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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"): x .

The Greek letter μ (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

x ¯ = 1 + 1 + 1 + 2 + 2 + 3 + 4 + 4 + 4 + 4 + 4 11 = 2.7
x ¯ = 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 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 n + 1 2 = 97 + 1 2 = 49 . The median is the 49th value in the ordered data. If the total number of data values is 100, then n + 1 2 = 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:

x ¯ = [ 3 + 4 + ( 8 ) ( 2 ) + 10 + 11 + 12 + 13 + 14 + ( 15 ) ( 2 ) + ( 16 ) ( 2 ) + ... + 35 + 37 + 40 + ( 44 ) ( 2 ) + 47 ] 40 = 23.6

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

n + 1 2 = 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 = 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.
x ¯ = 23.6 , M = 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?

x ¯ = 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 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 x ¯ is an example of a statistic which estimates the population mean μ .

Questions & Answers

how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
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Uday
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Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
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Kristine 2*2*2=8
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Differences Between Laspeyres and Paasche Indices
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
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J, combine like terms 7x-4y
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Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
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I start with an easy one. carbon nanotubes woven into a long filament like a string
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Porter
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AMJAD
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Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
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AMJAD
what is system testing
AMJAD
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Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
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Prasenjit
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Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
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Damian
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not now but maybe in future only AgNP maybe any other nanomaterials
Azam
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Uday
I'm interested in Nanotube
Uday
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Prasenjit
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Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Collaborative statistics-parzen remix. OpenStax CNX. Jul 15, 2009 Download for free at http://legacy.cnx.org/content/col10732/1.2
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