DescriptiveStats3

Summary

In this lecture, Erin Heerey delves into various measures of central tendency, including the arithmetic mean, median, and mode, explaining their significance in data analysis. The discussion extends to the use of histograms, box plots, and violin plots to visually represent data distributions and highlight key trends, such as shifts in demographics or financial data over time. Heerey emphasizes understanding how data is distributed to draw meaningful conclusions about broader populations.

Highlights

• The arithmetic mean, commonly known as the average, is the central focus of many data discussions 💬
• Histograms show the distribution of data and are useful for visualizing the mode 📊
• The Great Backyard Bird Count serves as a real-world example of data collection 🐦
• Box plots and violin plots are tools for displaying the median and understanding data spread 📊
• Changes in generational debt illustrate shifts in societal financial trends 💰
• Violin plots provide more detailed insights into data distribution than box plots 🎻
• A histogram example highlighted changes in ages of first-time mothers between 1980 and 2016 👶
• The median divides data into two equal parts and is often represented in a box plot 📦

Key Takeaways

• Measures of central tendency include mean, median, and mode 📊
• The arithmetic mean is calculated by dividing the sum of all numbers by the count 📐
• Median is the middle value when data points are ordered 🌟
• Mode is the most frequently occurring value ⚡
• Histograms visually display data distributions 📈
• Box plots and violin plots provide insights into data distribution and median 🎻
• The Great Backyard Bird Count is a practical example of data collection and analysis 🐦
• Histograms can reveal shifts in data trends over time, such as age of first-time mothers 👶

Overview

In this enlightening session with Erin Heerey, the focus is primarily on understanding measures of central tendency in statistics, with a particular emphasis on the arithmetic mean, median, and mode. Each of these measures provides unique insights into the data set, revealing different aspects of central values that aid in interpreting statistical data more effectively. Heerey also introduces students to the role of histograms, a key tool in illustrating the distribution of data points.

Heerey's lecture uses real-world examples like the Great Backyard Bird Count to illustrate the practical application of data analysis. By participating in this event, data about bird populations can be visualized and interpreted, offering insights into migratory patterns and population sizes. This example underscores the importance of data visualization tools such as histograms, which can reveal trends and distributions at a glance.

Moreover, the discussion on graphical data representation extends to the use of box plots and violin plots. These tools not only showcase the median and dispersion of data but also highlight shifts in statistical measures over time, such as changing trends in the ages of first-time mothers or generational differences in household debt. By the end of this lecture, students gain a comprehensive understanding of these essential statistical tools and the stories they can tell through data.

