Mean, Median, and Mode: Measures of Central Tendency: Crash Course Statistics #3

Hi I’m Adriene Hill, and welcome to Crash
Course Statistics. In the last video we tried to make sense of
ginormous numbers. And teeny-tiny numbers. Today we’re going to talk about less showy
numbers. The numbers stuck in the middle. The averages. The medians. The modes. They may not seem as mind-blowing. Or all that flashy. But, turns out they are really, really important. Those middle numbers are often the ones that
get ALL the press attention. Get tossed back in forth in political debate. And…they give us this little fun bit of
trivia… What’s the average…or mean… number of
feet people have? It’s not 2. Turns out the average number of feet people
has is a little less than 2. Cause the average takes into account the small
number of people out there with fewer than 2 feet. So, if you have two feet, you have more than
the average number of feet. `And wIth that…let’s get what into “measures of central tendency” are, and why they’re useful. INTRO If your boss asks you for a report on this
quarter’s sales numbers but is rushing to a meeting and only has time to listen to one
piece of information about the data, that piece of information you give her should probably
be a measure of central tendency. The center of a bunch of data points is usually
a good example (or summary) of the type of data we can expect from the group as a whole. One common measure of the middle is the mean. You’ve likely heard it called the average–though
all of these measures are sometimes called “averages”. Some people call it the “expectation”
of a set of data. The mean…or average…takes the sum of all
the numbers in a data set, and divides by the number of data points. So, if 10 pregnant dogs give birth to 50 total
puppies–the average litter size is 5 puppies. Each data point, in this case each litter
of puppies, contributes equally to the calculation. Awwwww. Here’s another example. Say you you have ten dollars and your best
friend has 20 dollars, the mean amount of cash you two have is 15 dollars. Ten-plus-twenty-divided by two. But saying that the mean is fifteen dollars–doesn’t
mean you each can buy that 12-dollar BFF necklace you’ve been eyeing…the one with the half-a-heart
that fits together. You personally only have ten dollars in your
pocket. The average of a set of data points tells
us something about the data as a whole, but it doesn’t tell us about individual data
points. The mean is good at measuring things that
are relatively “normally” distributed. “Normal” means a distribution of data
that has roughly the same amount of data on either side of the middle, and has its most
common values around the middle of the data. Data that are distributed normally will have
a symmetrical bell shape that you’ve probably seen before. A distribution shows us how often each value
occurs in our data set, which is also known as their frequency. Imagine you are trying to impress your new
college dorm mates by guessing how many times they’ve each seen Harry Potter and the Sorcerer’s
Stone. Your mom is in the entertainment industry
and you overheard, at her last dinner party, that 18 year olds, on average, had seen the
movie five times each. That’s a lot of quidditch. So you should guess your new friends have
seen the movie five times each. (Unless you can clearly see Slytherin tattoos.) You won’t be right each time, but it’s
your best guess. It might not be the best way to impress them
though. It’s not a great party trick. Sometimes the mean is misleading. For instance: life expectancy in the Middle
Ages. As we explored in Crash Course World History,
there was an incredibly high rate of infant mortality in the days before modern medicine,
but the people who made it to adulthood lived relatively long lives. Because of the high rate of infant and child
mortality, the average life expectancy was about thirty years. But things weren’t nearly as dire as all
that. Not if you actually made it to 30. In the 13th century a male who lived to 30–was
likely to make it into his fifties! To give unusually large or small values, also
called outliers, less influence on our measure of where the center of our data is, we can
use the median. Unlike the mean, the median doesn’t use
the value of every data point in it’s calculation. The median is the middle number if we lined
up our data from smallest to largest. For example, if you have two cats, Julian
has one cat, and Erik has three cats, the median number of cats in your little cat-loving
group would be two. When we put the number of cats in order from
least to most cats, two is in the middle. But what if there’s no middle number? You invite Will to join your cat group. He has an impressive …or is it
excessive…total of fourteen cats. Now there are four cat owners. There is no one middle number; both two and
three are in the middle. In this case there are differing opinions
on how to calculate the median, but most often we take the mean of the two middle numbers,
so our median would be 2.5 cats. Meow. Meow Me…. Let’s go to the thought bubble. Imagine ten artists have been working for
years, together, to come up with a new, fresh way to tie macrame knots. The standard square knot…just wasn’t inspiring
them the way it used to. And finally. They do. Viola! The abracadabra-doolittle knot! So these 10 artists go out to celebrate. And they go to a relatively modestly priced
restaurant…cause macrame artists don’t make all that much money. Each of them pulls in about $20,000 a year. So the average…or mean… income in around
the table is twenty-thousand dollars. And the median income is also twenty-thousand
dollars. Now, let’s imagine that Elon Musk gets wind
