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## AP®︎/College Statistics

### Course: AP®︎/College Statistics > Unit 2

Lesson 3: Describing the distribution of a quantitative variable# Classifying shapes of distributions

When we describe shapes of distributions, we commonly use words like symmetric, left-skewed, right-skewed, bimodal, and uniform. Not every distribution fits one of these descriptions, but they are still a useful way to summarize the overall shape of many distributions.

## Want to join the conversation?

- i mean do people mesure house flies?(18 votes)
- What are some applications of this? Never seen it used in real life? @0:00do baseball statisticians use this?(5 votes)
- you could use this in real life because it can tell you correlation and averages, like on the coffee graph you can look and see most people drink 3 cups a day. Another example is how you can see that in almost all skewed distribution you see correlation (ex. in left tailed as x goes up y goes up) so you use this in real life to be able to see things like how exercising every day relates to longer life span. Though while doing math memorizing distribution types can help with just being able to glance at the graph and getting the gist. (ps. I don't know much about baseball so wouldn't know if base ball statisticians use this but I would guess they do because almost all statisticians do.)(4 votes)

- Is a random distribution always uniform? What about the viceversa?

The first implication (random=>uniform) seems reasonable: if you toss a coin you expect a uniform distribution of tails and heads as the number of throws grows bigger and bigger, the same reasoning seems to apply for throwing dice or generating random numbers.

The second implication (uniform =>random) "feels" false but I can't come up with a meaningful counter example. Any ideas?(5 votes)- If graphed, a uniform distribution will look like what most people would describe as a "bell-shaped" curve, with the majority of the data points clustering around the central tendencies, like test scores. There will be a lot of people in a class that get a passing grade, while a small portion will not get a passing grade, and a small portion will get above passing grade. Meanwhile, a random distribution, if graphed, will look uniform. Because it is random, then any given data point will have an equal chance of landing in each bucket. For example, rolling the dice will yield a roughly uniform distribution graph because each time you roll the die, each number is equally likely to come up. But if you randomly survey a person's test scores, the probability that he or she getting a average, above-average, or below-average score will not be equivalent. That student will be much more likely to get a "C" instead of an "A" or "F". Hope that helps!(1 vote)

- The bi-modal graph example (to do with high temperatures), how many groups of data is in that graph, and how would one understand that graph?(3 votes)
- Each bar tells us the amount of days the daily high temperature was within a certain interval.

For example, the third bar from the left says that there were 8 days with a daily high temperature between 5 and 7.5 °C.

My guess is that the left half of the graph are mostly winter days, while the other half are mostly summer days, in which case the temperature a typical winter day would be somewhere around 5 °C, and in the summer more like 25 °C.(5 votes)

- How would trimodal look like?(4 votes)
- A trimodal graph, as the name suggests, would have three "peaks," or places where a lot more data points are clustered around.(1 vote)

- What is the exact meaning of an outlier?(3 votes)
- Can someone please explain the concept to me?(3 votes)
- Do you only describe the data as bimodal or unimodal if its symmetric or are there other instances that you would describe the data as bimodal or unimodal? (Basically, when would you use those certain shapes?)(3 votes)
- Neither a bimodal graph nor an uni-modal graph has to be symmetric. A uni-modal graph generally refers to all distributions where a significant percentage of data points cluster around a certain value. A bimodal graph refers to all distributions where a significant percentage of data points cluster around two certain values. They do not have to be symmetric.(1 vote)

- My guess is that the left half of the graph are mostly winter days,(2 votes)
- I am unsure how to solve a problem like this: Suppose that the value of a stock varies each day from $9.82 to $35.17 with a uniform distribution.

Find the upper quartile; 25% of all days the stock is above what value?(1 vote)

## Video transcript

- [Instructor] What we have here are six different distributions. And what we're gonna do with this video is think about how to classify them, or use the words people typically use to classify distributions. So, let's first look at this
distribution right over here, it's the distribution of
the lengths of houseflies. So, someone went out there and measured a bunch of houseflies. And then said, "Hey look, there's many houseflies that are between six tenths of a centimeter and six and a half
tenths of a centimeter." Looks like there's about
40 houseflies there. And then if you say between six
and a half and seven tenths, there's about 30 houseflies. And if you were to say
between five and a half tenths and six tenths, it looks like
it's about the same amount. This type of distribution
is usually described as being symmetric. Why is it called that? Because if you were to draw a line down the middle of this distribution, both sides look like mirror
images of each other. This one looks pretty exactly symmetric. But more typically when
you're collecting data, you'll see roughly
symmetric distributions. Now, here we have a distribution that gives us the dates on pennies. So, someone went out there, observed a bunch of pennies, looked at the dates on them. They saw many pennies, looks like a little bit
more than 55 pennies, had a date between 2010 and 2020. While very few pennies had a date older than 1980 on them. And this type of distribution when you have a tail to the left, you can see it right over here, you have a long tail to the left, this is known as a
left-skewed distribution. Left-skewed. Now in future videos,
we'll come up with more technical definitions of
what makes it left-skewed, but the way that you can recognize it is, you have the high points of
your distribution on the right, but then you have this long tail that skews it to the left. Now, the other side of a left-skewed, you might say, well, that would be a right-skewed distribution, and that's exactly what
we see right over here. This is a distribution
of state representatives, and as you can see, most of
the states in the United States have between zero and ten representatives. It looks like it's a little over 35. None of them actually have zero, they all have at least one representative, but they would fall into this bucket, while very few have more
than 50 representatives. So, here where the bulk of our
distribution is to the left, where we have this tail
that skews us to the right, this is known as a
right-skewed distribution. Now, if we look at this next distribution, what would this be? Pause this video and think about it. Well, this could be a
distribution of maybe someone went around
the office and surveyed how many cups of coffee each person drank, and if they found someone who drank one cup of coffee per day, maybe this would be them. If they found another person who drinks one cup of coffee, that's them, then they found three people who drank two cups of coffee. Well, this is a very similar situation to what we saw on the dates on pennies. A large amount of our data
fell into this right bucket of three cups of coffee, but then we have this tail to the left. So, this would be left-skewed. Now, these right two
distributions are interesting. One could argue that
this distribution here, which is telling us the number of days that we had different high temperatures, that this looks roughly symmetric, or actually even looks exactly symmetric. 'Cause if you did that little exercise of drawing a dotted line down the middle, it looks like the two sides are
mirror images of each other. Now, that would not be
technically incorrect. But typically when you
see these two peaks, this would typically be called
a bi-modal distribution. So, even though bi-modal distributions can sometimes be symmetric
or roughly symmetric, you wanna be more precise, and here when you have these two peaks, that's where the bi comes from. You'd call it bi-modal,
and this makes sense because you have a lot of days that are warm that might
happen during the summer and you might have a lot
of days that are cold that are happening during the winter. Now, this last distribution here, the results from die rolls, one could argue as well that
this is roughly symmetric. It's not exact, it's
not perfectly symmetric, but when you look at this dotted line here on the left and the right sides it looks roughly symmetric. But a more exact classification here would be that it looks
approximately uniform. So, rather than calling it
a symmetric distribution, or a roughly symmetric distribution, most people would classify this as an approximately uniform distribution.