WEBVTT mathematics/statistics/son
00:00:00.000 --> 00:00:02.400
Hi and welcome to www.educator.com.
00:00:02.400 --> 00:00:09.000
We are going to be talking about central tendency, mean, median, and mode.
00:00:09.000 --> 00:00:15.600
Mean, median, and mode are what most people think about when they think about statistics especially descriptive statistics.
00:00:15.600 --> 00:00:20.800
When I ask my students what is descriptive statistics about, they are like mean, median, and mode.
00:00:20.800 --> 00:00:24.500
Sometimes they also say standard deviation.
00:00:24.500 --> 00:00:28.800
I know this is largely what sticks on people’s minds.
00:00:28.800 --> 00:00:32.000
We are going to be talking about these three measures of central tendency.
00:00:32.000 --> 00:00:34.900
Then we are going to spend a little bit more time on mean.
00:00:34.900 --> 00:00:45.300
We are going to review the summation symbol Sigma (Σ) and we are also going to talk about the different formulas
00:00:45.300 --> 00:00:48.600
for population mean versus sample mean.
00:00:48.600 --> 00:00:58.300
Then we are going to talk about some Excel examples and finally we are going to compare these measures of central tendency.
00:00:58.300 --> 00:01:06.000
Central tendency, basically the idea is we want to be able to summarize a distribution and we have learned about
00:01:06.000 --> 00:01:08.300
some different ways you could summarize it like the shape.
00:01:08.300 --> 00:01:15.200
For example you say it is a uniform shape, it is a nice way to summarize a distribution.
00:01:15.200 --> 00:01:20.300
Shape is just one dimension that you could summarize a distribution on.
00:01:20.300 --> 00:01:21.900
A way that you can describe it on.
00:01:21.900 --> 00:01:25.700
Central tendency is another dimension that you could describe it on.
00:01:25.700 --> 00:01:30.000
The so central tendency you could think of as the middleness.
00:01:30.000 --> 00:01:34.300
How do you say that is the middle of a distribution?
00:01:34.300 --> 00:01:35.700
There is a couple of ways.
00:01:35.700 --> 00:01:47.200
When we talk about mode, we are going to be talking about most frequent value as sort of this center member.
00:01:47.200 --> 00:02:00.000
In the case with 1, 2, 2, 3, 4, 5, 7, 2 is the most frequent value and because of that we are going to say okay that is one way
00:02:00.000 --> 00:02:10.000
we could describe the center median is literally the middle value.
00:02:10.000 --> 00:02:16.800
In order to find the median it often helps to line up your distribution in order.
00:02:16.800 --> 00:02:22.100
We have the least on this side and the greatest on this side.
00:02:22.100 --> 00:02:37.100
We would just count to the center so since there is 7 numbers here, we know that is an odd number so here we just pick the middle value, the median.
00:02:37.100 --> 00:02:39.600
Let us talk about mean.
00:02:39.600 --> 00:02:48.600
Mean is what people often call the average, the average number, the average value.
00:02:48.600 --> 00:02:59.300
With mean we actually care what each value means, what is the extent of each value.
00:02:59.300 --> 00:03:10.300
Because of that what we are going to do is add up, sum up all the values in our distribution,
00:03:10.300 --> 00:03:22.900
in our sample and divide by the number of values you have in your sample.
00:03:22.900 --> 00:03:31.300
Here we are just going to add all these up so here is a 5, 10, and another 10, 20, 24.
00:03:31.300 --> 00:03:39.100
24 ÷ 7 and that will give you that mean.
00:03:39.100 --> 00:03:44.700
This is probably something like 3 and 3/7.
00:03:44.700 --> 00:03:49.000
That is our mean.
00:03:49.000 --> 00:03:58.700
I have changed that same distribution just slightly, all that I have done is I have added an extra number and let us see what changes.
00:03:58.700 --> 00:04:04.500
The way we describe the central tendency of our distribution change?
00:04:04.500 --> 00:04:05.900
Let us see.
00:04:05.900 --> 00:04:15.600
Remember mode is most frequent, does our most frequent value change?
00:04:15.600 --> 00:04:17.500
In this case, no.
00:04:17.500 --> 00:04:22.400
2 is still our most frequent value.
