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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 x ¯ , the variance of x is given by

var ( x ) = 1 n - 1 k = 1 n ( x k - x ¯ ) 2 = ( x 1 - x ¯ ) 2 + ( x 2 - x ¯ ) 2 + . . . + ( x n - x ¯ ) 2 n - 1 .

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

std ( x ) = var ( x ) .

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.

var ( x ) = ( 1 - 5 ) 2 + ( 7 - 5 ) 2 + ( 2 - 5 ) 2 + ( 5 - 5 ) 2 + ( 9 - 5 ) 2 + ( 6 - 5 ) 2 5 = 9 . 2
std ( x ) = var ( x ) 3 . 03

Example 3.2

Consider the data from "Example 2.2" , where the mean x ¯ = 300. The variance is

var ( x ) = 16 · ( 100 - 300 ) 2 + 3 · ( 900 - 300 ) 2 + ( 1700 - 300 ) 2 13 193 , 684

and the standard deviation is

std ( x ) = ( var ( x ) ) 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?


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 Δ x . The bins are the intervals [0, Δ x ], ( Δ x , 2 Δ x ], (2 Δ x , 3 Δ 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 Δ 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?

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
characteristics of micro business
for teaching engĺish at school how nano technology help us
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
what is biological synthesis of nanoparticles
Sanket Reply
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
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Source:  OpenStax, The art of the pfug. OpenStax CNX. Jun 05, 2013 Download for free at http://cnx.org/content/col10523/1.34
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