Chapters

• 00:00 - 00:30: Introduction to Measures of Central Tendency The chapter introduces the concept of measures of central tendency, which provide information about the average data point. It highlights that there are various kinds of averages, with a specific focus on the arithmetic mean. The arithmetic mean is explained as the sum of all numbers in a data set divided by the number of data points.
• 00:30 - 01:00: Arithmetic Mean, Median, and Mode The chapter discusses three primary measures of central tendency: mean, median, and mode. The mean, also known as the average, is the sum of all values divided by the number of values. The median is the middle value when all numbers are ordered from least to greatest, effectively placing it at the 50th percentile. This means half of the data is below the median and half is above. Lastly, the mode is identified as the most frequently occurring value in a set of data.
• 01:00 - 02:00: Understanding the Mode and its Visualization This chapter introduces the concept of the mode in statistical data analysis, highlighting it as the most frequently occurring data value in a dataset. An example is provided with a dataset where the number 6 is identified as the mode. The chapter also mentions that histograms are a common way to visualize modes and that data sets can have more than one mode. Histograms will be used frequently throughout the class for data visualization.
• 02:00 - 04:00: Histograms and Data Distribution The chapter titled 'Histograms and Data Distribution' introduces histograms as tools to display modes without using numeric terms. Histograms are described as plots that represent the distribution of data, available in various shapes. The x-axis of these plots typically represents the individual values a specific variable can take. More details on histograms and their applications are promised to be covered in subsequent lectures.
• 04:00 - 05:30: Histogram Example: Age of First-Time Mothers This chapter explains the concept of a histogram, specifically using the example of 'Age of First-Time Mothers'. It describes how histograms are used to count and display the frequency of various values within a dataset, with values plotted along the x-axis and frequency on the y-axis. The chapter briefly mentions the Great Backyard Bird Count as a hypothetical example to illustrate how data collection and plotting occur.
• 05:30 - 06:30: Shift in Distribution Over Time This chapter discusses the shift in bird distribution over time, focusing on how people contribute to studying this phenomenon. It highlights the role of birdwatchers and citizen scientists who use apps to record sightings, which assist researchers in estimating bird populations, distribution, and migratory patterns. This data collection is crucial for understanding changes in bird habitats and movements.
• 06:30 - 08:30: Introduction to Mean and Two Examples The chapter provides an introduction to the concept of 'mean' using examples from bird watching. It suggests counting common birds like cardinals and robins in a neighborhood to illustrate the idea. By observing and reporting the numbers of these birds, various data points can be collected.
• 08:30 - 10:00: Introduction to the Median This chapter introduces the concept of the median in statistical analysis. It starts by discussing common real-world examples such as counting birds in a yard to illustrate how data can be collected and summarized. The chapter then transitions to explaining the use of histograms, which are tools that help us understand the distribution of data within a sample. It touches on the importance of determining whether data is normally distributed or follows a different kind of distribution. This foundational knowledge serves as an entry point to more advanced statistical concepts.
• 10:00 - 14:00: Box Plot and Violin Plot for Median The chapter titled 'Box Plot and Violin Plot for Median' discusses how histograms can provide insights into the data distribution within a larger population. Histograms can indicate if a sample is skewed, suggesting possible skewness in the population itself. This provides valuable information about the distribution characteristics of the data.
• 14:00 - 15:30: Comparison: Box Plot vs Violin Plot The chapter discusses methods for understanding statistical distribution, specifically focusing on the comparison between box plots and violin plots.
• 15:30 - 17:00: Conclusion on Violin Plot The chapter titled 'Conclusion on Violin Plot' discusses the changes in the age distribution of first-time mothers from 1980 to 2016. Initially, the most common age for first-time mothers was between 18 and 20, primarily among 19-year-olds who graduated from high school, got married, and started families. By 2016, this mode has shifted to around 20 years old, demonstrating a clear bimodal distribution.
• 17:00 - 18:00: Preview of Next Section: Dispersion The chapter discusses the concept of multiple modes in data, particularly focusing on how data shape indicates trends rather than just numeric counts. A specific example is given, showing a graph illustrating a substantial shift in the age at which people have their first child. Compared to 1980, the data reveals that people are now waiting longer, indicating interesting trends in societal behavior.