00:04:22.400 --> 00:04:27.700
What about the median, the middle value? has that changed?
00:04:27.700 --> 00:04:37.400
Here we have 8 numbers in our distribution and because of that there is no exact middle point.
00:04:37.400 --> 00:04:40.400
The middle is in between two numbers.
00:04:40.400 --> 00:04:46.700
In the case what you do is you take those two numbers in the middle and you divide by 2.
00:04:46.700 --> 00:04:48.300
You add them up and you divide by 2.
00:04:48.300 --> 00:05:03.800
Basically you find the average of the two middle values or average of two middle values.
00:05:03.800 --> 00:05:13.500
In this case it would be 2 + 3 ÷ 2, something like 2, 2 ½ or 2.5.
00:05:13.500 --> 00:05:15.400
That would be our median.
00:05:15.400 --> 00:05:25.300
Notice that our median has changed a little bit because we added something over the smaller side of our distribution, our median has shifted over.
00:05:25.300 --> 00:05:31.400
Let us think about the mean or the average.
00:05:31.400 --> 00:05:34.700
Let us add them up and see.
00:05:34.700 --> 00:05:44.200
0, because I added a 0 the actual value of the sum does not change, that is 24.
00:05:44.200 --> 00:05:48.300
We are just dividing by 8.
00:05:48.300 --> 00:05:54.600
Our mean is now just 3 instead of 3 and 3/7.
00:05:54.600 --> 00:06:08.200
Once again because we added a number that is small in our distribution, our mean has shifted over a little bit towards the smaller side.
00:06:08.200 --> 00:06:13.800
That is basically the ideas of central tendency.
00:06:13.800 --> 00:06:24.700
Before we go on to talk about the actual formula, the formal algebraic notation for mean, I want to talk a little bit about the summation symbol.
00:06:24.700 --> 00:06:29.500
What we want is some kind of symbol that will tell us add up all the numbers.
00:06:29.500 --> 00:06:33.700
I do not care how many numbers you have, add them all up.
00:06:33.700 --> 00:06:37.200
It will be nice if we have a symbol that could do that.
00:06:37.200 --> 00:06:40.700
Here is the summation symbol (Σ).
00:06:40.700 --> 00:06:49.100
It looks like that sigma (Σ), upper case.
00:06:49.100 --> 00:06:53.100
We are going to be using lower case sigma for something else.
00:06:53.100 --> 00:07:03.700
Upper case sigma and people will write some variable here to represent which variable set you want to take from.
00:07:03.700 --> 00:07:06.700
Add up all of age.
00:07:06.700 --> 00:07:21.600
If it is age and I will represent that with letter A, if A = age and I want to get the average age then maybe I will put A here.
00:07:21.600 --> 00:07:25.000
Sum up all of A.
00:07:25.000 --> 00:07:28.200
Sometimes the summation symbol is written like this.
00:07:28.200 --> 00:07:36.600
There is always a little more detail that sometimes they do not show you what you cannot see.
00:07:36.600 --> 00:07:40.300
Here is often what is hidden underlying this.
00:07:40.300 --> 00:07:46.900
Usually this is saying add up everybody in your set A.
00:07:46.900 --> 00:07:57.600
Take them all the way from I, the index from the first one all the way to the last one in your set.
00:07:57.600 --> 00:08:05.200
Remember how many are in our set is usually represented by n.
00:08:05.200 --> 00:08:12.900
From 1 all the way to n, 1, 2, 3, 4, 5, 6, 7, all the way to n, whatever n is.
00:08:12.900 --> 00:08:16.100
Add up A.
00:08:16.100 --> 00:08:19.600
Here we are going to put a little A sub I.
00:08:19.600 --> 00:08:21.200
This means this.
00:08:21.200 --> 00:08:35.600
Here A is age, and let us say A is actually the set 5, 10, and 15.
00:08:35.600 --> 00:08:43.000
Here is the corresponding I, 1, 2, and 3.
00:08:43.000 --> 00:08:57.400
What the summation symbol is telling us is if I =1 for this one this index twice to, add up A sub I, the first A.
00:08:57.400 --> 00:09:02.100
Then add to it A sub 2, which is 10.
00:09:02.100 --> 00:09:07.200
Then add to is A sub 3, which is 15.
