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Statistics intro: Mean, median, & mode

The mean (average) of a data set is found by adding all numbers in the data set and then dividing by the number of values in the set. The median is the middle value when a data set is ordered from least to greatest. The mode is the number that occurs most often in a data set. Created by Sal Khan.

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  • orange juice squid orange style avatar for user blindmewithscience
    I've heard of both the arithmetic mean and the geometric mean. What's the difference?
    (578 votes)
    • leaf blue style avatar for user Matthew Daly
      Think about it this way. The arithmetic mean of a bunch of numbers is the number a that satisfies
      x₁ + x₂ + x₃ + .... = a + a + a + ... + a

      The geometric mean is the number b that satisfies
      x₁ * x₂ * x₃ * ... = b * b * b * ... * b

      There is also a harmonic mean, which is the number h that satisfies

      1/x₁ + 1/x₂ + 1/x₃ + ... = 1/h + 1/h + ... + 1/h.
      (551 votes)
  • spunky sam blue style avatar for user Sachin
    Sal, can you please answer my question?
    If the set of numbers were (2 , 4 , 6 , 8 , 10) , how would you find the mode?
    (There are no numbers repeated in the above question.)
    (138 votes)
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  • leafers seedling style avatar for user Amanda Cardey
    is centeral tendancy the same thing as mean?? What is the difference??
    (66 votes)
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    • leaf blue style avatar for user Matthew Daly
      The arithmetic mean is one example of a statistic that describes the central tendency of a dataset. But any other formula or process that takes a dataset and generates a single number that represents a "typical" value is also a measure of central tendency. That includes the median and mode as well as more exotic things like the midrange or the arithmetic mean when you ignore the largest and smallest value. All of these numbers attempt to capture the spirit of a dataset by giving you a sense of a single "usual" value, and that is what makes them measures of central tendency..
      (113 votes)
  • piceratops ultimate style avatar for user Willie J
    if there is a question such as:
    what is the mode of 2,2,3,5,6,5?
    would it be 2 or 5?
    (71 votes)
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  • stelly blue style avatar for user 𝘤𝘩𝘪𝘰𝘮𝘢ゴジラ
    Here are some quick definitions.

    Statistics: the study and manipulation of data (basically it's just data)

    Descriptive statistics: Showing data while summarizing or using a smaller set of numbers.

    Average: (these are just words to describe average) In general, mean, typical, middle, central tendency.

    Arithmetic mean (or just mean, I just like arithmetic mean more): The process of adding a set of numbers, then dividing by the amount of numbers. (Example: 4+3+1+6+1+7= 22 divided by 6. The answer to that is 3 2/3.)

    Median: The "middle" number in a set. Rule: When faced with a set of numbers where the amount of numbers in it are all even, you take the two middle numbers and add them. Then you divide them by two to get the median. (In the set of numbers we had earlier [1, 1, 3, 4, 6, 7], 3 and 4 were the middle numbers. We add them together to get 7, then divide it by 2 to get a median of 3.5.) However, this DOES NOT apply to a set of numbers with an odd set of numbers in it.

    Mode: The most common number in a set (the number that repeats itself the most). Rule: If all the numbers are represented equally (basically if there are the same amount of each number, if that makes sense,) there is no mode. For example, in the set of numbers from earlier (1, 1, 3, 4, 6, 7), the mode would be 1. But if we remove one of the ones, since every number would be represented equally, there would be no mode.

    Hope this helped.
    (29 votes)
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  • old spice man blue style avatar for user AdityaRajesh16
    If two numbers are the most common in a set ( example: 1,2,3,3,4,5,6,6,7), what would be the mode?
    (13 votes)
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  • duskpin ultimate style avatar for user 💕🐱-CatCakes-🐱💕
    How would you use average in real life?
    (10 votes)
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    • aqualine ultimate style avatar for user famousguy786
      There are countless applications. I'll give some examples. The normal body temperature is 98.6 degrees Fahrenheit. How was this exact temperature chosen?This number was given by a German doctor Carl Reinhold August Wunderlich, after examining millions of readings taken from 25,000 German patients and taking their average. The mileage of automobiles is calculated by finding the average volume of fuel consumed by the automobile. Each and every science experiment done in the lab involves calculation of the average reading after repeating the experiment many times, so that error is minimized. In fact, calculating the average is one of the most essential mathematical skills. One would need this knowledge regardless of which field he/she works in.
      (24 votes)
  • piceratops tree style avatar for user ivan
    What if the numbers are 1,3,5,6,7,8,23,42,76,83,93 how do you find the median
    (10 votes)
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    • blobby green style avatar for user e.b.morran
      You put the numbers in order (as you've done) and count how many numbers there are. If there's an odd number of numbers (as in this case), you pick the number in the middle of the list, and that's the median. If there's an even number of numbers, you take the two numbers in the middle, add them together, and divide them by two.
      (18 votes)
  • duskpin sapling style avatar for user 18mertens
    Does anyone know an easy way (such as a song or rhyme) to memorize what mean, median, and mode are?
    (6 votes)
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  • hopper happy style avatar for user HI :) DO NOT READ MY BIO
    what if the numbers only happen once and you are trying to find a mode
    (9 votes)
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Video transcript

