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### Course: Statistics and probability > Unit 3

Lesson 2: More on mean and median- Calculating the mean: data displays
- Calculating the median: data displays
- Comparing means of distributions
- Means and medians of different distributions
- Impact on median & mean: removing an outlier
- Impact on median & mean: increasing an outlier
- Effects of shifting, adding, & removing a data point
- Mean as the balancing point
- Missing value given the mean
- Missing value given the mean
- Median & range puzzlers
- Median & range puzzlers

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# Median & range puzzlers

In a car dealership with 11 salespeople, the median number of cars sold is six, and the range of cars sold is four. This means the difference between the highest and lowest number of cars sold is four. Therefore, it's impossible for any salesperson to have sold more than 10 cars.

## Want to join the conversation?

**Range**:

Range is the difference between the largest amount with the smallest amount.*Ex.*

Here is the cups of lemonade Shelly sold each day

2, 5, 7, 1, 3, 4

The biggest value is 7 if you lay out the list in order, the least value is 1, so the range of this list is 6, because 7 - 1 = 6

Answer: 6**Median**:

Median is the middle number when you lay out the numbers in order.*Ex.*

Here is Anna's number of lemonade cups sold on each day.

4, 9, 2, 6, 3

Here is the list if you lay out the numbers in order.

2, 3, 4, 6, 9

The middle number in the list is 4.**Mode**:

Mode is the number that repeats the most in a list.*Ex.*

This is Jhon's pay every day. He earns 69 dollars at max every hour. This is the amount of money he earned every hour today.

54, 23, 76, 69, 69, 69, 23, 56, 43

The number that is repeated the most is 69(intentional number).

If there was 2 or more numbers that were repeated the most the same amount, like 23, 23, 42, 42, 69, 69, the mode is 23, 42, and 69.**Mean**(The worst one):

Mean is when you add every number in the list and divide it by the amount of numbers in the list.*Ex.*

The list is shown below

(Sorry! There is no available story problem).

1,2,3,4,5

(This number list was chosen because it is easier to calculate mean)

Step 1: Add all the numbers, it is 15

Step 2: Divide 15 by the amount of numbers in the list, which is 5

Step 3: Divide 15 by 3, it is 5

Step 4: Write your answer down:**5**

*Thanks for reading this tutorial! For more like this please Upvote!*

**Goal:*Get to 20 upvotes!*

Peace out.

All material is written by aarnav.school. Pls don't copy and comment again. This can be used in a presentation, when used, include this text.

Year written: 2023

Month written: May

Written by: aarnav.school(43 votes)- excuse me, you need to post an answer like this one for you to get the badge(5 votes)

- That was the most confusing video on here!(28 votes)
- What were you confused about? Maybe I can help :)(18 votes)

- I think that when you have an odd number of people is easy, but what about even numbers? Like in this video, it said 11 salespeople, but what about 10 salespeople? Maybe you can do a video about an even amount of people or things?(15 votes)
- It certainly does make the question a lot harder.

Just for fun, lets try to solve it anyway.

Instead of 11 salespersons, lets have 10!*min, _, _, _, A ,B, _, _, _ ,max*

We know that*(A + B)/2 = 6*, so*A + B = 12*

We also know that the range in 4, so*max - min = 4*

To test the claim that at least one salesperson sold more than 10 cars, lets make the max = 11.

As,*max - min = 4*,*min = 7**7, _, _, _, A ,B, _, _, _ ,11*

The lowest possible value of A is 7 (*7, 7, 7, 7, A ,B, _, _, _ ,11*).

If*A = 7*, then*B = 5*because we knew*A + B =12*

But B cannot be less than A, therefore the statement that at least one salesperson sold more than 10 cars is**false**.

The key lies in the fact that if there is an even number of items in the data set, then the median is found by taking the mean (average) of the two middlemost numbers.

I hope this helps.(26 votes)

- Sal's drawing skills on point 🚘😎(16 votes)
- We can have 10 as maximum, and actually we can even have 5 10 in total.

e.g. 6 6 6 6 6 6 10 10 10 10 10

Median is 6

Range is 4(16 votes)- I think you could be right(1 vote)

- Can't we use this method? We know the median equals (min+max)/2 so (min+max)/2 = 6 => min + max = 12

and we know the range = max - min. and the range is 4. so max = 4 + min. so we have two equations. min + max =12 and max = 4 + min. If we substitute in place of max we will have. min + 4 + min =12 =>2min=8 => min = 4 and max = 8. now we have the exact number. isn't this a more efficient way?(6 votes)- I wonder if ur still active on khan bc this was a while ago(1 vote)

- I’m still confused :D(6 votes)
- Why does the video have to be like 7 minutes long when on the lesson only has 4 questions.(4 votes)
- hello i have a question I need more videos I do not get this subject(5 votes)