00:09:07.200 --> 00:09:10.700
That is how the summation symbol works, al the way up to n.
00:09:10.700 --> 00:09:15.900
N in this case is 3 because there is 3 numbers in our set.
00:09:15.900 --> 00:09:20.800
Let us do one more example.
00:09:20.800 --> 00:09:29.800
Here is my summation symbol and now I’m going to say add up x.
00:09:29.800 --> 00:09:45.700
Let us say x is number of books read this summer.
00:09:45.700 --> 00:09:54.900
X is going to be 2, 4, 6, 8, 10.
00:09:54.900 --> 00:10:09.500
We now that it is saying go from I sub 1 all the way to n, n in this case is 5.
00:10:09.500 --> 00:10:25.200
All the way to I here, this is x sub 1, x sub 2, x sub 3, x sub 4, x sub 5.
00:10:25.200 --> 00:10:33.900
Here, this is telling us go all the way from x sub 1 to x sub 5 and add them all up.
00:10:33.900 --> 00:10:48.500
This is just a formal algebraic notation to say add up all the numbers in your set no matter how big or small your set is.
00:10:48.500 --> 00:10:59.500
Now that you know the summation symbol we can start creating the formula for mean, samples, and population.
00:10:59.500 --> 00:11:02.400
Let us think about the mean for the sample.
00:11:02.400 --> 00:11:11.600
The mean of the sample is always represented by the symbol x bar.
00:11:11.600 --> 00:11:14.100
How do we calculate x bar?
00:11:14.100 --> 00:11:20.400
We have to add up all the numbers in our set and divide by the number of items in our set.
00:11:20.400 --> 00:11:25.500
We know that number of items is n.
00:11:25.500 --> 00:11:32.700
We could use our summation symbol.
00:11:32.700 --> 00:11:37.700
We know that it is the summation symbol but all of x.
00:11:37.700 --> 00:11:41.400
This is one way that you could write it and that is a very simple way.
00:11:41.400 --> 00:11:48.000
It is implicit but it is telling you go to i=1 all the way to n.
00:11:48.000 --> 00:11:55.700
But just for our purposes, I’m just going to put in the hidden stuff just to show you as well.
00:11:55.700 --> 00:12:08.500
It is implicitly saying go from I =1 all the way up to n, however many n is and because of that out of each x sub i.
00:12:08.500 --> 00:12:12.500
X sub 1, x sub 2, x sub 3, all the way to n.
00:12:12.500 --> 00:12:18.300
That is one way that you could see it but you do not need all of this complicated stuff.
00:12:18.300 --> 00:12:21.100
That is optional.
00:12:21.100 --> 00:12:28.800
If you have the distribution of the actual population which is almost impossible to get
00:12:28.800 --> 00:12:32.800
but let us say from some reason you want to write a formula for it.
00:12:32.800 --> 00:12:39.800
Actually you do want to write a formula for it because it is going to be handy if we do not.
00:12:39.800 --> 00:12:42.600
How would we write the formula for that?
00:12:42.600 --> 00:12:51.100
For population, we do not call the population mean x bar, instead we call it μ.
00:12:51.100 --> 00:12:54.100
Our population mean is called μ.
00:12:54.100 --> 00:13:06.900
Here we want to add up all of the x but here we use an upper case X because we are saying draw nail from the population distribution.
00:13:06.900 --> 00:13:15.800
Instead of the lower case n, we are going to write upper case n because upper case N means how many values
00:13:15.800 --> 00:13:21.900
you have in your population not just your sample.
00:13:21.900 --> 00:13:32.900
I’m going to put in the invisible stuff, go from X sub 1 all the way up to N.
00:13:32.900 --> 00:13:35.400
Here I’m going to put that index.
00:13:35.400 --> 00:13:36.900
There you have it.
00:13:36.900 --> 00:13:44.200
This is the population formula and this is the sample formula.
00:13:44.200 --> 00:13:51.600
Although their sums like differences and notations, for instance here we use the Greek letter,
00:13:51.600 --> 00:13:56.600
we use the roman letter, here we use upper case, here we use lower case.
00:13:56.600 --> 00:13:58.500
Except for I, I just means index.
