# 2.6 Measures of the center of the data

 Page 1 / 4
This chapter discusses measuring descriptive statistical information using the center of the data

The "center" of a data set is also a way of describing location. The two most widely used measures of the "center" of the data are the mean (average) and the median . To calculate the mean weight of 50 people, add the 50 weights together and divide by 50. To find the median weight of the 50 people, order the data and find the number that splits the data into two equal parts (previously discussed under box plots in this chapter). The median is generally a better measure of the center when there are extreme values or outliers because it is not affected by the precise numerical values of the outliers. The mean is the most common measure of the center.

The words "mean" and "average" are often used interchangeably. The substitution of one word for the other is common practice. The technical term is "arithmetic mean" and "average" is technically a center location. However, in practice among non-statisticians, "average" is commonly accepted for "arithmetic mean."

The mean can also be calculated by multiplying each distinct value by its frequency and then dividing the sum by the total number of data values. The letter used to represent the sample mean is an $x$ with a bar over it (pronounced " $x$ bar"): $\overline{x}$ .

The Greek letter $\mu$ (pronounced "mew") represents the population mean. One of the requirements for the sample mean to be a good estimate of the population mean is for the sample taken to be truly random.

To see that both ways of calculating the mean are the same, consider the sample:

• 1
• 1
• 1
• 2
• 2
• 3
• 4
• 4
• 4
• 4
• 4

$\overline{x}=\frac{1+1+1+2+2+3+4+4+4+4+4}{11}=2.7$
$\overline{x}=\frac{3×1+2×2+1×3+5×4}{11}=2.7$

In the second calculation for the sample mean, the frequencies are 3, 2, 1, and 5.

You can quickly find the location of the median by using the expression $\frac{n+1}{2}$ .

The letter $n$ is the total number of data values in the sample. If $n$ is an odd number, the median is the middle value of the ordered data (ordered smallest to largest). If $n$ is an even number, the median is equal to the two middle values added together and divided by 2 after the data has been ordered. For example, if the total number of data values is 97, then $\frac{n+1}{2}$ = $\frac{97+1}{2}$ = $49$ . The median is the 49th value in the ordered data. If the total number of data values is 100, then $\frac{n+1}{2}$ = $\frac{100+1}{2}$ = $50.5$ . The median occurs midway between the 50th and 51st values. The location of the median and the value of the median are not the same. The upper case letter $M$ is often used to represent the median. The next example illustrates the location of the median and the value of the median.

AIDS data indicating the number of months an AIDS patient lives after taking a new antibody drug are as follows (smallest to largest):

• 3
• 4
• 8
• 8
• 10
• 11
• 12
• 13
• 14
• 15
• 15
• 16
• 16
• 17
• 17
• 18
• 21
• 22
• 22
• 24
• 24
• 25
• 26
• 26
• 27
• 27
• 29
• 29
• 31
• 32
• 33
• 33
• 34
• 34
• 35
• 37
• 40
• 44
• 44
• 47

Calculate the mean and the median.

The calculation for the mean is:

$\overline{x}=\frac{\left[3+4+\left(8\right)\left(2\right)+10+11+12+13+14+\left(15\right)\left(2\right)+\left(16\right)\left(2\right)+\text{...}+35+37+40+\left(44\right)\left(2\right)+47\right]}{40}=\mathrm{23.6}$

To find the median, M , first use the formula for the location. The location is:

$\frac{n+1}{2}=\frac{40+1}{2}=20.5$

Starting at the smallest value, the median is located between the 20th and 21st values (the two 24s):

• 3
• 4
• 8
• 8
• 10
• 11
• 12
• 13
• 14
• 15
• 15
• 16
• 16
• 17
• 17
• 18
• 21
• 22
• 22
• $24$
• $24$
• 25
• 26
• 26
• 27
• 27
• 29
• 29
• 31
• 32
• 33
• 33
• 34
• 34
• 35
• 37
• 40
• 44
• 44
• 47

$M=\frac{24+24}{2}=24$

The median is 24.

## Using the ti-83,83+,84, 84+ calculators

Calculator Instructions are located in the menu item 14:Appendix (Notes for the TI-83, 83+, 84, 84+ Calculators).
• Enter data into the list editor. Press STAT 1:EDIT
• Put the data values in list L1.
• Press STAT and arrow to CALC. Press 1:1-VarStats. Press 2nd 1 for L1 and ENTER.
• Press the down and up arrow keys to scroll.
$\overline{x}=\mathrm{23.6}$ , $M=\mathrm{24}$

Suppose that, in a small town of 50 people, one person earns $5,000,000 per year and the other 49 each earn$30,000. Which is the better measure of the "center," the mean or the median?

$\overline{x}=\frac{5000000+49×30000}{50}=129400$

$M=30000$

(There are 49 people who earn $30,000 and one person who earns$5,000,000.)

The median is a better measure of the "center" than the mean because 49 of the values are 30,000 and one is 5,000,000. The 5,000,000 is an outlier. The 30,000 gives us a better sense of the middle of the data.

Another measure of the center is the mode. The mode is the most frequent value. If a data set has two values that occur the same number of times, then the set is bimodal.

## Statistics exam scores for 20 students are as follows

Statistics exam scores for 20 students are as follows:

• 50
• 53
• 59
• 59
• 63
• 63
• 72
• 72
• 72
• 72
• 72
• 76
• 78
• 81
• 83
• 84
• 84
• 84
• 90
• 93

Find the mode.

The most frequent score is 72, which occurs five times. Mode = 72.

Five real estate exam scores are 430, 430, 480, 480, 495. The data set is bimodal because the scores 430 and 480 each occur twice.

When is the mode the best measure of the "center"? Consider a weight loss program that advertises a mean weight loss of six pounds the first week of the program. The mode might indicate that most people lose two pounds the first week, making the program less appealing.

The mode can be calculated for qualitative data as well as for quantitative data.

Statistical software will easily calculate the mean, the median, and the mode. Some graphing calculators can also make these calculations. In the real world, people make these calculations using software.

## The law of large numbers and the mean

The Law of Large Numbers says that if you take samples of larger and larger size from any population, then the mean $\overline{x}$ of the sample is very likely to get closer and closer to $µ$ . This is discussed in more detail in The Central Limit Theorem .

The formula for the mean is located in the Summary of Formulas section course.

## Sampling distributions and statistic of a sampling distribution

You can think of a sampling distribution as a relative frequency distribution with a great many samples. (See Sampling and Data for a review of relative frequency). Suppose thirty randomly selected students were asked the number of movies they watched the previous week. The results are in the relative frequency table shown below.

# of movies Relative Frequency
0 5/30
1 15/30
2 6/30
3 4/30
4 1/30

If you let the number of samples get very large (say, 300 million or more), the relative frequency table becomes a relative frequency distribution .

A statistic is a number calculated from a sample. Statistic examples include the mean, the median and the mode as well as others. The sample mean $\overline{x}$ is an example of a statistic which estimates the population mean $\mu$ .

#### Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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
characteristics of micro business
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
1 It is estimated that 30% of all drivers have some kind of medical aid in South Africa. What is the probability that in a sample of 10 drivers: 3.1.1 Exactly 4 will have a medical aid. (8) 3.1.2 At least 2 will have a medical aid. (8) 3.1.3 More than 9 will have a medical aid.