DescriptiveStats3 Transcription

• 00:00 - 00:30 So let's talk about measures of central tendency.  Measures of central tendency tell us about the   average data point. Now, there are different kinds  of averages that we can calculate and we'll talk   about a bunch of them in today's lecture. The one  that we are going to, that you're probably most   familiar with, and that we are going to focus on  most heavily is what we call the arithmetic mean.   The arithmetic mean is the sum of the numbers in  a data set, divided by the number of data points
• 00:30 - 01:00 that we have in that data set. It's often  referred to as the 'mean' or the 'average'. We will also talk about the median, which is  the middle value in the data set. It's the 50th   percentile, so it's right where 50 of the data  are lower than that number and 50 of the data   are higher than that number. And then we have the  mode which is the most common value in a data set.
• 01:00 - 01:30 Let's talk about the mode first mostly because  it's the easiest one to describe. it is simply   the most frequent data value that exists.  So if I give you this pretend data set here,   with just a few data points in it what you can see  is that it's really clear that 6 is our mode. Six   is the most frequently occurring value. One of the  ways we can see modes, is in fact, one of the most   common displays of these data are histograms.  We can also have data sets with more than one   mode. By and large we'll be looking at a lot of  histograms in the in the context of the class
• 01:30 - 02:00 and histograms are how we display modes  when we don't want to use numeric terms.   Histograms are simply plots of distributions  that come in many shapes, and we'll talk more   about these in future lectures, and they provide  a picture of the distribution of the data. The   x-axis typically plots the individual values a  specific variable can take for example you might
• 02:00 - 02:30 have a variable things that you can count and  so you plot how many of each of the values occur   on the x-axis and the y-axis is the frequency  with which each of these individual values   occurs. So let's say you're participating  in, a good example is the Great Backyard   Bird Count. This is a worldwide event that  happens, I think it happens, in February,
• 02:30 - 03:00 where all kinds of people who put bird feeders  in their yards and like to go around with their   binoculars, walk around and they count birds and  there are apps that you can download and you can   put in birds and what kind of bird you saw in  the location where you saw it. And that allows people who study birds, and their  migratory patterns, and their habits,   to estimate how many of these birds there are,  where they're living, where they're migrating,
• 03:00 - 03:30 and that tells us things like about like climate  change and so forth. But what we might do is let's   say you saw one of a really common bird in our  area, like cardinals. Robins are another really   common bird in our area as well. And there are  many of these birds. And so you could count the   number of robins that you see coming into your  yard and you could report that on your app and so   you could have 1 Robin you could have 5 Robins you  could have 26 Robins but you can count the number
• 03:30 - 04:00 of each of these birds that shows up in your yard,  and that would give people a count of the number   of times people you you saw that bird or you  could count the number of different birds you saw   from cardinals to robins to  finches to sparrows to whatever. Histograms tell us about the way the  data in a sample are distributed. Are   they normally distributed or what kind of  distribution do they have if they're not
• 04:00 - 04:30 normal? And another interesting thing that we  can learn from the histogram of a sample is it   can tell us something about how the data are  likely distributed in the larger population.   So histograms give an estimate of this. If we  have a histogram that looks skewed for example,   that might suggest that the population  itself has a skewed distribution,   and that can tell us something. We can sort of  look at the degree to which a sample might be
• 04:30 - 05:00 representative of a larger population or might  be generalizable to another population with a   same/different kind of distribution. it also tells  us about whether certain values are more frequent   than others. Here's an example of a histogram.  This is a great histogram that was produced by the   New York Times a number of years ago and this is  the age of first-time mothers in 1980 compared to   the age of first-time mothers in 2016. These are  all women who got pregnant the good old-fashioned   way, not using any high-tech methods, and what  you can see is that distribution of their ages has
• 05:00 - 05:30 changed dramatically between 1980 and 2016. From  a peak where the greatest proportion of first-time   mothers were getting pregnant sometime between  the ages of 18 and 20. So these are 19 year olds,   people who graduated from high school got married  and got pregnant and you can see that that same   mode has now increased to 20 at least in 2016  and there's a clear bimodal distribution here
• 05:30 - 06:00 where there are probably multiple modes - not  necessarily by the exact numbers of counts. but   certainly by the shape. What you can see  is this graph shows a substantial shifted,   so that people are waiting longer to  have their first child than they did   back in 1980. So, we can see these  really interesting data trends from the
• 06:00 - 06:30 distribution of scores that are presented by these  histograms. This tells a really interesting story.   The mean is our next statistic, our next measure  of central tendency. It's the balance point in   a data set. So here's the data the little  data set that I made up - that I showed you   before and it's mean it's 3.9 and you can think  about that mean as a place where if we were to   take the blocks just as they're balanced on this  number line here the place the balance point would