00:13:58.500 --> 00:14:03.800
It just means a little counter or pointer to each thing in the set at a time.
00:14:03.800 --> 00:14:10.200
I is just, you could use j or whatever you want nut we usually use I for index.
00:14:10.200 --> 00:14:15.200
Other than those little characteristics, the mean is the same.
00:14:15.200 --> 00:14:22.600
It means add up all values in your set and divide by however many you have in your set.
00:14:22.600 --> 00:14:25.900
I want to point out one other thing here.
00:14:25.900 --> 00:14:33.800
Here one thing you could see is that if you have the mean of your sample.
00:14:33.800 --> 00:14:39.600
If you happen to have x bar and you multiply n to each side.
00:14:39.600 --> 00:14:42.800
I’m going to multiply n to each side invisibly here.
00:14:42.800 --> 00:14:45.300
I’m going to multiply n.
00:14:45.300 --> 00:14:52.200
Then you will get the sum of all of your x.
00:14:52.200 --> 00:14:56.000
That is just algebraic transformation.
00:14:56.000 --> 00:14:58.600
I have not done anything to change the formula.
00:14:58.600 --> 00:15:00.200
The formula stays the same.
00:15:00.200 --> 00:15:05.000
This little trick is going to come in handy later on.
00:15:05.000 --> 00:15:10.300
Just keep that in mind in the back of your head.
00:15:10.300 --> 00:15:13.700
Let us do some Excel example.
00:15:13.700 --> 00:15:23.900
Here we see that this is asking us about our data once again from our 100 www.facebook.com friends.
00:15:23.900 --> 00:15:31.100
It is asking us find the mode, median, and mean in Excel for height as well as male height.
00:15:31.100 --> 00:15:36.700
Just to remind you, here is our frequency distribution that we looked at in previous lessons.
00:15:36.700 --> 00:15:48.500
We thought that this was a bimodal distribution, if you just consider height of everybody.
00:15:48.500 --> 00:16:03.000
This one is actually a little bit taller because we have to add up all these little guys on.
00:16:03.000 --> 00:16:08.200
They are always asking us to do is find the mode, median, and mean in Excel.
00:16:08.200 --> 00:16:10.400
Excel is going to make life a little bit easier for us.
00:16:10.400 --> 00:16:11.900
We do not have to put them in order.
00:16:11.900 --> 00:16:15.500
We do not have to spend time adding them up.
00:16:15.500 --> 00:16:18.300
Excel will do it automatically for us.
00:16:18.300 --> 00:16:20.800
Let us open it up to Excel.
00:16:20.800 --> 00:16:25.400
Here is our data once again and remember it is asking us the height.
00:16:25.400 --> 00:16:31.500
I’m going to go to our height variable, our height column.
00:16:31.500 --> 00:16:38.600
It is height in inches and I’m going to click on the height sheet.
00:16:38.600 --> 00:16:41.800
Here I have put in some labels for us.
00:16:41.800 --> 00:16:44.500
Height, find the mode, median, and mean.
00:16:44.500 --> 00:16:49.800
Just the height of males, find the mode, median, and mean.
00:16:49.800 --> 00:16:53.300
Excel makes it easy for us with their functions.
00:16:53.300 --> 00:16:59.200
Their functions for mode is simply mode.
00:16:59.200 --> 00:17:03.500
We are just going to go and select our data to find our mode from.
00:17:03.500 --> 00:17:09.600
Excel will go ahead and count which is the most frequent.
00:17:09.600 --> 00:17:12.300
I’m going to close my parentheses and hit enter.
00:17:12.300 --> 00:17:17.900
It turns out that our most frequent mode is 64.
00:17:17.900 --> 00:17:20.300
Let us find the median.
00:17:20.300 --> 00:17:25.600
Excel makes it easy for us once again, it is just =median.
00:17:25.600 --> 00:17:32.800
If you are ever at lost for how to find formulas, one thing I do is www.google.com it over.
00:17:32.800 --> 00:17:42.900
I look it up on the Excel help or function help and they have a whole bunch of list of functions categorized into different types.
00:17:42.900 --> 00:17:48.100
It is pretty easy to find things that you need.
00:17:48.100 --> 00:17:54.300
We should have saved whatever our data is but we could just drag it again.