## Video transcript

- [Voiceover] Let's say that
we run a car dealership, and we have 11 salespeople, and we record the number of cars each of the 11 salespeople
sold in the past week. So that's one of them,
two, three, four, five, six, seven, eight, nine, 10, 11. So we record how much each of them sold. So maybe one person sold,
maybe they sold five cars. Maybe the next person sold seven cars. Maybe the next person sold 10 cars. And so that's how we would do it. So we're going to record all of that. But then we're told that
the median car sold is six. So if we ordered all of these numbers, so let's just assume
that we ordered them all from least to greatest. So maybe, and I'm just
making up numbers here, maybe this is four. This is four. Maybe this is five,
five, five, six, seven, seven, eight, nine, 10. The median car sold, meaning
that the middle number here is going to be six. Now we don't know whether
all of these other numbers are the actual numbers. I just made those up. But we know that the middle
number needs to be six. So let me just fill out
only the middle number. So the middle number, if
you have 11 data points, the middle number's going
to have five on either side. So that's that one right over there. One, two, three, four, five. One, two, three, four, five. So we know that the median,
we know the median-- Actually, let me do that a purple color. The median is going to be six. Now the other thing we know is that the range of cars sold is four. Now let's remind ourselves, the range is the maximum
number of cars sold minus the minimum number of cars sold, and if we have sorted all of these out, if we take-- If this is our max right over here, because we've sorted them all out, and this is our min right over here, our range is going to be
our max minus our min. So let me write that down. So you take your min and your max. You have your range. Range is equal to the
maximum minus the minimum. And they're telling us that that is four. This range is four. So the maximum minus the minimum, the difference between the number of cars that the most productive salesperson sold and the number of cars that the least productive
salesperson sold, that difference is going to be four. So given all of those assumptions, giving all of that information, I'm now going to give you a statement, and our challenge is if we assume everything I just said is true, is this statement going to be true, or is this statement going to be false, or do we not know? So let me write this down. So is this going to be true, false, or do we not know? Do we not have enough information to say it's for sure going to be true, for sure going to be false, or for sure we don't know? And so I encourage you
to pause the video now and try it out. All right, so let's work
through it together. This is kind of a fun puzzle here, where we've been given some clues, and then we have a statement, and we figure out can
we make this statement. Can we say it's true or can we
say for sure that it's false, or do we just not know? So a couple of ways to tackle it. One way to tackle it is, well look, one of the salespeople
must have sold six cars. If the median sold is six, and there's 11 salespeople here, the middle number here,
that if we order it, one of them must have sold six cars. If we had an even number here, then that would be the
average of the two middles, but if we have an odd number here, that is literally the
middle number of cars sold. So someone sold six cars. So if we assume-- We're trying to find a
world where someone sold more than 10 cars. So if someone sold six cars, is there a way, with a range of four, that someone sold more than 10? Well let's just think about it. Let's just assume that the min is six. We don't know that the min is six, but let's just try it out. If the minimum is six,
if the minimum is six, what's the maximum going to be? Well remember, the range is equal to-- We know that the range is four. The range is equal to the
maximum minus the minimum, maximum minus the minimum, and so the maximum in this
case would be 10, would be 10. The max would be equal
to, would be equal to 10. If the min is equal to
six, and instead of colons I'm going to write an equals sign, if the min is equal to six, then the max at most can be equal to 10. And we can't take a higher min, because we know that the six is going to be one of the values. We could try a lower minimum value. We could say the minimum
value is five or four or three or two or one, but then the maximum
would go down even more, because remember, the
maximum is no more than four larger than the minimum. So if we assume six, and we know that one of the salespeople sold six cars, then the maximum that
any of the salespeople could have sold is 10. And so the statement, at
least one of the salespeople sold more than 10 cars,
that's got to be false. That's got to be false. Now there's another way that
you could think about it. You could assume that
someone sold more than 10. You could assume. In the last example, or the
last way of thinking about it, we assumed that the min was six, but now let's just assume, let's just assume that someone sold-- Let's just try it out. Let's see if it's possible. Let's just assume that the max is 11, that someone sold more than 10 cars. If the max is equal to 11, what's the min going to need to be? Well we just have to remind ourselves, range is equal to max minus min. So four is equal to the maximum, 11, minus the minimum. So 11 minus what is equal to four? Well 11 minus seven is equal to four. So if we assume that the maximum is 11, then the minimum is going to have to be, the minimum is going to
have to be equal to seven. Now can the minimum be seven
when your median is six? No. If your median is six,
that means you have-- That means if you have an
odd number of data points, that means one of the data points is six, and if you had an even
number of data points, that means that the middle two are going to average out to be six, which means you even have data points that are less than six. So your minimum can't be seven. You're going to have a value
that is at least as low, at least as low as six. So your assumption can't
be true, can't be true, so once again, the assumption
based on the statement can't be true. This statement is going to be false.