When my oldest son was born, I remember the pediatrician using a chart similar to the one below to let me know his height and weight percentile. That is, how he measured up relative to other babies his age. This is a type of cumulative relative frequency distribution. These charts help determine relative position of one data point to the rest of the dataset, showing an accumulating percent of observations for each value. In this case, the chart helps determine how a child is growing relative to other babies his age.

I decided to figure out how to create one in Tableau. Based on the types of cumulative frequency distributions I was used to when I taught AP Stats, I first determined I wanted the value of interest on the horizontal axis and the percents on the vertical axis.

Make a histogram

Using a simple example – US President age at inauguration – I started with a histogram so I could look at the overall shape of the distribution:

Adjust bin size appropriately

From here I realized I already had what I needed in my view – discrete ages on the x-axis and counts of ages on the y-axis. For a wider range of values I would want a wider bin size, but in this situation I needed to resize bins to 1, representing each individual age.

Change the marks from bars to a line

Create a table calculation

Click on the green pill on the rows (the COUNT) and add a table calculation.

Actually, TWO table calculations

First choose “Running Total”, then click on the box “add secondary calculation”:

Next, choose “percent of total” as the secondary calculation:

Polish it up

Add drop lines…

…and CTRL drag the COUNT (age in years) green pill from the rows to labels. Click on “Label” on the marks card and change the marks to label from “all” to “selected”.

And there you have it.

Interpreting percentiles

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. Using the baby weights example, the percentile is the percent of all babies of the same age and gender weighing less than your baby.

Back to the US president example.

Since I know Barack Obama was 47 when inaugurated, let’s look at his age relative to the other US presidents’ ages at inauguration:

And another way to look at this percentile: 87% of US presidents were older than Barack Obama when inaugurated.

As a follow-up to last week’s webinar with Andy Kriebel and Eva Murray, I’ve put together just a few common examples of means other than the ubiquitous arithmetic mean. A great deal of work on each of these topics can be found throughout the interwebs if your Googling fingers get itchy.

The Weighted Mean

My favorite of all the means. Sometimes called expected value, or the mean of a discrete random variable.

Grades

When computing a course grade or overall GPA, the weighted mean takes into account each possible outcome and how often that outcome occurs in a dataset. A weight is applied to each possible outcome — for example, each type of grade in a course — then added together to return the overall weighted mean. And since Econ was my favorite course in college…

If you have an exam average of 80, quiz/homework average of 65 and lab average of 78, what is your final grade? (Hint: Don’t forget to change percentages to decimals.)

Vegas

Weighted means are also effective for assessing risk in insurance or gambling. Also known as the expected value, it considers all possible outcomes of an event and the probability of each possible outcome. Expected values reflect a long-term average. Meaning, over the long run, you would expect to win/lose this amount. A negative expected value indicates a house advantage and a positive expected value indicates the player’s advantage (and unless you have skills in the poker room, the advantage is never on the player’s side). An expected value of $0 indicates you’ll break even in the long-run.

I’ll admit my favorite casino game is American roulette:

As you can see, the “inside” of the roulette table contains numbers 1-36 (18 of which are red, the other 18 black). But WAIT! Here’s how they fool you — see the numbers “0” and “00”? 0 and 00 are neither red nor black, though they do count towards the 38 total outcomes on the roulette board. When the dealer spins the wheel, a ball bounces around and chooses from numbers 1 thru 36, 0 AND 00 — that’s 38 possible outcomes.

Let’s say you wager $1 on “black”. And if the winning number is, in fact, black, you get your original dollar AND win another (putting you “up” $1). Unsuspecting victims new to the roulette table think they have a 50/50 shot at black; however, the probability of “black” is actually 18/38 and the probability of “not black” is 20/38″.

Here’s how it breaks down for you:

Just as in the grading example, each outcome (dollars made or lost) is first multiplied by its weight, where the weight here is the theoretical probability assigned to that outcome. After multiplying, add each product (outcome times probability) together. Note: Don’t divide at the end like you’d do for the arithmetic mean – it’s a common mistake, but easy to remedy if you check your work.

Some Gambling Advice: The belief that casino games adhere to some “law of averages” in the short run is called the Gambler’s Fallacy. Just because the ball on the roulette wheel landed on 5 red numbers in a row doesn’t mean it’s time for a black number on the next spin! I watched a guy lose $300 on three spins of the wheel because, as he exclaimed, “Every number has been red It’s black’s turn! It’s the law of averages!”