00:17:54.300 --> 00:18:00.300
Here is our data.
00:18:00.300 --> 00:18:04.200
I’m going to close my parentheses and hit enter.
00:18:04.200 --> 00:18:07.200
Here we see that the median is different from the mode.
00:18:07.200 --> 00:18:09.100
The median is actually 66.
00:18:09.100 --> 00:18:10.600
That is the middle.
00:18:10.600 --> 00:18:13.900
But the most frequent values is 64.
00:18:13.900 --> 00:18:17.200
Now let us calculate the mean.
00:18:17.200 --> 00:18:22.500
In Excel, mean would not mean anything.
00:18:22.500 --> 00:18:30.100
Instead, you have to type in average.
00:18:30.100 --> 00:18:39.200
Let us put in our data, and I’m going to close in my parentheses.
00:18:39.200 --> 00:18:40.600
Hit enter.
00:18:40.600 --> 00:18:46.900
What we find is our average or mean is 67.
00:18:46.900 --> 00:18:48.800
It is not 66 or 64.
00:18:48.800 --> 00:18:57.700
Remember height was bimodal distribution.
00:18:57.700 --> 00:19:10.900
One of the things that we see here in this bimodal distribution is that mode, median, and mean are not necessarily of the same value.
00:19:10.900 --> 00:19:22.100
Remember how does males, when we looked at it before, this was actually approximately normal distribution.
00:19:22.100 --> 00:19:26.400
Normal distribution for male, we are going to find that more in detail later.
00:19:26.400 --> 00:19:33.100
Normal distribution basically means it has a one month in look.
00:19:33.100 --> 00:19:38.300
It has axis of symmetry, that is also the mode.
00:19:38.300 --> 00:19:50.400
It is unimodal, symmetrical, and the mode of inflection is about the size of a standard deviation.
00:19:50.400 --> 00:19:52.300
Let us look at height of males.
00:19:52.300 --> 00:19:55.100
The mode, median, and mean.
00:19:55.100 --> 00:19:59.900
In order to find that just for males, one thing we may want to do is sort our data.
00:19:59.900 --> 00:20:03.600
We probably want to sort it so that all the males height are grouped together.
00:20:03.600 --> 00:20:08.900
I have already sorted it for you but I’m just going to color the heights of males.
00:20:08.900 --> 00:20:23.500
I’m just going to color this blue so that we can remember to ourselves these are the heights of males.
00:20:23.500 --> 00:20:30.900
We could just use the same formulas, mode.
00:20:30.900 --> 00:20:39.800
I’m only going to select these blue ones and I’m going to close my parentheses.
00:20:39.800 --> 00:20:57.400
For males, the mode is 69 and I’m just going to copy control c so that I do not have to go back to my data all the time.
00:20:57.400 --> 00:21:08.100
Let us put in median and I’m just going to command v, copy and paste in my data, hit enter.
00:21:08.100 --> 00:21:13.700
How do you know? The mode and the median are the same number.
00:21:13.700 --> 00:21:25.000
Finally I am putting in average and I’[m just copying my data and here we find that the average is also the same.
00:21:25.000 --> 00:21:26.500
The mean is also the same.
00:21:26.500 --> 00:21:31.200
That is largely what you find from normal way of distribution.
00:21:31.200 --> 00:21:35.200
You find that the mean, median, and mode are the same values.
00:21:35.200 --> 00:21:42.100
That is one thing handy about a normal distribution.
00:21:42.100 --> 00:21:47.100
That is our Excel example.
00:21:47.100 --> 00:21:54.700
Let us contrast median and mean, two of the measures of central tendency.
00:21:54.700 --> 00:22:02.000
One of the things about median that you should know is that it is handy because it is less affected by outliers.
00:22:02.000 --> 00:22:08.000
Means are more affected by outliers and you could think about why.
00:22:08.000 --> 00:22:22.900
They are more affected because they are actual extent of the values matter.
00:22:22.900 --> 00:22:32.100
If you have a very large number, that value is exactly how much that value is, it gets added in.
00:22:32.100 --> 00:22:36.900
Because of that means are more affected by outliers or one extreme score.
00:22:36.900 --> 00:22:46.600
Medians are less affected even if you add in one extreme score, it does not usually change the median by too much.
