Median (Q2) – Roughly the 50th percentile, marks the position of the middle value where half fall above and half fall below this point
Upper Quartile (Q3) – Roughly the 75th percentile. 75% of the data falls below this point and 25% lies above.
Quartiles are found using these steps:
Arrange the data from least to greatest.
Find the median, the middle number. (If there are two numbers in the middle, find the average of the two.)
Using the median as the midpoint, the data is now split in half. Now find the middle value in the bottom half of values (this is Q1).
Lastly, find the middle number of the top half of values (this is Q3).
Example: The following values represent the lifespan of a sample of animals. What values break these lifespans into quartiles?
Often quartiles are displayed visually with a box-and-whisker plot:
A z-score indicates how many standard deviations a value falls above or below the mean (average). A value with a positive z-score lies above the mean while a value with a negative z-score falls below the mean. Z-scores are a way of standardizing values in order to compare them using relative position.
To calculate a z-score, subtract the average of the population/dataset (mean) from the data point (observation), then divide by the standard deviation of the population/dataset:
In 1927, Babe Ruth made history hitting 60 home runs in one Major League Baseball season. Only four people have been able to break Ruth’s record (though Mark McGwire and Sammy Sosa have broken that record 2 and 3 times, respectively). In 2001, Barry Bonds set the most recent record, hitting 73 home runs in a single season.
But just how does Barry’s home run performance compare to Babe’s? Many outside factors such as bat quality and pitcher performance could impact the number of homeruns hit by a MLB player. So how did these athletes compare to their peers of the time?
The 1927 league home run average was 7.2 home runs with a standard deviation of 9.7 home runs. While the 2001 league average was an astounding 21.4 home runs, with a standard deviation of 13.2 home runs.
To properly compare these heavy hitters, we need to determine how they preformed relative to the peers of their era by standardizing their absolute HR numbers into z-scores:
While both athletes displayed phenomenal performances, Babe Ruth could still argue his status of home run champion when comparing in relative terms.
If you automatically think of a bell-shaped, normal curve when you hear “z-score”, you’re not alone. That’s a common connection because of the way we initially introduce z-scores in stats courses:
But z-scores can apply to ANY distribution because they are a way to compare data values using relative position. That is, z-scores “standardize” the data values from absolute to relative metrics.
Statisticsare values or calculations that attempt to summarize a sampleof data. If you’re talking about a population of data, you’re usually dealing with parameters. (Easy to remember: Statistics and Samples start with the letter S, and Populations and Parameters start with the letter P.)
And if you know all there is to know about a population, then you wouldn’t be concerned with statistics. However, we almost never know about an entire population of anything, which is why we focus on studying samples and the statistics that describe those samples.
Statistics and summary statistics, as I mentioned, help us summarize a sample of data so that we can wrap our mind around the entire dataset. Some statistics are better than others, and which statistic you choose to summarize a dataset depends entirely upon the type of data you’re working with and your end goals.
Here, I will briefly discuss the different types, definitions, calculations, and uses of basic summary statistics known as “Measures of Central Tendency” and “Measures of Variation.”
A weighted mean is helpful when all values in the dataset do not contribute to the average in the same way. For example, course grades are weighted based on what the type of assignment (test, quiz, project, etc). A test might count more towards your final grade than a homework assignment.
Weighted means are also applied when calculating the expected value of an outcome, such as in gambling and actuarial science. An expected value in gambling is what you’d expect to lose (because you’re gonna lose) over the long-run (many, many trials). To calculate this, a probability is applied to each possible outcome and then multiplied by the value of the outcome — pretty much the same way you calculated your grade back in high school:
Note: Weighted Means and Expected Values will reappear in a later post discussing discrete probability distributions.
The middle number of the dataset is only one statistic used the summarize the dataset — the spread of the data is also extremely important to understanding what is happening within your data. Measures of variation tell us how the data varies from end to end and/or within the middle of the dataset and because of this, some measures of spread help in identifying outliers.
For the following examples I took a non-random sample of animal lifespans, put them in order from shortest to longest lifespan. You’ll see the median is 11 years:
The range only considers the variation from the smallest to largest value in the dataset. Here, the animal’s lifespans range 30 years (or, 35 – 5 years). Unfortunately, range doesn’t give us much information about the variation within the datset.
Interquartile range, or IQR, is the range of the middle 50% of the dataset. In this situation, it represents the middle 50% of animal lifespans.