The Geometric Mean

A Geometric mean is useful when you’re looking to average a factor (multiplier) applied over time – like investment growth or compound interest.

I enjoyed my finance classes in school, especially the part about how compound interest works. If you think about compound interest over time, you may recall the growth is exponential, not linear. And exponential growth indicates that in order to grow from one value to the next, a constant was multiplied (not added).

As a basic example, let’s say you invest $100,000 at the beginning of 4 years. For simplicity, let’s say the growth rate followed the pattern +40%, -40%, +40%, -40% over the 4 years. At the end of 4 years, you’ve got $70,560 left.

So you know your 4-year return on the investment is: (70,560 – 100,000)/100,000 = -.2944 or -29.44%. But if you averaged out the 4 growth rates using the arithmetic mean, you’d have 0%. Which is why the arithmetic mean doesn’t make sense here.

Instead, apply the geometric mean:

The Harmonic Mean

You drive 60 mph to grandma’s house and 40 mph on the return trip. What was your average speed?

Let’s dust off that formula from physics class: speed = distance/time

Since the speed you drive plays into the time it takes to cover a certain distance, that formula may clue you in as to why you can’t just take an arithmetic mean of the two speeds. So before I introduce the formula for harmonic mean, I’ll combine those two trips using the formula for speed to determine the average speed.

The set-up Distance doesn’t matter here so we’ll use 1 mile. Feel free to use a different distance to verify, but you’d be reducing fractions a good bit along the way and I’m all about efficiency. Use a distance of 1 mile for each leg of the journey and the two speeds of 40mph and 60 mph.

First determine the time it takes to go 1 mile by reworking the speed formula:

To determine the average speed, we’ll combine the two legs of the trip using the speed formula (which will return the overall, or average, speed of the entire trip):

The formula for the harmonic mean looks like this:

Where n is the number of 1-mile trips, in this example, and the rates are 40 and 60 mph:

If you scroll up and check out that last step using the speed formula (above), you’ll see the harmonic mean formula was merely a clean shortcut.

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I was a high school math and statistics teacher for 14 years. And my stats course always began by visualizing the distribution of a variable using a simple chart or graph. One variable at a time we’d focus on creating, interpreting, and describing appropriate graphs. For quantitative variables, we’d use histograms or dot plots to discuss the distribution’s specific physical features. Why? Data visualization helps students draw conclusions about a population using sample data than summary statistics alone.

This post aims to review the basics of how measures of central tendency — mean, median, and mode — are used to measure what’s typical. Specifically, I’ll show you how to inspect distributions of variables visually and dissect how mean, median, and mode behave, in addition to common ways they are used. Ultimately it may be difficult, impossible, or misleading to describe a set of data using one number; however, I hope this journey of data exploration helps you understand how different types of data can effect how we describe what’s typical.

Remember Middle School?

Fair enough — I too try to forget the teased hair and track suit years. But I do recall learning to calculate mean, median, mode, and range for a set of numbers with no context and no end game. The math was simple, yet painfully boring. And I never fully realized we were playing a game of Which One of These is Not Like the Other.

It wasn’t until my first college stats course that I realized descriptive statistics serve a purpose – to attempt to summarize important features of a variable or dataset. And mean, median, mode – the measures of central tendency – attempt to summarize the typical value of a variable. These measures of typical may help us draw conclusions about a specific group or compare different groups using one numerical value.

To check off that middle school homework, here’s what we were programmed to do:

Mean: Add the numbers up, divide by the total number of values in the set. Also known as the arithmetic mean and informally called the “average”.

Median: Put the numbers in order from least to greatest (ugh, the worst part) and find the middle number. Oh, there’s two middle numbers? Average them. Did you leave out a number? Start over.

Mode: The number(s) that appear the most.

Repeat until you finish the worksheet.

Because we arrive at mean, median, and mode using different calculations, they summarize typical in different ways. The types of variables measured, the shape of the distribution, the context, and even the size of the set of data can alter the interpretation of each measure of central tendency.

Visually Inspecting Measures of Typical

What do You Mean When You Say, “Mean”?

We’re programmed to think in terms of an arithmetic mean, often dubbed the average; however, the geometric and harmonic means are extremely useful and worth your time to learn. Furthermore, when you want to weigh certain values in a dataset more than others, you’ll calculate a weighted mean. But for simplicity of this post, I will only use the arithmetic mean when I refer to the “mean” of a set of values.