We will now begin our journey into the world of statistics, which is really a way to understand or get our head around data. So statistics is all about data. And as we begin our journey into the world of statistics, we will be doing a lot of what we can call descriptive statistics. So if we have a bunch of data, and if we want to tell something about all of that data without giving them all of the data, can we somehow describe it with a smaller set of numbers? So that's what we're going to focus on. And then once we build our toolkit on the descriptive statistics, then we can start to make inferences about that data, start to make conclusions, start to make judgments. And we'll start to do a lot of inferential statistics, make inferences. So with that out of the way, let's think about how we can describe data. So let's say we have a set of numbers. We can consider this to be data. Maybe we're measuring the heights of our plants in our garden. And let's say we have six plants. And the heights are 4 inches, 3 inches, 1 inch, 6 inches, and another one's 1 inch, and another one is 7 inches. And let's say someone just said-- in another room, not looking at your plants, just said, well, you know, how tall are your plants? And they only want to hear one number. They want to somehow have one number that represents all of these different heights of plants. How would you do that? Well, you'd say, well, how can I find something that-- maybe I want a typical number. Maybe I want some number that somehow represents the middle. Maybe I want the most frequent number. Maybe I want the number that somehow represents the center of all of these numbers. And if you said any of those things, you would actually have done the same things that the people who first came up with descriptive statistics said. They said, well, how can we do it? And we'll start by thinking of the idea of average. And in every day terminology, average has a very particular meaning, as we'll see. When many people talk about average, they're talking about the arithmetic mean, which we'll see shortly. But in statistics, average means something more general. It really means give me a typical, or give me a middle number, or-- and these are or's. And really it's an attempt to find a measure of central tendency. So once again, you have a bunch of numbers. You're somehow trying to represent these with one number we'll call the average, that's somehow typical, or middle, or the center somehow of these numbers. And as we'll see, there's many types of averages. The first is the one that you're probably most familiar with. It's the one-- and people talk about hey, the average on this exam or the average height. And that's the arithmetic mean. Just let me write it in. I'll write in yellow, arithmetic mean. When arithmetic is a noun, we call it arithmetic. When it's an adjective like this, we call it arithmetic, arithmetic mean. And this is really just the sum of all the numbers divided by-- this is a human-constructed definition that we've found useful-- the sum of all these numbers divided by the number of numbers we have. So given that, what is the arithmetic mean of this data set? Well, let's just compute it. It's going to be 4 plus 3 plus 1 plus 6 plus 1 plus 7 over the number of data points we have. So we have six data points. So we're going to divide by 6. And we get 4 plus 3 is 7, plus 1 is 8, plus 6 is 14, plus 1 is 15, plus 7. 15 plus 7 is 22. Let me do that one more time. You have 7, 8, 14, 15, 22, all of that over 6. And we could write this as a mixed number. 6 goes into 22 three times with a remainder of 4. So it's 3 and 4/6, which is the same thing as 3 and 2/3. We could write this as a decimal with 3.6 repeating. So this is also 3.6 repeating. We could write it any one of those ways. But this is kind of a representative number. This is trying to get at a central tendency. Once again, these are human-constructed. No one ever-- it's not like someone just found some religious document that said, this is the way that the arithmetic mean must be defined. It's not as pure of a computation as, say, finding the circumference of the circle, which there really is-- that was kind of-- we studied the universe. And that just fell out of our study of the universe. It's a human-constructed definition that we found useful. Now there are other ways to measure the average or find a typical or middle value. The other very typical way is the median. And I will write median. I'm running out of colors. I will write median in pink. So there is the median. And the median is literally looking for the middle number. So if you were to order all the numbers in your set and find the middle one, then that is your median. So given that, what's the median of this set of numbers going to be? Let's try to figure it out. Let's try to order it. So we have 1. Then we have another 1. Then we have a 3. Then we have a 4, a 6, and a 7. So all I did is I reordered this. And so what's the middle number? Well, you look here. Since we have an even number of numbers, we have six numbers, there's not one middle number. You actually have two middle numbers here. You have two middle numbers right over here. You have the 3 and the 4. And in this case, when you have two middle numbers, you actually go halfway between these two numbers. You're essentially taking the arithmetic mean of these two numbers to find the median. So the median is going to be halfway in-between 3 and 4, which is going to be 3.5. So the median in this case is 3.5. So if you have an even number of numbers, the median or the middle two, the-- essentially the arithmetic mean of the middle two, or halfway between the middle two. If you have an odd number of numbers, it's a little bit easier to compute. And just so that we see that, let me give you another data set. Let's say our data set-- and I'll order it for us-- let's say our data set was 0, 7, 50, I don't know, 10,000, and 1 million. Let's say that is our data set. Kind of a crazy data set. But in this situation, what is our median? Well, here we have five numbers. We have an odd number of numbers. So it's easier to pick out a middle. The middle is the number that is greater than two of the numbers and is less than two of the numbers. It's exactly in the middle. So in this case, our median is 50. Now, the third measure of central tendency, and this is the one that's probably used least often in life, is the mode. And people often forget about it. It sounds like something very complex. But what we'll see is it's actually a very straightforward idea. And in some ways, it is the most basic idea. So the mode is actually the most common number in a data set, if there is a most common number. If all of the numbers are represented equally, if there's no one single most common number, then you have no mode. But given that definition of the mode, what is the single most common number in our original data set, in this data set right over here? Well, we only have one 4. We only have one 3. But we have two 1's. We have one 6 and one 7. So the number that shows up the most number of times here is our 1. So the mode, the most typical number, the most common number here is a 1. So, you see, these are all different ways of trying to get at a typical, or middle, or central tendency. But they do it in very, very different ways. And as we study more and more statistics, we'll see that they're good for different things. This is used very frequently. The median is really good if you have some kind of crazy number out here that could have otherwise skewed the arithmetic mean. The mode could also be useful in situations like that, especially if you do have one number that's showing up a lot more frequently. Anyway, I'll leave you there. And we'll-- the next few videos, we will explore statistics even deeper.