To find the IQR, the values in the dataset must first be ordered least to greatest with the median identified (as I did above). Since the median cuts the dataset in half, we then look for the middle value of the bottom half of the dataset (that is, the middle lifespan between the kangaroo and the cat). That represents the lower quartile, or Q1. Then look for the middle value of the top of the dataset, (that is, the middle value between the dog and the elephant). The second number represents the upper quartile, or Q3.
The interquartile range is found by subtracting Q3 – Q1, or 17.5 – 8.5 = 9 which tells me that the middle 50% of these animals’ lifespans range by 9 years. Unfortuantely, IQR only gives information about the spread of the middle of the dataset.
You can calculate the IQR in Excel using the QUARTILE function to find the first quartile and the third quartile, then subtracting like in the above example.
One common method for calculating an outlier threshold in a
dataset depends on the IQR. Once the IQR is calculated, it is then multiplied
by 1.5. Find the low outlier threshold by subtracting the IQR*1.5 from Q1.
Find upper outlier threshold by adding the IQR*1.5 to Q3. This
is the method used to show outliers in box-and-whisker plots.
Standard deviation measures the typical departure, or distance, of each data point from the mean. (Recall, the “mean” is just the average.) So ultimately, the calculation for standard deviation relies on the value of the mean.
The formula below specifically calculates the standard deviation of the sample. I wouldn’t worry too much about using this formula; however, understanding how standard deviation is calculated might help you understand what it calculates, so I’ll walk you through it:
In the numerator within the parentheses, the mean is subtracted from each data point. As you can imagine, the mean is the center of the data so half of the resulting differences are negative (or 0), and the other half are positive (or 0). (Adding these differences up will always result in 0.)
Those differences are then squared (making all values positive).
The Greek uppercase letter Sigma in front of the parentheses means, “sum” — so all the squared differences are added up.
Divide that value by the samples size (n) MINUS 1. Your resulting answer is the variance of the dataset.
Since the variance is a squared measure, take the square root at the end. Now you have the standard deviation.
An easier way, of course, is to use Excel’s standard deviation functions for samples: STDEV or STDEV.S
Just like the mean, standard deviation is easily affected by outlying (extreme) values.
the MEDIAN is to INTERQUARTILE RANGE as MEAN is to STANDARD DEVIATION
– The Stats Ninja
Several approaches are used to calculating the outlier
threshold using standard deviations – which method is employed typically
depends on the use case. Many software packages default to 3 standard
deviations. If the outlier threshold is calculated using IQR (from above), 2.67
standard deviations mark that boundary.
The year was 2002. It was the first time I ever stood in front of a classroom of “grown-ups”. The students didn’t know me or care who I was — a TA at LSU covering a College Algebra class. The topic was logarithms. Specifically, an introduction to logs as the inverse of an exponent. I may have been slightly older than the median age of those students and I was terrified, nervous, and profusely sweating.
Up until that moment, I thought deep content knowledge was the secret sauce of teaching. But in the 17 years of experiences that followed, I’ve learned how much MORE there is to teaching than merely knowing your stuff. Student buy-in is the key to student engagement and, ultimately, student learning.
As a corporate trainer I’ve found adults are no different from kids in how they learn and how they engage. It all depends on the trainer’s ability to read the room and adapt as needed.
I compiled the list below after 17 years of total instruction including college algebra and statistics, high school math and AP Statistics, and corporate training for data analysts. I’m sure I will update this list in the future, but at this point, reflecting on my own feedback and observing other trainers, these are the top points I find trainers miss.
Please note this list is not exhaustive and assumes you follow the basic tenants of instruction such as: knowing your audience, knowing your content, preparing x 10, taking breaks every 60 – 75 minutes, beginning promptly after a break, being approachable, avoiding excessive talk and rabbit holes, Rule of 3, minimizing powerpoint, summarizing major points, etc. So here you go – 5 ways to improve student engagement.
1. Be Adaptable.
“It is not the strongest of the species that survives, nor the most intelligent. It is the one that is most adaptable to change.”
– Charles Darwin
I’ve entered a classroom to discover the WiFi down. Many times. There have been a few unannounced fire drills, some medical emergencies, that time the projector bulb blew — all derailing my meticulously-planned lesson. But the show must go on!
If something goes wrong, keep calm but think on your feet. Focus on keeping the students engaged first. So if the students have working laptops and the only problem is a projector (or your laptop), get them started. Walk around the room teaching the concepts you’d planned to teach from the front. Ask the students questions, have them come up with solutions.