Think of the mean as the balancing point of a distribution. That is, imagine you have a solid histogram of values and you must balance it on one finger. Where would you hold it? For all symmetric distributions the balancing point – the mean – is directly in the center.

The Median

Just like the median in the road (or, “neutral ground” if you’re from Louisiana), the median represents that middle value, cutting the set of values in half — 50% of the data values fall below and 50% lie above the median. No matter the shape of the distribution, the median is the measure of central tendency reflecting the middle position of the data values.

The Mode(s)

The mode describes the value or category in a set of data that appears the most often. The mode is specifically useful when asking questions about categorical (qualitative) variables. In fact, mode is the only appropriate measure of typical for categorical variables. For example: What is the most common college mascot? What type of food do college students typically eat? Where are most 4+ Year colleges and universities located?

Modes are also used to describe features of a distribution. In large sets of quantitative data, values are binned to create histograms. The taller “peaks” of the histogram indicate where more common data values cluster, called modes. A cluster of tall bins is sometimes called a modal range. A histogram having one tall peak is called unimodal while two peaks is referred to as bimodal. Multiple peaks = multimodal.

You may notice multiple tall peaks of varying heights in one histogram — despite some bins (and clusters of bins) containing fewer values, they are often described as modes or modal ranges since they contain local maximums.

When the Mean and the Median are Similar

The histogram above shows a distribution of heights for a sample of college females. The mean, median, and mode of this distribution are equal at about 66.5 inches. When the shape of the distribution is symmetric and unimodal, the mean, median, and mode are equal.

Now I want to see what happens when I add male heights into the histogram:

This histogram shows the distribution of heights of both male and female college students. It is symmetric, so the mean and median are equal at about 68.5 inches. But you’ll notice two peaks, indicating two modal ranges — one from 66 – 67 inches and another from 70 – 71 inches.

Do the mean and median represent the typical college student height when we are dealing with two distinctly different groups of students?

When the Mean and the Median Differ

In a skewed distribution, the median remains the center of the values; however, the mean is pulled away from the median from extreme values and outliers.

For example, the histogram above shows the distribution of college enrollment numbers in the United States from 2013. The shape of the distribution is skewed to the right — that is, most colleges reported enrollment below 5,000 students. However, the “tail” of the distribution is created by a small number of larger universities reporting much higher enrollment. These extreme outlying values pull the mean enrollment to the right of the median enrollment.

Reporting an average enrollment of 7,070 students for colleges in 2013 exaggerates the typical college enrollment since most US colleges and universities reported enrollment under 5,000 students.

The median, on the other hand, is resistant to outliers since it is based on position relative to the rest of the data. The median helps you conclude that half of all colleges enrolled fewer than 3,127 students and half of the colleges enrolled more than 3,127 students.

Depending on your end goal and context, median may provide a better measure of typical for skewed set of data. Medians are typically used to report salaries and housing prices since these distributions include mostly moderate values and fewer on the extremely high end. Take a look at the salaries of NFL players, for example:

Are we to only report medians for skewed distributions?

The median is not a good description of typical for a very small dataset (eg, n<10, depending on context).

The median is helpful when you want to ignore (or lessen effects of) outliers. Of course, as Daniel Zvinca* points out, your data could contain significant outliers that you don’t want to ignore.

In school, our grades are reported as means. However, students’ grade distributions can be symmetric or skewed. Let’s say you’re a student with three test grades, 65, 68, 70. Then you make a 100 on the fourth test. The distribution of those 4 grades is skewed to the right with a mean of 75.8 and median of 69. Despite the shape of the distribution, you may argue for the mean in this situation. On the other hand, if you scored a 30 on the fourth test instead of 100, you’d argue for the median. With only 4 data points, the median is not a good description of typical so here’s hoping you have a teacher who understands the effects of outliers and drops your lowest test score.

Inserting my opinion: As a former teacher, I recognize that when averaging all student grades from an assignment or test, the result is often misleading. In this case, I believe the median is a better description of the typical student’s performance because extreme values usually exist in a class set of grades (very high or very low) and will affect the calculation of the mean. After each test in AP statistics, I would post the mean, median, 5 number summary and standard deviation for each class. It didn’t take long for students to draw the same conclusion.

