# Mean, variance, and histograms

 Page 2 / 2

## Exercise 2.2

Suppose you have set the goal of making an A in your math class. If your class grades consist of 4 tests, and you have made a 98, 80, and 90 on your first three tests, what do you need to make on your last test so that the mean of your grades is 90?

## Exercise 2.3

(for the advanced) Suppose that, for the same class, you have already computed the mean of the first three tests when you receive your fourth test grade. Instead of computing the mean of all four tests from scratch, it's possible to update the mean that you've already computed. Write a Matlab code that takes two inputs, the mean of your first three tests and the grade of your fourth test, and computes the mean of all four tests.

## Variance and standard deviation

As you saw in "Example 2.2" , the mean is not always representative of the data, and other measures are needed to analyze the spread of the data. The variance is a measure of the distance of each number from the mean. Given a vector x of n numbers and mean value $\overline{x},$ the variance of x is given by

$\mathrm{var}\left(\mathbf{x}\right)=\frac{1}{n-1}\sum _{k=1}^{n}{\left({\mathbf{x}}_{k}-\overline{x}\right)}^{2}=\frac{{\left({\mathbf{x}}_{1}-\overline{x}\right)}^{2}+{\left({\mathbf{x}}_{2}-\overline{x}\right)}^{2}+...+{\left({\mathbf{x}}_{n}-\overline{x}\right)}^{2}}{n-1}.$

The standard deviation of the data is related to the variance and is given by

$\mathrm{std}\left(\mathbf{x}\right)=\sqrt{\mathrm{var}\left(\mathbf{x}\right)}.$

You can compute the variance and standard deviation of x in Matlab by typing the commands var(x) and std(x).

## Example 3.1

Consider the vector given in "Example 2.1" , x = [1, 7, 2, 5, 9, 6]. Recall that the mean of x = 5.

$\mathrm{var}\left(\mathbf{x}\right)=\frac{{\left(1-5\right)}^{2}+{\left(7-5\right)}^{2}+{\left(2-5\right)}^{2}+{\left(5-5\right)}^{2}+{\left(9-5\right)}^{2}+{\left(6-5\right)}^{2}}{5}=9.2$
$\mathrm{std}\left(\mathbf{x}\right)=\sqrt{\mathrm{var}\left(\mathbf{x}\right)}\simeq 3.03$

## Example 3.2

Consider the data from "Example 2.2" , where the mean $\overline{x}$ = 300. The variance is

$\mathrm{var}\left(\mathbf{x}\right)=\frac{16·{\left(100-300\right)}^{2}+3·{\left(900-300\right)}^{2}+{\left(1700-300\right)}^{2}}{13}\simeq 193,684$

and the standard deviation is

$\mathrm{std}\left(\mathbf{x}\right)=\sqrt{\left(\mathrm{var}\left(\mathbf{x}\right)\right)}\simeq 440$

Because the standard deviation is considerably larger than the mean, the variance tells us that the mean is not very representative of the data.

## Exercise 3.1

Compute the variance and standard deviation of y = [3, 8, 2, 5, 5, 7], using both the formulas and the Matlab commands.

## Exercise 3.2

Suppose that in the situation of "Example 2.2" , there are 50 general exmployees instead of 16. Compute the mean and variance of the daily salary. Is the mean more or less representative of the data than it was in Example 2.2?

## Histograms

Although the mean, variance, and standard deviation provide information about the data, it is often useful to visualize the data. A histogram is a tool that allows you to visualize the proportion of numbers that fall within a given bin, or interval. To compute the histogram of a set of data, x , follow the algorithm below.

1. Choose the bin size $\Delta x$ . The bins are the intervals [0, $\Delta x$ ], ( $\Delta x$ , 2 $\Delta x$ ], (2 $\Delta x$ , 3 $\Delta x$ ], and so on.
2. For each bin, count the number of data points that lie within the bin.
3. Create a bar graph showing the number of data points within each bin.

## Example 4.1

Consider again the vector from "Example 2.1" , x = [1, 7, 2, 5, 9, 6]. Using a bin size $\Delta x$ = 2, there are 5 bins.

• Bin 1 = [0, 2] has 2 elements of x
• Bin 2 = (2, 4] has 0 elements of x
• Bin 3 = (4, 6] has 2 elements of x
• Bin 4 = (6, 8] has 1 element of x
• Bin 5 = (8, 10] has 1 element of x

In Matlab, you can plot the histogram of a vector x by typing hist(x). Matlab will automatically use 10 bins. If you'd like to specify the bin centers, type hist(x,c), where c is a vector of bin centers. The histogram of "Example 4.1" was generated by the Matlab command hist(x, [1, 3, 5, 7, 9]).

## Exercise 4.1

Plot the histogram of the vector y = [3, 8, 2, 5, 5, 7], both on paper and in Matlab.

## Exercise 4.2

Plot the histogram of the daily salaries from "Example 2.2" . For this example, does the histogram or the mean give you a better idea of what salary you would be making if you got the job?

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Got questions? Join the online conversation and get instant answers!