Tech completely down? Ask yourself, “What is the goal of this class?” It’s not easy to teach a tech-driven class with a whiteboard, but it can be done (and here’s hoping the WiFi will be up soon). In fact, if you teach a workshop using a software or only technology, I would urge you to get in the habit of adding in low-tech activities for those “just in case” moments.
Pro Tip: Oftentimes you can lead the students to an a-ha moment or two, then request IT support at the next break.
According to The Oxford Review, adaptability in the workplace is related to one’s emotional intelligence and emotional resilience. And, of course, mindset.
Lastly, being adapatable also means being coachable. Everyone gets frustrated from negative evaluations/feedback at times. But try to step back and ask yourself if you could have improved the delivery. Making tweaks to your performance based on student feedback can help YOU in the long run. Being coachable does NOT mean you give up confidence. You are the professional, but all great professionals learn from feedback and reflection.
2. Don’t Fake it.
It’s okay to admit you don’t know the answer to a question. Saying, “I’ll find out and get back to you” is not a weakness. What’s not okay is making up an answer. “Fake it til you make it” is NOT a mantra of teaching. Especially when you have Google.
“When you tell a lie, you steal someone’s right to the truth.”
– Khaled Hosseini
If you’re worried about questions, I recommend giving everyone sticky notes at the beginning of the class. Encourage students to ask questions. If a question comes up that is not relevant to the topic at hand OR if you don’t know the exact answer, ask the student to write down their question. Create a space on the wall for participants to stick these questions up (some trainers call this a “parking lot”) and on breaks, take some time to research and answer the questions. I would strongly recommend you DON’T take class time to do said research.
And oh the mistakes I’ve made when teaching. Some embarrassing. It happens. And it is important to own those mistakes, especially if you can turn it into a “teaching moment.” For example, I once observed a new math teacher square a binomial incorrectly. A COMMON mistake among students. So instead of (x+2)^2 = (x+2)*(x+2), she (without thinking about it) squared both terms, making (x+2)^2 = x^2 + 4. Whoops. A big mistake in the math world, but really not a big deal when she stopped herself, realizing her mistake, and laughed about it. Then explained that was an example of what NOT to do.
Squaring a binomial incorrectly may not be your mistake, but you will make one (probably many). But to err is human. And there is plenty of research out there to suggest an HONEST teacher is a TRUSTWORTHY teacher. People LIKE honest teachers, especially when it comes to their own flaws.
3. Use People’s Names.
“Go the extra mile. It’s never crowded.”
Names are powerful. Dale Carnegie once said, “A person’s name is, to him or her, the sweetest and most important sound in any language.” When someone takes the time to learn and use your name, you feel important. Which means using a person’s name in conversation is the quickest way to connect with them on a personal level — and therefore promotes positive classroom engagement.
Generally, people also enjoy talking about themselves. Which is a great way to learn their name. On the very first day, after I introduce myself, I give participants the opportunity to introduce themselves and say a few words. You probably already do this. And I use this as an opportunity to learn their name — I write it down then say their name aloud (so they hear it AND to help me remember). Creating a blank seating chart ahead of time is always helpful – this way I can jot down the name and an interesting fact while they speak, creating a reference for later in the course.
My friend and colleague Ryan Nokes remembers names much better than I do, impressing his classes by learning every name immediately! After preliminary introductions he says each person’s name, first to last person, and states their name (without notes). And then Ryan does it again at the start of the next day. People enjoy hearing their own names and are pleased when you remember them later
I teach hands-on courses and encourage constant interaction. When calling on students, I use their first name, careful not to just point to them. When talking to them one on one, I use their name. And by the way, please use the name they gave you. NOT their government name. FYI, I cringe when people call me, “Annamarie.”
Moving around the room, when done correctly, increases participant engagement.
When I teach, I rarely sit down. Moving around the room allows me to interact with each student one-on-one and check for their understanding. This proximity also allows the more reluctant talker/questioner to ask their burning question when they know the entire class won’t hear them. And, dare I say it: Moving around the room keeps you in control.
This is why you hear grade school teachers say they never sit down. K-12 teachers use physical proximity to manage their classrooms. Being interested in each student’s learning promotes positive behaviors and keep students on task. In the same way, walking around keeps adults out of their inbox. And you won’t ever hear me criticize a training participant about their email/phone use in class (despite it being a bit irritating — I mean, you DID sign up to be here) because when they expect me to move their direction, they self monitor and correct these behaviors themselves, often apologizing.