Ultimately, context can guide you in this decision of mean versus median but consider the existence of outliers and the distribution shape.

Using Modality to Find the Story

By investigating a distribution’s physical features, students are able to connect the numbers with a story in the data. In quantitative data, unusual features can include outliers, clusters, gaps and “peaks”. Specifically, identifying causes of the multimodality of a distribution can build context behind the metrics you report.

When I investigated the distribution of college tuition, I expected the shape to appear skewed. I did not expect to find the smaller peak in the middle. So I filtered the data by type of college (public or private) and found two almost symmetric distributions of tuition:

The existence of the modes in this data makes it difficult to find a typical US college tuition; however, they did point to the existence of two different types of colleges mixed into the same data.

Now I’m not confident that one number would represent the typical college tuition in the U.S., though I can say, “The typical tuition for 4+ year colleges in the US for the 2013-14 school year was about $7,484 for public schools and $27,726 for private schools.”

Oh and did you notice the slight peaks on the right side of both private and public tuition distributions? Me too. Which prompted me to look deeper:

Measuring what’s Typical

So here’s the thing: Summarizing a set of values for a variable with one numerical description of “center” can help simplify a reporting process and aid in comparisons of large sets of data. However, sometimes finding this measure proves difficult, impossible, or even misleading.

As I suggest to my students, visualizing the distribution of the variable, considering its context and exploring its physical features will add value to your overall analysis and possibly help you find an appropriate measure of typical.

*Special thank you to Daniel Zvinca for providing feedback for this post with his domain knowledge and extensive industry expertise.

Since our primary audience tends to be those in data visualization, I used the regression output in Tableau to highlight the p-value in a test for regression towards the end. However, I spent the majority of the webinar discussing p-values in general because the logic of p-values applies broadly to all those tests you may or may not remember from school: t-tests, Chi-Square, z-tests, f-tests, Pearson, Spearman, ANOVA, MANOVA, MANCOVA, etc etc.

I’m dedicating the remainder of this post to some “rules” about statistical tests. If you consider publishing your research, you’ll be required to give more information about your data for researchers to consider your p-value meaningful. In the webinar, I did not dive into the assumptions and conditions necessary for a test for linear regression and it would be careless of me to leave it out of my blog. If you use p-values to drive decisions, please read on.

More about Tests for Linear Regression

Cautions always come with statistical tests – those cautions do not fall solely on the p-value “cut-off” debate.

Do you plan to publish your findings?

To publish your findings in a journal or use your research in a dissertation, the data must meet each condition/assumption before moving forward with the calculations and p-value interpretation, else the p-value is not meaningful.

Each statistical test comes with its own set of conditions and assumptions that justify the use of that test. Tests for Linear Regression have between 5 and 10 assumptions and conditions that must be met (depending on the type of regression and application).

Below I’ve listed a non-exhaustive list of common assumptions/conditions to check before running a test for linear regression (in no particular order).

The independent and dependent variables are continuous variables.

The two variables exhibit a linear relationship – check with scatterplot.

No significant outliers present in the residual plot (AKA points with extremely large residuals) – check with residual plot.

Observations are independent of each other (as in, the existence of one data point does not influence another) – test with Durbin-Watson statistic.

The data shows homoscedasticity (which means the variances remains the same along entire line of best fit) – check the residual plot, then test with Levene’s or Brown-Forsythe’s tests.

Note: Check with your publication and/or dissertation chair for complete list of assumptions and conditions for your specific situation.

Can I use p-values only as a sniff test?

Short answer: Yes. But I recommend learning how to interpret them and their limitations. Glancing over the list of assumptions above can give a good indication of how sensitive regression models are to outliers and outside variables. I’d also be hesitant to draw conclusions based on a p-value alone for small datasets.

In this BrightTalk webinar with Eva Murray and Andy Kriebel, I discussed how to use residual plots to help determine the fit of your linear regression model. Since residuals show the remaining error after the line of best fit is calculated, plotting residuals gives you an overall picture of how well the model fits the data and, ultimately, its ability to predict.

For simplicity, I hard-coded the residuals in the webinar by first calculating “predicted” values using Tableau’s least-squares regression model. Then, I created another calculated field for “residuals” by subtracting the observed and predicted y-values. Another option would use Tableau’s built in residual exporter. But what if you need a dynamic residual plot without constantly exporting the residuals?