00:22:46.600 --> 00:22:57.600
Now let us talk about the relationship between the parameter, the parameter mean, and the statistic mean or the parameter median or statistic median.
00:22:57.600 --> 00:23:06.400
The sample mean or x bar is actually the best predictor of the population mean or μ.
00:23:06.400 --> 00:23:17.900
Usually in medians though, we do not use the median of a sample which is not very many people use
00:23:17.900 --> 00:23:48.200
that in order to predict the median of the population.
00:23:48.200 --> 00:23:55.900
Not many people use that to predict this and largely it is because the relationship is less stable than this one.
00:23:55.900 --> 00:24:02.600
We are going to learn more about why this relationship holds later on in the lessons.
00:24:02.600 --> 00:24:15.100
Different types of measurements, it will be better to use median or mean for different types of measurements.
00:24:15.100 --> 00:24:25.800
For mean, usually you want to use any type of measurements that are either ratio or interval.
00:24:25.800 --> 00:24:34.500
Those are going to be best for using mean as an indicator of central tendency.
00:24:34.500 --> 00:24:45.000
Median is best for measurements that are ordinal.
00:24:45.000 --> 00:24:51.800
For nominal measurements, median and mean do not mean very much.
00:24:51.800 --> 00:24:56.700
For nominal measurements, you may want to use mode.
00:24:56.700 --> 00:24:59.900
Which distribution you want to use this with?
00:24:59.900 --> 00:25:08.600
Median is most frequently use for describing the center of skewed distributions.
00:25:08.600 --> 00:25:12.300
That is what you want to think about when you think about median.
00:25:12.300 --> 00:25:20.500
When you think of mean, this can be used for a lot of distributions but for skewed distributions the mean will be a little bit off.
00:25:20.500 --> 00:25:24.500
For all others.
00:25:24.500 --> 00:25:27.500
The mean is going to be pretty flexible for us.
00:25:27.500 --> 00:25:32.200
We are going to be using that quite a bit.
00:25:32.200 --> 00:25:34.800
Let us move on to an example.
00:25:34.800 --> 00:25:39.500
Sometimes people think that mean, median, and mode is pretty easy.
00:25:39.500 --> 00:25:40.700
It is true.
00:25:40.700 --> 00:25:43.900
It is pretty easy to calculate but there are going to be problems
00:25:43.900 --> 00:25:48.400
that do not necessarily ask you just to straight up calculate the mean, median, and mode.
00:25:48.400 --> 00:25:53.400
But asks you to think about it more flexibly.
00:25:53.400 --> 00:25:55.700
Here is an example of that.
00:25:55.700 --> 00:26:01.500
There were 9 people in a room who made an average salary of 40,000 per year.
00:26:01.500 --> 00:26:06.900
When someone walks in who makes 84,000 per year, what happens to the mean?
00:26:06.900 --> 00:26:13.100
Here you do not know each individuals salary of these 9 people.
00:26:13.100 --> 00:26:18.200
All you know is that their average is 40,000.
00:26:18.200 --> 00:26:22.300
Then you know that somebody also walks in who makes 84,000.
00:26:22.300 --> 00:26:28.000
Can you calculate the new mean of the 10 people in this room?
00:26:28.000 --> 00:26:29.400
Yes you can.
00:26:29.400 --> 00:26:30.800
Here is how.
00:26:30.800 --> 00:26:39.500
The previous mean of the sample, we call it x bar, was 40,000.
00:26:39.500 --> 00:26:43.700
I will just write 40 for now and later will have in to add that k.
00:26:43.700 --> 00:26:47.700
$40,000.
00:26:47.700 --> 00:27:00.000
Since there were 9 people in the room, what we do not know is this.
00:27:00.000 --> 00:27:04.900
I going from 1 all the way to 9.
00:27:04.900 --> 00:27:09.300
What we do not know is the sum?
00:27:09.300 --> 00:27:13.900
We do know the average ends up being 40 for this 9 people.
00:27:13.900 --> 00:27:19.100
Remember there is that algebraic transformation that we do.
00:27:19.100 --> 00:27:24.700
I can multiply both sides by n and get the value of this.
00:27:24.700 --> 00:27:29.800
I do not know each individual value but I know what the value if it is all added up.