Note: I do try to give every group/person an equal amount of “attention” without lingering anywhere too long.
Educational research also promotes student movement around the room. So when delivering instruction, I like to create activities that make students/groups visualize data by hand – on a white board or big 3M sticky poster. Or even a post-it. “Around the room” activities could also include giving other groups positive feedback or presenting a new discovery in their data.
5. Seat People with Similar Experience-Levels Together.
When dealing with heterogeneous groups of participants.
“We are more powerful when we empower each other.”
After years of hoopla over the concept of “tracking students”, this tip might surprise you. How many times have you heard someone say, “Pair a low with a high?” And, while this strategy could work in certain courses and situations, it is, overall, an outdated practice.
Imagine. You have a grasp of the basics of a particular data visualization tool and use it weekly. A colleague in the same course has only installed the software that morning. Your instructor teams you up so you can “help” your colleague. How does this make you feel? At first, it might feel rewarding — you know the answers! However, in many situations the person doing the “helping” ends up feeling like they didn’t grow in their domain while the person being helped can eventually feel inadequate and frustrated.
No matter how well we market a course (“beginner”, “advanced”, etc) there will always be a heterogeneous group of abilities when I walk in to start instruction. And this is the way it always is — K12 or corporate training. So I can either roll my eyes and teach the outline as prescribed, pacing the middle of all abilities, or I can help all learners by differentiating my instruction a bit.
I’m not talking about group projects here — I only mean seating participants of like abilities near each other for an improved user experience. Of course in a training situation, this arrangement has more to do with experience levels — of which they can self-sort. I generally ask students “new” to the software to sit in the front, and others to sit behind them. That’s all that is usually needed.
But let’s look at the origins of this thought: When used appropriately, “flexible grouping” — pairing and grouping students based on need — can aid student learning on both ends of the experience spectrum. This can be homogeneous or heterogeneous groups. And how you utilize it matters. If you must pair high/low, do it only for a short time. (Because I’ve had students ask me, “Am I the dumb one or the smart one?”)
In the long run, research suggests pairing/seating students with similar abilities/experience in a domain (or software) engages all students. And if done correctly, actually improves their learning experience and accelerates their growth. How? If you’re already moving around the room (see #4 above), then it should make sense that you can tailor your instruction much easier to pairs/groups of similar background knowledge than if they are scattered around the room. Think about it — when you are helping a group of participants who are relatively new to the topic or software, you can give a “you try” practice problem to enrich or even accelerate the other group to work on their own, and vice versa. Peer pairing/grouping on similar experience levels also encourages those students to develop a deeper understanding of the topic together, rather than the back-and-forth waiting that occurs when unlike abilities are grouped.
Personally, I mix up my delivery — some whole group instruction, some partner work, maybe an activity in a group, and solo work. Pairing them up encourages dialog about the concepts while they work through a challenge. Groups can offer multiple points of view. Solo work helps the student think through the problem on their own. Since my classes are always hands-on I incorporate the process of I do, we do, you do. But I do start with seating like experience levels together.
Being a teacher (or instructor, or coach) does require multiple skill sets including: entertainer, orchestra conductor, problem-solver, mind reader, therapist, referee, and cheerleader. However, promoting student engagement (teens and adults alike) goes beyond preparing a “fun lesson.” Student engagement results from student buy-in. And student buy-in results from the little things that create a positive atmosphere.
I’m going to add to this list over time. Do you have any suggestions on how you promote adult student engagement?
How do you get a group of 15 to 18-year-old students interested in data prep and analysis? Why, you take them to play laser tag, of course!
That’s right, on a cold January day I loaded up two buses of teens and piloted them to an adventure at our local Stars and Strikes. And this is no small feat — this particular trip developed out of months of planning, and after years proclaiming that I will never ever ever ever EVER coordinate my own field trip for high school kids. I mean, you should SEE the stack of paperwork. And the level of responsibility itself made me anxious.
So there I was, field trip money in one hand, clipboard in another: Imagine a caffeinated Tracy Flick. But thanks to the help of two parent chaperones and the AP Psychology teacher (Coach B), we ran the smoothest data-related field trip modern education has ever known.
What Does Laser Tag Have to do With Statistics?
Statistics textbooks are full of canned examples and squeaky clean data that often have no bearing on a students’ interests. For example, there is an oh-so-relatable exercise computing standard error for D-glucose contained in a sample of cockroach hindguts. In my experience I’ve learned when students can connect to the data, they are able to connect to the concept. We’re all like that, actually — to produce/collect our own data enables us to see what we otherwise would have missed.