Note: “least-squares regression model” is merely a nerdy way of saying “line of best fit”.

How to create a dynamic residual plot in Tableau

In this post I’ll show you how to create a dynamic residual plot without hard-coding fields or exporting residuals.

Step 1: Always examine your scatterplot first, observing form, direction, strength and any unusual features.

Step 2: Calculated field for slope

The formula for slope: [correlation] * ([std deviation of y] / [std deviation of x])

correlation doesn’t mind which order you enter the variables (x,y) or (y,x)

y over x in the calculation because “rise over run”

be sure to use the “sample standard deviation”

Step 3: Calculated field for y-intercept

The formula for y-intercept: Avg[y variable] – [slope] * Avg[x variable]

Step 4: Calculated field for predicted dependent variable

The formula for predicted y-variable = {[slope]} * [odometer miles] + {[y-intercept]}

Here, we are using the linear equation, y = mx + b where

y is the predicted dependent variable (output: predicted price)

m is the slope

x is the observed independent variable (input: odometer miles)

b is the y-intercept

Since the slope and y-intercept will not change value for each odometer mile, but we need a new predicted output (y) for each odometer mile input (x), we use a level of detail calculation. Luckily the curly brackets tell Tableau to hold the slope and y-intercept values at their constant level for each odometer mile.

Step 5: Create calculated field for residuals

The formula for residuals: observed y – predicted y

Step 6: Drag the independent variable to columns, residuals to rows

Step 7: Inspect your residual plot.

Don’t forget to inspect your residual plot for clear patterns, large residuals (possible outliers) and obvious increases or decreases to variation around the center horizontal line. Decide if the model should be used for prediction purposes.

The horizontal line in the middle is the least-squares regression line, shown in relation to the observed points.

The residual plot makes it easier to see the amount of error in your model by “zooming in” on the liner model and the scatter of the points around/on it.

Any obvious pattern observed in the residual plot indicates the linear model is not the best model for the data.

In the plot below, the residuals increase moving left to right. This means the error in predicting 4Runner price gets larger as the number of miles on the odometer increase. And this makes sense because we know more variables are affecting the price of the vehicle, especially as mileage increases. Perhaps this model is not effective in predicting vehicle price above 60K miles on the odometer.

To recap, here are the basic equations we used above:

Thank you to Makeover Monday‘s Eva Murray and Andy Kriebel for allowing me to grace their BrightTALK air waves with my love language of statistics yesterday! If you missed it, check out the recording.

With 180 school days and 5 classes (plus seminar once/week), you can imagine a typical U.S. high school math teacher has the opportunity to instruct/lead between 780 and 930 lectures each year. After 14 years teaching students in Atlanta-area schools (plus those student-teaching hours, and my time as a TA at LSU), I’ve instructed somewhere in the ballpark of 12,000 to 13,500 lessons in my lifetime.

So let’s be honest. Yesterday I was nervous to lead my very first webinar. After all, I depend on my gift of crowd-reading to determine the pace (and the tone) of a presentation. Luckily, I’m an expert at laughing at my own jokes so after the first few slides (and figuring out the delay), I felt comfortable. So Andy and Eva, I am ready for the next webinar on December 20th — Audience, y’all can sign up here.

Fun Fact: In 6th grade I was in the same math class as Andy Kriebel’s sister-in-law. It was also the only year I ever served time in in-school suspension (but remember, correlation doesn’t imply causation).

Webinar Questions and Answers

I was unable to get to all the questions asked on the webinar but rest assured I will do my best to field those here.

Q: Can you provide the dataset? A: Here’s a link to the 4Runner data I used for most of the webinar. Let me know if you’d like any others.

Q: Do you have the data that produced the cartoon in the beginning slide? A: A surprising number of people reproduced the data and the curves from the cartoon within hours of its release. Here is one person’s reproduction in R from this blog post

Q: Do you have any videos on the basics of statistics? A: YES! My new favorite is Mr. Nystrom on YouTube, we use similar examples and he looks like he loves his job. For others, Google the specific topic along with the words “AP Statistics” for some of the best tutorials out there.

Q: Could you explain about example A with r value -0.17, it seems as 0. A: The picture when r = -.17 is slightly negative — only slightly. This one is very tricky because we tend to think r = 0 if it’s not linear. But remember correlation is on a continuous scale of weak to positive – which means r = -.17 is still really, really weak. r = 0 is probably not very likely to be observed in real data unless the data creates a perfect square or circle, for example.