00:27:29.800 --> 00:27:37.600
If I do 40 × 9 then I will get the sum for 9.
00:27:37.600 --> 00:27:40.800
I will get that sum automatically.
00:27:40.800 --> 00:27:50.600
In order to get the mean for everybody in the room, this is going to be my x bar of the 9 people.
00:27:50.600 --> 00:27:54.500
What about my x bar for 10 people?
00:27:54.500 --> 00:28:13.500
What I need is the sum of all 9 of my people and add in 84, and divide that whole thing by 10.
00:28:13.500 --> 00:28:17.800
Because this is now the sum of all 10 people.
00:28:17.800 --> 00:28:23.100
I have the sum of 9 + my last guy.
00:28:23.100 --> 00:28:39.300
Because we know this guy it is 40 × 9 ÷ 10.
00:28:39.300 --> 00:28:57.500
I’m just going to use just random Excel sheet to help me do this calculation.
00:28:57.500 --> 00:29:00.500
You could do this in your calculator, in our head.
00:29:00.500 --> 00:29:10.900
Here is (40 × 9 + 84) ÷ 10.
00:29:10.900 --> 00:29:13.400
I get 44.4.
00:29:13.400 --> 00:29:19.500
My new mean is 44.4.
00:29:19.500 --> 00:29:25.000
Before our mean was 40,000, now it is 44,400.
00:29:25.000 --> 00:29:28.800
That is our new mean.
00:29:28.800 --> 00:29:34.900
This is what you mean by it is not just only a straight forward calculation of the mean, this is what I want you to think
00:29:34.900 --> 00:29:37.300
about what the mean is made up of.
00:29:37.300 --> 00:29:45.300
It is made up of two pieces, the sum of all the values and it is divided by n.
00:29:45.300 --> 00:29:47.300
It is those two pieces.
00:29:47.300 --> 00:29:52.600
Can you play with these two pieces?
00:29:52.600 --> 00:29:57.000
Here is another example.
00:29:57.000 --> 00:30:08.000
The mean of x and y is 20, the mean of x, y, z is 17 ½ , what is the value of x?
00:30:08.000 --> 00:30:20.300
The mean if x and y, it is just x bar for two of these guys.
00:30:20.300 --> 00:30:30.300
Goes from I all the way up to 2.
00:30:30.300 --> 00:30:34.100
That is equals 20.
00:30:34.100 --> 00:30:45.800
But the x bar of sub 3, the 3 of these values is I goes from 1 all the way to 3.
00:30:45.800 --> 00:30:49.900
X sub I / 3.
00:30:49.900 --> 00:30:54.400
This is 17 ½
00:30:54.400 --> 00:31:11.400
In order to get this, it will be nice to know (x + y + z) ÷ 3 = 17 ½ .
00:31:11.400 --> 00:31:21.800
We actually have x + y, because this is actually x + y.
00:31:21.800 --> 00:31:29.200
That is simply x + y = 20 × 2.
00:31:29.200 --> 00:31:35.500
I’m just going to multiply 2 on each side to get 40.
00:31:35.500 --> 00:31:38.200
We already have x + y.
00:31:38.200 --> 00:31:48.700
I will put in my 40 + z / 3 = 17 ½ .
00:31:48.700 --> 00:32:09.500
I will just multiply both sides by 3 and then subtract 40.
00:32:09.500 --> 00:32:24.900
I will just do that here, 17.5 × 3 – 40 = 12.5.
00:32:24.900 --> 00:32:30.700
Z=12.5 or 12 ½ .
00:32:30.700 --> 00:32:37.300
That makes sense because here when it is just x and y, the mean was higher.
00:32:37.300 --> 00:32:45.200
When we added z in, the mean became lower so we know that z must have been something low to drag down the mean.
00:32:45.200 --> 00:32:52.000
It is indeed 12.5.
00:32:52.000 --> 00:33:00.400
Next it says the average number of calories in a BK burger at their 15 burger menu is 700 calories.
00:33:00.400 --> 00:33:09.600
The average number of calories of 12 burgers in Mc Donald’s burger menu is 670.
00:33:09.600 --> 00:33:16.800
What is the average calorie count when combining across BK and Mc Donald’s menus?