(I can assure you confidence intervals constructed from D-glucose in coachroach hindguts did little for understanding standard error.)
The real world is made up of messy data. It’s full of unknowns, clerical errors, bias, unnecessary columns, confusing date formats, missing values; the list goes on. Laser Tag was suggested to me as a way to collect a “large” amount of data in a relatively short amount of time. And because of the size of the dataset, it required the student to input their own data — creating their own version of messy data complete with clerical errors. From there they’d have to make sense of the data, look for patterns, form hypotheses.
Students entered their data into a Google doc — you can find the complete data here.
Each partner team developed two questions for the data: One involving 1-variable analysis, another requiring bivariate analysis.
The duos then had to explore, clean, and analyze all 47 rows and 48 columns. At this point in the school year, students had been exposed to data up to about 50 rows, but never had they experienced “wide” data.
Analyses and presentations required a visualization, either using Excel or Tableau.
Playing the Games
Methodology: Each student was randomly assigned to a team using a random number generator. Teams of 5 played each other twice during the field trip. The teams were paired to play each other randomly. If, by chance, a team was chosen to play the same team twice, that choice would be ignored and another random selection would be made until a new team was chosen.
Before each game, I recorded which student wore which laser tag vest number. From the set-up room (see above picture), I could view which vest numbers were leading the fight and which team had the lead. It was entertaining. As the students (and Coach B — we needed one more player for even teams) finished their games, score cards were printed and I handed each student their own personal results. The words, “DON’T lose this” exited my lips often.
Upon our return to school (this only took a few hours, to the students’ dismay), results were already pouring the into the Google doc I’d set up ahead of time.
Teaching Tableau and Excel Skills
The AP Statistics exam is held every year in May, hosted by The College Board. On the exam, students are expected to use a graphing calculator but have no access to a computer or Google. Exactly the opposite of the real world.
Throughout the course, I taught all analysis first by hand, or using the TI-83/84. As students became proficient, I added time in the computer lab to teach basic skills using Excel and Tableau (assignments aligned to the curriculum while teaching skills in data analysis). It was my goal for students to have a general understanding of how to use these “real world” analytics tools while learning and applying AP Statistics curriculum.
After the field trip, we spent three days in the computer lab – ample time to work in Tableau and Excel with teacher guidance. Students spent time exploring the 48-column field trip dataset with both Excel and Tableau. They didn’t realize it, but by deciding which chart type to use for different variables, they were actually reviewing content from earlier in the year.
Most faculty members had never heard of Tableau. At lunch one day I sat down with Coach B to demonstrate Tableau’s interface with our field trip dataset.
“What question would you ask this set of data?” I asked.
“A back shot is a cheap shot. I wonder who is more likely to take a cheap shot, males or females?”
So I proceeded to pull up a comparison and used box-and-whiskers plots to look for outliers. Within seconds, a large outlier was staring back at us within the pool of male students:
“Ha. I wonder who that was.” – Coach B
“That’s YOU.” – Me
From there, I created a tongue-in-cheek competitive analysis from the data:
I’ve been teaching since 2004. Over the years, this was probably the most successful project I’ve seen come through my classroom. By “successful”, I’m talking the proportion of students who were able to walk outside of their comfort zone and into a challenging set of data, perform in-depth analyses, then communicate clear conclusions was much higher than in all previous years.
At the end of the year, after the AP Exam, after grades were all but inked on paper, students still talked excitedly about the project. I’d like to think it was the way I linked a fun activity to real-world analysis, though it most likely has to do with getting out of school for a few hours. Either way, they learned something valuable.
One student, Abby, gave me permission to share her work adding, “This is the project that tied it all together. This was the moment I ‘got’ statistics.”
Interestingly, students were less inclined to suggest the female outlier of 2776 shots was a clerical mistake (which it was). I found there were two camps: Students who didn’t want to hurt feelings, and students who think outliers in the wild need no investigation. Hmmm.
What I Learned
When you teach, you learn. Earlier I said the project was a success based on the students’ results. That’s only partially true; it was also a success because I grew as an educator. After years of playing by the rules I realized that sometimes you need to get outside your comfort zone. For me that was two-fold: 1) Sucking it up and planning a field trip and 2) Losing the old, tired TI-83 practice problems and teaching real-world analytics tools.
Note: Before proceeding, a great recap of probability concepts can be found here, written by Paul Rossman.