of this macrame milestone. Turns out…he’s a huge macrame fan himself. Beauty in design. He couldn’t miss up a chance to celebrate…So
he decides to show up at the restaurant. Musk’s total annual compensation…including
his salary and stock options…is reportedly in the neighborhood of 100-million dollars. As soon as Musk walks in the door. The average income in the room…skyrockets…to
a little over 9 million dollars. But…nobody else in the room is ACTUALLY
richer…nobody feels any richer. The median income of the macrame artists and
Musk is still $20,000 because most of our group is still making $20,000. And…this isn’t just the stuff of make-believe
macrame world…it happens in REAL life too…the “average” is distorted by outliers. Thanks Thought Bubble! Alright, now say there’s a controversial
book on Amazon called Pineapple Belongs on Pizza, with 400 reviews; 200 five-star reviews,
and 200 one-star reviews. The mean number of stars given was 3, but
no one in our sample actually gave the book 3 stars, just like no one could actually have
the median of 2.5 cats. In both of these situations, it can be useful
to look at the mode. The word mode comes from the Latin word modus,
which means “manner, fashion, or style” and gives us the French expression a la mode,
meaning fashionable. Just like the most popular and fashionable
trends, the mode is the most popular value. But not popular like Despacito. When we refer to the “mode” of our data,
we mean the value that appears most in our data set. For our Amazon book review of Pineapple Belongs
on Pizza the modes are both 5 and 1, which give us a better understanding of how people
feel about the book. These reviews are called “bimodal” because
there are two values that are most common. Bimodal data is an example of “Multimodal”
data which has many values that are similarly common. Usually multimodal data results from two or
more underlying groups all being measured together. In the case of our book, the two groups were
the “love it” five-star group, and the “hate it” one-star group. Or for another example, if we made a graph
of the times customers went to IN-N-OUT, we’d probably see two peaks because there’s two
groups of people: one around lunch time, and one around dinnertime. The mode is useful here because it’s an
actual value that occurs in our data set, unlike the median and mean which can give
us numbers that wouldn’t actually occur and don’t describe our data very well. The mean time people come into In-N-Out may
very well be 3:30pm, but that doesn’t suggest you should expect an overflowing restaurant
in the middle of the afternoon. You should be able to get your animal style
burger …without too much of a wait. The mode is most useful when you have a relatively
large sample so that you have a large number of the popular values. One other benefit of the mode is that it can
be used with data that isn’t numeric. Like, if I ask everyone their favorite color,
I could have a mode of “blue”. There’s no such thing as a “mean” or
average favorite color. The relationship between the mean, median,
and mode can tell us a lot about the distribution of data. In normal distribution that we mentioned earlier
they’re all the same. We know that the middle value of the data
(the median) is also the most common (the mode) and is the peak of the distribution. The fact that the median and mean are the
same tells us that the distribution is symmetric: there’s equal amounts of data on either
side of the median, and equal amounts on either side of the mean. Statisticians say the normal distribution
has zero skew, since the mean and median are the same. When the median and mean are different, a
distribution is skewed, which is a way of saying that there are some unusually extreme
values on one side of our distribution, either large or small in our data set. With a skewed distribution, the mode will
still be the highest point on the distribution, and the median will stay in the middle, but
the mean will be pulled towards the unusual values. So, if the mean is a lot higher than the median
and mode, that tells you that there’s a value (or values) that are relatively large
in your data set. And a mean that’s a lot lower than your
median and mode tells you that there’s a value (or values) that are relatively small
in your dataset. Let’s go to the News Desk. The average income of a US family GREW 4 percent
between 2010 and 2013. Those average paychecks expanded from 84-thousand-dollars
to over 87-thousand dollars. But not everyone is cheering. The median income FELL five percent during
those same years. Median family income dropped from 49-thousand dollars to just over 46 and a half thousand dollars. This really happened, back in the years after
the financial crisis. The mean income rose at the same time the
median income fell. That’s because families at the tip-top of
the income distribution…we’re making more money. And pushing the mean up. While many other families were making less. And even though unscrupulous politicians could
accurately claim “average incomes are rising”–and pat themselves on the back–it would be misleading. For most Americans during that stretch incomes
were flat or falling. This points to another really important point
about statistics, a point we’ll come back to time and time again during this series. Statistics can be simultaneously true …and
deceptive. And an important part of statistics is understanding
which questions you are trying to answer. And whether or not the information you have
is answering those questions. Statistics can help us make decisions. But we’ve all gotta use our common sense. And a little skepticism. Thanks for watching. I’ll see you next time.

