2.10 Measures of the spread of the data (Page 3/7)

Collaborative statistics (custom Page 3 / 7

In these formulas, $f$ represents the frequency with which a value appears. For example, if a value appears once, $f$ is 1. If a value appears three times in the data set or population, $f$ is 3.

Sampling variability of a statistic

The statistic of a sampling distribution was discussed in Descriptive Statistics: Measuring the Center of the Data . How much the statistic varies from one sample to another is known as the sampling variability of a statistic . You typically measure the sampling variability of a statistic by its standard error. The standard error of the mean is an example of a standard error. It is a special standard deviation and is known as the standard deviation of the sampling distribution of the mean. You will cover the standard error of the mean in The Central Limit Theorem (not now). The notation for the standard error of the mean is

\frac{σ}{\sqrt{n}}

where

σ

is the standard deviation of the population and

n

is the size of the sample.

In practice, USE A CALCULATOR OR COMPUTER SOFTWARE TO CALCULATE THE STANDARD DEVIATION. If you are using a TI-83,83+,84+ calculator, you need to select the appropriate standard deviation $σ_{x}$ or $s_{x}$ from the summary statistics. We will concentrate on using and interpreting the information that the standard deviation gives us. However you should study the following step-by-step example to help you understand how the standard deviation measures variation from the mean.

In a fifth grade class, the teacher was interested in the average age and the sample standard deviation of the ages of her students. The following data are the ages for a SAMPLE of $n = 20$ fifth grade students. The ages are rounded to the nearest half year:

9
9.5
9.5
10
10
10
10
10.5
10.5
10.5
10.5
11
11
11
11
11
11
11.5
11.5
11.5

\bar{x} = \frac{9 + 9.5 \times 2 + 10 \times 4 + 10.5 \times 4 + 11 \times 6 + 11.5 \times 3}{20} = 10.525

The average age is 10.53 years, rounded to 2 places.

The variance may be calculated by using a table. Then the standard deviation is calculated by taking the square root of the variance. We will explain the parts of the table after calculating $s$ .

Data	Freq.	Deviations	${Deviations}^{2}$	(Freq.)( ${Deviations}^{2}$ )
$x$	$f$	$(x - x)$	${(x - \bar{x})}^{2}$	$(f) {(x - \bar{x})}^{2}$
$9$	$1$	$9 - 10.525 = - 1.525$	${(- 1.525)}^{2} = 2.325625$	$1 \times 2.325625 = 2.325625$
$9.5$	$2$	$9.5 - 10.525 = - 1.025$	${(- 1.025)}^{2} = 1.050625$	$2 \times 1.050625 = 2.101250$
$10$	$4$	$10 - 10.525 = - 0.525$	${(- 0.525)}^{2} = 0.275625$	$4 \times .275625 = 1.1025$
$10.5$	$4$	$10.5 - 10.525 = - 0.025$	${(- 0.025)}^{2} = 0.000625$	$4 \times .000625 = .0025$
$11$	$6$	$11 - 10.525 = 0.475$	${(0.475)}^{2} = 0.225625$	$6 \times .225625 = 1.35375$
$11.5$	$3$	$11.5 - 10.525 = 0.975$	${(0.975)}^{2} = 0.950625$	$3 \times .950625 = 2.851875$

The sample variance, $s^{2}$ , is equal to the sum of the last column (9.7375) divided by the total number of data values minus one (20 - 1):

$s^{2} = \frac{9.7375}{20 - 1} = 0.5125$

The sample standard deviation $s$ is equal to the square root of the sample variance:

$s = \sqrt{0.5125} = . 0715891$ Rounded to two decimal places, $s = 0.72$

Typically, you do the calculation for the standard deviation on your calculator or computer . The intermediate results are not rounded. This is done for accuracy.

Verify the mean and standard deviation calculated above on your calculator or computer.

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

Enter data into the list editor. Press STAT 1:EDIT. If necessary, clear the lists by arrowing up into the name. Press CLEAR and arrow down.
Put the data values (9, 9.5, 10, 10.5, 11, 11.5) into list L1 and the frequencies (1, 2, 4, 4, 6, 3) into list L2. Use the arrow keys to move around.
Press STAT and arrow to CALC. Press 1:1-VarStats and enter L1 (2nd 1), L2 (2nd 2). Do not forget the comma. Press ENTER.
$x$ =10.525
Use Sx because this is sample data (not a population): $Sx$ =0.715891

Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?

Aislinn Reply

tijani

what is titration

John Reply

what is physics

Siyaka Reply

A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance

Jude Reply

Can you compute that for me. Ty

Jude

what is the dimension formula of energy?

David Reply

what is viscosity?

David

what is inorganic

emma Reply

what is chemistry

Youesf Reply

what is inorganic

emma

Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes

Adjei

please, I'm a physics student and I need help in physics

Adjanou

chemistry could also be understood like the sexual attraction/repulsion of the male and female elements. the reaction varies depending on the energy differences of each given gender. + masculine -female.

Pedro

A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity

Krampah Reply

2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.

Sahid Reply

you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s

Samuel Reply

can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?

Joseph Reply

Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time

Joseph

Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing

Joseph

"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true

Ryan

what's motion

Maurice Reply

what are the types of wave

Maurice

answer

Magreth

progressive wave

Magreth

hello friend how are you

Muhammad Reply

fine, how about you?

Mohammed

Mujahid

A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?

yasuo Reply

Who can show me the full solution in this problem?

Reofrir Reply

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< Chapter < Page Page > Chapter >>

Practice Key Terms 2

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

Source: OpenStax, Collaborative statistics (custom online version modified by t. short). OpenStax CNX. Jul 15, 2013 Download for free at http://cnx.org/content/col11476/1.5

Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Collaborative statistics (custom online version modified by t. short)' conversation and receive update notifications?

Ask

	Are you part of the R5 Family? By Anonymous User Start Quiz
	23 AP Key Terms 23 The Digestive System By OpenStax Start Key Terms
	4 Neuroanatomy 04 Spinal Cord Brain Stem By Stephen Voron Start Quiz
	23 AP 23 Digestive System Essay By OpenStax Start Flashcards
	2 Neuroanatomy 02 Brain External Features By Stephen Voron Start Quiz
	Chemistry Final By Briana Hamilton Start Flashcards
	NCE Ch 06 Groups By Anh Dao Start Quiz
©flickr: Justin	Music Appreciation Final Practice By Madison Christian Start Exam
©flickr: Hey	Anatomy Physiology Final By Joli Julianna Start Test
©flickr: quinn	Neuroanatomy By George Turner Start Quiz