• 06:30 - 07:00 be is 3.9. This is where the blocks weigh the same  amount on one side as they do on another side. How do we find the mean in the sample? Well the  mean is often times represented by the name of a   variable in this case the variable is called X  and it has a little bar sitting right over the   top of it. We call this 'x-bar'. So it's the  mean of the variable X and it's presented by
• 07:00 - 07:30 an equation that looks terrible but really  isn't as bad as it looks. In plain English   this is the sum of each of the numbers  from the beginning of the data set (the   first number) all the way to the last number  in the data set. We add up all those numbers and then we take the count of the numbers (how  many there are) and we divide the sum of the   numbers by the count. So it looks terrible but  it's not nearly as bad as it looks. The mean
• 07:30 - 08:00 of of a variable is the sum of the data that it  contains, divided by how many data points there   are and here's a good map, a good graphic. This  is an infographic showing means of U.S household   debt across different generations of people. So  the silent generation, the Baby Boomers, Gen X,   Millennials, Gen Z and so forth. This shows  you the increase between 2019 and 2020; or
• 08:00 - 08:30 decrease depending there depending on who you're  looking at, of debt across these communities.   so it's an interesting visualization of  what's happening in the context of covid   for people who have who are these different, who  are born in different time eras. The infographic   also tells you when these time eras are,  just so that you can get a feel for that,
• 08:30 - 09:00 and how much increase in debt  they have now relative to the   beginning of the pandemic so it's a  really interesting infographic that   tells us something about how unequally the  pandemic has affected various individuals. The median is the center point of our data set.  It's the 50th percentile and in order to find it
• 09:00 - 09:30 we put the numbers in order; if there's an  odd number of data points we take the very   middle value in the data set; if there's an  even number of data points we take the mean   of the two middle values. So what we do is the  numbers are in order and we work our way to the   very middle of the data set and it happens,  in this data set, the median is three. Now,   if we add in a value, let's say we added a  4 to this, that number that would cause our   mean to shift ever so slightly and it would  cause our median to shift to 3.5 because
• 09:30 - 10:00 when there's an even number of data points  (originally this data set had an odd number)   but if we had an even number of data points  we take the mean of the two middle values. The median is often displayed using what we call  a box plot so a box plot is a very fancy plot I'm   not going to walk you through all the elements  here, but you probably should know what they are.
• 10:00 - 10:30 A box plot is a plot that has a box of the central  distribution of the data so if it runs from,   the box usually runs from the 25th percentile, the  median is the 50th percentile there, and the top   of the box is the 75th percentile. So the median  is usually displayed as a horizontal line in the   middle of the box plot if the box plot is oriented  this way. Then you can see, you know, what we
• 10:30 - 11:00 would expect for the lower value of the data what  we would expect for the upper values of the data   and then whether there are outliers as well, so a  box plot is how we typically represent the median. And there's another plot that we often also use  that you will see a lot in this class called the   violin plot. It's another plot that also shows  us the median, but it shows us something else. It   shows us a fancy thing called the 'kernel density  estimate'. This is called the violin plot because
• 11:00 - 11:30 it actually is sort of a bit wiggly - it has  wiggly outsides, instead of these nice, square   tidy boxplot bars here. But what this shows is how  the data are distributed across these different   bands so it can provide a bit more information  than what you might anticipate from a simple box   plot. The median is this dot right here. It's this  central value in the interquartile range. So this
• 11:30 - 12:00 box here marks exactly the same space out that  this box here marks with a median and the same   point, but what it's doing here is it's showing  you where and how the data are distributed.   We see these often in research as  well. Here's another version of   these same plots kind of together so here's the  box plot, which you can obviously see - you can   see there are some outliers in this box plot.  Here's the kernel density estimate, based on
• 12:00 - 12:30 a histogram. A kind of smooth histogram is one  way to think about the kernel density estimate   and it shows you how the data are distributed  across the graph, from the lower bound of the   data set to the upper bound of the data set. And  so you can see where the median is on these plots   and you can see how the data are distributed  so these are all plots of the median.
• 12:30 - 13:00 Here's an interesting GIF that shows how much  more informative violin plots are than box plots   specifically each one of these bars in the raw  data produce, no matter how they're arranged here,   produce box plots that look like this and the  violin plots themselves look very different   depending on how the data are distributed.  so when the data distribution changes the
• 13:00 - 13:30 violin plot really changes shape, even  though the box plot is unaffected by that. And we often see these in scientific or  research presentations they can look,   this is a very nicely formatted one, looking  at social preferences in infants and showing   how these data are distributed. This  one's interesting as well because what   the researchers have done here is they've  drawn the individual data points right   inside of the plots. These are nice violin  plots that are there rather descriptive.
• 13:30 - 14:00 The next section we'll talk about  is dispersion and we'll talk about   that in the final element of this week's lecture.