00:33:16.800 --> 00:33:27.200
Here what we could do is get the sum of the BK calories and the sum of all the Mc Donald’s calories and
00:33:27.200 --> 00:33:34.400
divide by the total number of burgers when we combine across their menus.
00:33:34.400 --> 00:33:47.500
For the BK burger menu it is 15 burgers and their average is 700 calories.
00:33:47.500 --> 00:33:57.400
700 × 15 should give us the summed up number of calories.
00:33:57.400 --> 00:34:08.800
Let us add that to the sum of all the Mc Donald’s calories, 670 × 12.
00:34:08.800 --> 00:34:12.800
To get the average, we have added all the calories up.
00:34:12.800 --> 00:34:33.800
To get the average we want to divide by number of total burgers we are talking about now which is 15 + 12.
00:34:33.800 --> 00:34:39.600
Remember you could always do it in your head, on a paper, calculator, whatever you want.
00:34:39.600 --> 00:34:45.700
I’m just going to do it in Excel just to show you.
00:34:45.700 --> 00:35:12.600
You need to make sure about your parentheses, that is (700 × 15 + 670 × 12) ÷ 27.
00:35:12.600 --> 00:35:25.100
That is something and 2/3.
00:35:25.100 --> 00:35:27.800
I forgot, I cannot keep this up at the same time.
00:35:27.800 --> 00:35:47.100
686 2/3 is the average number of calories when we average all the burgers on both Burger King and Mc Donald’s menus.
00:35:47.100 --> 00:35:48.500
Here is example 4.
00:35:48.500 --> 00:35:54.300
It says create an example set where n =5, where the median is greater than the mean.
00:35:54.300 --> 00:35:58.300
Modify that set so that the mean is greater than the median.
00:35:58.300 --> 00:36:06.700
Let us start off by just putting down 5 slots.
00:36:06.700 --> 00:36:18.100
Basically when looking for the median here, what we want the mean to fall somewhere down here.
00:36:18.100 --> 00:36:31.500
In order to pull that mean downward what we would do is simply create more numbers down here.
00:36:31.500 --> 00:36:35.000
These numbers to be farther away from the median.
00:36:35.000 --> 00:36:43.200
Let us make them 0, and we will make this 5 to make that distance farther.
00:36:43.200 --> 00:36:47.900
We will make these 6 and 7.
00:36:47.900 --> 00:36:50.800
The median is easy, we already have them it is 5.
00:36:50.800 --> 00:36:53.900
It is in the middle.
00:36:53.900 --> 00:36:55.800
What is the mean?
00:36:55.800 --> 00:36:58.000
Let us add these up and divide by 5.
00:36:58.000 --> 00:37:13.200
5 + 6 = 11 + 7 = 18 ÷ 5 = 3 3/5.
00:37:13.200 --> 00:37:22.400
3 3/5 or 3.6 that is going to fall below 5.
00:37:22.400 --> 00:37:28.000
Here we have the median being greater than the mean.
00:37:28.000 --> 00:37:30.400
5 is greater than 3.35.
00:37:30.400 --> 00:37:41.200
Let us modify that set so that the mean is greater than the median now.
00:37:41.200 --> 00:37:48.500
What I would do is I just think about the mean as being more influenced than the actual little number in here.
00:37:48.500 --> 00:37:55.900
I could probably keep a lot of these the same but all we want to do is weight down one side.
00:37:55.900 --> 00:37:58.900
If I do that let us see what happens.
00:37:58.900 --> 00:38:05.700
The median obviously does not change but let us see what happens to the mean.
00:38:05.700 --> 00:38:08.300
Does the mean change?
00:38:08.300 --> 00:38:19.300
11 + 20 =31 ÷ 5 = 61/5 or 6.2.
00:38:19.300 --> 00:38:24.000
Here we see now the mean is greater than the median.
00:38:24.000 --> 00:38:33.200
What I have to do is whatever you want your mean, you want to weight one side or the other.
00:38:33.200 --> 00:38:46.500
Here is a skewed but they are skewed in different ways.
00:38:46.500 --> 00:38:48.500
That is the end of central tendency.
00:38:48.500 --> 00:38:50.000
Thanks for joining us on www.educator.com.