Q: Question for Anna, does she also use Python, R, other stats tools? A: I am learning R! R-studio makes it easier. When I coach doctoral candidates on dissertation defense I use SPSS and Excel; one day I will learn Python. Of course, I am an expert on the TI-84. Stop laughing.

6. Q: So with nonlinear regression [is it] better to put the prediction on the y-axis? A: With linear and nonlinear regression, the variable you want to predict will always be your y-axis (dependent) variable. That variable is always depicted with a y with a caret on top : And it’s called “y-hat”

Other Helpful Links

If you haven’t had time to go through Andy’s Visual Vocabulary, take a look at the correlation section.

At the end of the webinar I recommended Bora Beran’s blog for fantastic explanations on Tableau’s modeling features. He has a statistics background and explains the technical in a clear, easy-to-understand format.

Don’t forget to learn about residual plots if you are using regression to predict.

As you’d imagine, I’ve been introspective. Am I living my best life? If I go out tomorrow, will I have regrets?

Regrets.

If you’ve asked my advice, I’ve recommended the adventure over the status-quo; the challenge over the straight path; the Soup Number 5 over the chicken fingers. I’ve told you challenges grow you and discomfort makes you stronger. I come from personal experience — I’ve found regrets only inside the comfortable and the “supposed tos”.

My good friend Ericka lost her brother Daniel 5 years ago yesterday in a base-jumping accident. To honor his memory, Ericka posted a video he’d “have others watch for inspiration.”

So watch the video. And afterwards, if you want, you can keep reading this short post. But I won’t be offended if you choose to act on the emotions of the question, “What if you just had one more day? What are you going to do today?”

The day I had regrets.

Thursday, December 3rd 2015 started off with a strange, uncomfortable internal pain – which I chose to ignore. I was married to my schedule and my routine. So I went to the gym, got ready for work, dropped the kids off at their schools, and was at my school by 7:50am. A little voice told me, “Go to the hospital” as I walked into my classroom, but my duty was to my students, to my school, and to my responsibility to everyone else. “I’m sure it’s nothing,” I told my brain.

The pains would come and go every 5 minutes by that time. I taught (or, I tried to teach) my first period pre-calculus class. (I’ll never forget – it was a lesson on the Law of Sines ambiguous case. The “ASS” case. A tricky lesson involving logic, geometry, and effective cursing.) But I hit a point where the pain was so intense I’d have to stop the lesson and sit to take deep breaths. A student suggested I had appendicitis. And I remembered from natural childbirth the indication of real pain was the inability to walk or talk when it hit – and THIS is when I decided to step off that path of expectations of others and ask for help.

A coworker drove me to the hospital and a short while later, I was lying on a stretcher receiving an ultrasound on my abdomen. Stunned techs ran around me loudly relaying their confusion to each other. I’d had enough pain medicine to take the edge off the intensity, but at this point my stomach was beginning to protrude near my belly button and I understood the voices around me were screaming, “Emergency!”

When you think you are on your death bed, or when you’re given terrible news, or when you are in your last moments, I think the thoughts are the same: Have I said and done enough? Do the people I love know how I feel about them? Will my children remember me?

No regrets.

At 40 my regrets are now the words I didn’t say to the people I love.

But few are given a second chance to change how they live their life.

For the record, I had an intussusception – my small intestine telescoped into my large intestine and was sucked further and further until emergency surgery saved my life. It took a while for the doctors to solve my mystery because an intussusception is so rare in adults, especially in females. I was told at first it was most likely caused by colon cancer, though thankfully, the pathology report came back clear a few days later. After a full 6 days in the hospital and 26 staples down my abdomen, I was released into the arms of my loving family.

What if you just had one more day? What are you going to do today?

Rest in peace, Daniel Moore.

For what it’s worth … it’s never too late, or in my case too early, to be whoever you want to be. There’s no time limit. Start whenever you want. You can change or stay the same. There are no rules to this thing. We can make the best or the worst of it. I hope you make the best of it. I hope you see things that startle you. I hope you feel things you never felt before. I hope you meet people who have a different point of view. I hope you live a life you’re proud of, and if you’re not, I hope you have the courage to start all over again.

– F. Scott Fitzgerald, The Curious Case of Benjamin Button