But First, Conditional Probability
When I teach conditional probability, I tell my students to pay close attention to the vertical line in the formula above. Whenever they see it, they must imagine the loud baritone behind-the-scenes announcer voice from Bill Nye saying, “GIVEN!”
This symbol | always indicates we assume the event that follows it has already occurred. The formula above, then, should be read: The probability event A will occur given event B has already occurred.
A simple example of conditional probability uses the ubiquitous deck of cards. From a standard deck of 52, what is the probability you draw an ace on the second draw if you know an ace has already been drawn (and left out of the deck) on the first draw?
Since a deck of 52 playing cards contains 4 aces, the probability of drawing the first ace is 4/52. But the probability of drawing an ace given the first card drawn was an ace is 3/51 — 3 aces left in the deck with 51 total cards remaining. Hence, conditional probability assumes another event has already taken place.
False Positives and False Negatives: What They’re Not
Tests are flawed.
According to MedicineNet, a rapid strep test from your doctor or urgent care has a 2% false positive rate. This means 2% of patients who do not actually have Group A streptococcus bacteria present in their mouth test positive for the bacteria. The rapid strep test also indicates a negative result in patients who do have the bacteria 5% of the time — a false negative.
Another way to look at it: The 2% “false positive” result indicates the test displays a true positive in 98% of patients. The 5% “false negative” result means the test displays a true negative in 95% of patients.
It’s common to hear these false positive/true positive results incorrectly interpreted. These rates do not mean the patient who tests positive for a rapid strep test has a 98% likelihood of having the bacteria and a 2% likelihood of not having it. And a negative result does not indicate one still has a 5% chance of having the bacteria.
Even more confusing, but important is the idea that while a 2% false positive does indicate that 2% of patients who do not have strep test positive, it does not mean that of all positives, 2% do have strep. There is more to consider in calculating those kinds of probabilities. Specifically, we would need to know how pervasive strep is for that population in order to come close to the actual probability that someone testing positive has the bacteria.
Enter: Bayes’ Theorem
Bayes’ Theorem considers both the population’s probability of contracting the bacteria and the false positives/negatives.
I know, I know — that formula looks INSANE. So I’ll start simple and gradually build to applying the formula – soon you’ll realize it’s not too bad.
Example: Drug Testing
Many employers require prospective employees to take a drug test. A positive result on this test indicates that the prospective employee uses illegal drugs. However, not all people who test positive actually use drugs. For this example, suppose that 4% of prospective employees use drugs, the false positive rate is 5%, and the false negative rate is 10%.
Here we’ve been given 3 key pieces of information:
The prevalence of drug use among these prospective employees, which is given as a probability of 4% (or 0.04). We can use the complement rule to find the probability an employee doesn’t use drugs: 1 – 0.04 = 0.96.
The probability a prospective employee tests positive when they did not, in fact, take drugs — the false positive rate — which is 5% (or 0.05).
The probability a prospective employee tests negative when they did, in fact, take drugs — the false negative rate — which is 10% (or 0.10).
It’s helpful to step back and consider the two things are happening here: First, the prospective employee either takes drugs, or they don’t. Then, they are given a drug test and either test positive, or they don’t.
I recommend a visual guide for these types of problems. A tree diagram helps you take these two pieces of information and logically draw out the unique possibilities.
Tree diagrams are also helpful to show us where to apply the multiplication principle in probability. For example, to find the probability a prospective employee didn’t take drugs and tests positive, we multiply P(no drugs) * P(positive) = (.96)*(.05) = 0.048.
An important note: The probability of selecting a potential employee who did not take drugs and tests negative is not the same as the probability an employee tests negative GIVEN they did not take drugs. In the former, we don’t know if they took drugs or not; in the latter, we know they did not take drugs – the “given” language indicates this prior knowledge/evidence.
What’s the probability someone tests positive?
We can also use the tree diagram to calculate the probability a potential employee tests positive for drugs.
A potential employee could test positive when they took drugs OR when they didn’t take drugs. To find the probabilities separately, multiply down their respective tree diagram branches:
Using probability rules, “OR” indicates you must add something together. Since one could test positive in two different ways, just add them together after you calculate the probabilities separately:
P(positive) = 0.048 + 0.036 = 0.084
Given a positive result, what is the probability a person doesn’t take drugs?