100 thoughts on “Mean, Median, and Mode: Measures of Central Tendency: Crash Course Statistics #3

  1. Awaiting a video on chi square tests of GOF, homogeneity, and independence/association. I can't tell which test to use given a problem :/

  2. This is a huge pet peeve of mine. I keep hearing politicians saying that the "average" income is rising, and I'm always like, "Mine isn't!"

  3. In A-level we were taught that "average" is an umbrella term for these three measures of central tendency: mean (which is what non-mathematicians normally intend when they say average), median and mode.

  4. Of course there is a mean color, just add up the wavelengths and probably get some yellow-greenish stuff for the mean. Unless someone says ultraviolet and then all gets thrown away :)!

  5. Some of the examples are taken from the book The Tiger that isn’t by Michael Blastland and Andrew Dilnot. Worth the read.

  6. Mode – Most
    Median – Middle
    Range – Subtract the smallest from the largest
    Mean – Add and count and divide

    This is a little song I learned in the fifth grade. Huh. I didn’t know it was statistics.

  7. Dear video editor, please allow for a 1.5 sec lapse after each term definition, before moving to the next subject, in order to let those who needs to make a screenshot, and for better digestion of the definition))

  8. If median is the middle of the low and high data values, then the salary example is wrong. The median salary of the macrame artists and Elon Musk should be a bit less than $50 million.

  9. THANK YOU SO MUCH!!! I love this!!! So much better than re-reading the same sentence in my textbook. Can you make one for Z-Scores and Standard Deviation?

  10. Statistics are currently my favorite field of mathematics.They can be very easily applied, and best of all, they can really help me understand a situation. I feel as though statistics have really helped me make better decisions in my life, and in some ways, made me a smarter individual. Thank you Crash Course for making these truly enlightening videos.

  11. I am preparing for a teacher certification test in another state and this video was very helpful and easy to understand. Thank-You!

  12. Loved it! Hoping I can make a suggestion. I just watched the entire video and I would love to use it in my middle school classes, however, I now live in a place that the one part of the video that shows all the macrame artist out having drinks makes it a no go. I am thinking the video would have been just as informative with shall we say "sweet tea". Just a thought to broaden the audience. Thanks for the good work!

  13. I can observe that this channel is sexist as it only examples showing women and no examples representing men thus subtly discourages men

  14. I feel like most students (at least where I'm from) would be better prepared after watching all crash courses than they are now after finishing high school.

  15. When Elon Musk joins the table, doesn't the median become the average of 20k and 10mil (instead of 20k as said in the video). There's only two distinct values, right?

  16. This reminded me of that one Cookie Clicker news joke, that said "the average person bakes 8 octillion cookies" or something(because the player is an extreme outlier), I think they could use a Median in this one!

  17. I found the "speed" of talking & information given for "learning" about some of this was so fast it was to "fast". I left like she was just trying to go over information she was super familiar with for so long, with others in that similar category of the relation to "her" knowledge on this subject…. "Slow it down some so we can remember this stuff well". 😉

  18. You make people that wouldnt search for subjects like statistics, actually watch and learn statistics, because you make it clear, and fun! Thank you. I will recommend you all over 🙂

  19. You help my child💕💕💕

  20. Hang on, wouldn't modal income be the most accurate way to assess a country's financial situation, since it corresponds to a real value and shows us what most people are earning?

  21. 2:55–3:13 wouldn't mode be your best guess in that situation? That is the most frequent number in the data set and therefore most likely to be an individual data point.
    For instance the foot example said the average person has less than one foot. Let's assign it's value as 1.7 for this situation. If I had 5 friends online, and i were to try guessing how many feet they have; 2 would be a better answer than 1.7 . I know this particularly works in the example since you only have 3 options (0,1,2); but that makes mode the best option since it indicates the most frequent answer. Median isn't necessarily a data point in the set, since when it's an even number you usually average the two to get median (meaning that value:IS NOT part of the set). This displays that Mean and Median aren't even necessarily equivalent to a data point in a set.

    Question: If there is only one value for mode; is that always going to be the best metric for guessing someones data point from a set?

  22. Just for the sake of learning this, it would be cool if mean, median, and mode didn't all being with the same letter. It makes the mnemonics hard for me.

  23. 8:04 – 8:15

    No such thing as an average favorite color? In the digital age, I wouldn't be too sure about that. You could theoretically find the mean R, mean G, and mean B value and composite those three components into a sort of "mean" favorite color. In a very simple example, Roy likes red (255, 0, 0), Bill likes blue (0, 0, 255), Greg likes green (0, 255, 0), Yolanda likes yellow (255, 255, 0), and Pete likes purple (255, 0, 255). This would make the "mean" a color of about (153, 102, 102), which I believe would be a dull reddish tone. Just a bit of fun!

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