Which brings us to Bayes’ Theorem:
Let’s find all of the pieces:
P(positive | no drugs) is merely the probability of a false positive = 0.05
P(no drugs) = 0.96
So we already calculated the numerator above when we multiplied 0.05*0.96 = 0.048
We also calculated the denominator: P(positive) = 0.084
which simplifies to
This means, if we know a potential employee tested positive for drug use, there is a 57.14% probability they don’t actually take drugs — which is MUCH HIGHER than the false positive rate of 0.05. In other words, if a potential employee (in this population with 4% drug use) tests positive for drug use, the probability they don’t take drugs is 57.14%
How is that different from a false positive? A false positive says, “We know this person doesn’t take drugs, but the probability they will test positive for drug use is 5%.” While if we know they tested positive, the probability they don’t take drugs is 57%.
Why is this probability so large? It doesn’t seem possible! Yet, it takes into account the likelihood a person in the population takes drugs, which is only 4%.
In math terms:
P(positive | no drugs) = 0.05 while P(no drugs | positive) = 0.5714
Which also means that if a potential employee tests positive, the probability they do indeed take drugs is lower than what you might think. You can find this probability by taking the complement of the last calculation: 1 – 0.5714 = 0.4286. OR, recalculate using the formula:
Teaching statistics year after year prepped me for the most common misinterpretations of confidence intervals and confidence levels. Confusion such as:
Incorrectly interpreting a 99% interval as having a “99% probability of containing the true population parameter”
Finding significance because “the sample mean is contained in the interval”
Applying a confidence interval to samples that do not meet specific assumptions
What are Confidence Intervals?
Confidence intervals are like fishing nets to an analyst looking to capture the actual measure of a population in a pond of uncertainty. The margin of error dictates the width of the “net”. But unlike fishing scenarios, whether or not the confidence interval actually captures the true population measure typically remains uncertain. Confidence intervals are not intuitive, yet they are logical once you understand where they start.
So what EXACTLY, are we confident about? Is it the underlying data? Is it the result? Is it the sample? The confidence is actually in the procedures used to obtain the sample that was used to create the interval — and I’ll come back to this big idea at the end of the post. First, let’s paint the big picture in three parts: The data, the math, and the interpretation.
As I mentioned, a confidence interval captures a “true” (yet unknown) measure of a population using sample data. Therefore, you must be working with sample data to apply a confidence interval — you’re defeating the purpose if you’re already working with population data for which the metrics of interest are known.
It’s important to investigate how the sample was taken and determine if the sample represents the entire population. Sampling bias means a certain group has been under- or over- represented in a sample – in which case, the sample does not represent the entire population. A common misconception is that you can offset bias by increasing the sample size; however, once bias has been introduced to the sample, a larger sample using the same procedure will ensure the sample is much different from the population. Which is NOT a representative sample.
Examples of sampling bias:
Excluding a group who cannot be reached or does not respond
Only sampling groups of people who can be conveniently reached
Changing sampling techniques during the sampling process
Contacting people not chosen for sample
Statistic vs Parameter
A statistic describes a sample. A parameter describes a population. For example, if a sample of 50 adult female pandas weigh an average of 160 pounds, the sample mean of 160 is known as the statistic. Meanwhile, we don’t actually know the average of all adult female pandas. But if we did, that average (mean) of the population of all female pandas would be the parameter. Statistics are used to estimate parameters. Since we don’t typically know the details of an entire population, we rely heavily on statistics.
Mental Tip: Look at the first letters! AStatistic describes a Sample and a Parameter describes a Population
All confidence intervals take the form:
A common example here is polling reports — “The exit polls show John Cena has 46% of the vote, with a margin of error of 3 points.” Most people without a statistics background can draw the conclusion: “John Cena likely has between 43% and 49% of the vote.”
The Statistic, AKA “The Point Estimate”
The “statistic” is merely our estimate of the true parameter.
The statistic in the voting example is the sample percent from exit polls — the 46%. The actual percent of the population voting for John Cena – the parameter – is unknown until the polls close, so forecasters rely on sample values.
A sample mean is another example of a statistic – like the mean weight of an adult female panda. Using this statistic helps researchers avoid the hassle of traveling the world weighing all adult female pandas.
The Margin of Error
With confidence intervals, there’s a trade off between precision and accuracy: A wider interval may capture the true mean accurately, but it’s also less precise than a more narrow interval.
The width of the interval is decided by the margin of error because, mathematically, it is the piece that is added to and subtracted from the statistic to build the entire interval.
How do we calculate the margin of error? You have two main components — a t or z value derived from the confidence level,and the standard error. Unless you have control over the data collection on the front end, the confidence level is the only component you’ll be able to determine and adjust on the back end.
The confidence level
“Why can’t we just make it 100% confidence?” Great question! And one I’ve heard many times. Without going into the details of sampling distributions and normal curves, I’ll give you an example:
Assume the “average” adult female panda weighs “around 160 pounds.” To be 100% confident that we’ve created an interval that includes the TRUE mean weight, we’d have to use a range that includes all possible values of mean weights. This interval might be from, say 100 to 400 pounds – maybe even 50 to 1000 pounds. Either way, that interval would have to be ridiculously large to be 100% confident you’ve estimated the true mean. And with a range that wide, have you actually delivered any insightful message?
Again consider a confidence interval like a fishing net, the width of the net determined by the margin of error – more specifically, the confidence level (since that’s about all you have control over once a sample has been taken). This means a LARGER confidence level produces a WIDER net and a LOWER confidence level produces a more NARROW net (everything else equal).
For example: A 99% confidence interval fishing net is wider than a 95% confidence interval fishing net. The wider net catches more fish in the process.
But if the purpose of the confidence interval is to narrow down our search for the population parameter, then we don’t necessarily want more values in our “net”. We must strike a balance between precision (meaning fewer possibilities) and confidence.
Once a confidence level is established, the corresponding t* or z* value — called an uppercritical value — is used in the calculation for the margin of error. If you’re interested in how to calculate the z* upper critical value for a 95% z-interval for proportions, check out this short video using the Standard Normal Distribution.
The standard error
This is the part of the margin of error you most likely won’t get to control.
As I mentioned, you will most likely NOT have much control over the standard error portion of the margin of error. But if you did, keep this PRO TIP in your pocket: a larger sample size (n) will reduce the width of the margin of error without sacrificing the level of confidence.
Back to the panda weights example here. Let’s assume we used a 95% confidence interval to estimate the true mean weight of all adult female pandas:
Interpreting the Interval
Typically the confidence interval is interpreted something like this: “We are 95% confident the true mean weight of an adult female panda is between 150 and 165 pounds.”
Notice I didn’t use the word probability. At all. Let’s look at WHY:
Interpreting the Level
The confidence level tells us:, “If we took samples of this same size over and over again (think: in the long run) using this same method, we would expect to capture the true mean weight of an adult female panda 95% of the time.” Notice this IS a probability. A 95% probability of capturing the true mean exists BEFORE taking the sample. Which is why I did NOT reference the actual interval values. A different sample would produce a different interval. And as I said in the beginning of this post, we don’t actually know if the true mean is in the interval we calculated.
Well then, what IS the probability that my confidence interval – the one I calculated between the values of 150 and 165 pounds – contains the true mean weight of adult female pandas? Either 1 or 0. It’s either there, or it isn’t. Because — and here’s the tricky part — the sample was already collected before we did the math. NOTHING in the math can change the fact that we either did or didn’t collect a representative sample of the population. OUR CONFIDENCE IS IN THE DATA COLLECTION METHOD – not the math.
The numbers in the confidence interval would be different using a different sample.
Visualizing the Confidence Interval
Let’s assume the density curve below represents the actual population mean weights of all adult female pandas. In this made up example the mean weight of all adult pandas is 156.2 pounds with a population standard deviation of 13.6 pounds.
Beneath the population distribution are the simulation results of 300 samples of n = 20 pandas (sampled using an identical sampling method each time). Notice that roughly 95% of the intervals cover the true mean — capturing 156.2 within the interval (the green intervals) while close to 5% of intervals do NOT capture the 156.2 (the red intervals).
Pay close attention to the points made by the visualization above:
Each horizontal line represents a confidence interval constructed from a different sample
The green lines “capture” or “cover” the true (unknown) mean while the red lines do NOT cover the mean.
If this was a real situation, you would NOT know if your interval contained the true mean (green) or did not contain the true mean (red).
The logic of confidence intervals is based on long-run results — frequentist inference. Once the sample is drawn, the resulting interval either does or doesn’t contain the true population parameter — a probability of 1 or 0, respectively. Therefore, the confidence level does not imply the probability the parameter is contained in the interval. In the LONG run, after many samples, the resulting intervals will contain the mean C% of the time (where C is your confidence level).
So in what are we placing our confidence when we use confidence intervals? Our confidence is in the procedures used to find our sample. Any sampling bias will affect the results – which is why you don’t want to use confidence intervals with data that may not represent the population.