Are The Measures Of Center The Best Statistics To Use With Theseã¢â‚¬â€¹ Data?

iii.ane: Measures of Heart

Last updated
Save as PDF

Page ID: 5171

This section focuses on measures of central trend. Many times y'all are asking what to wait on average. Such as when you pick a major, you would probably ask how much you expect to earn in that field. If you lot are thinking of relocating to a new town, you might ask how much you tin can expect to pay for housing. If you lot are planting vegetables in the spring, you might want to know how long it will be until you tin can harvest. These questions, and many more, tin can exist answered by knowing the center of the information set up. There are three measures of the "middle" of the data. They are the mode, median, and mean. Whatever of the values tin be referred to as the "average."

The mode is the data value that occurs the most frequently in the information. To discover it, you count how often each data value occurs, and then make up one's mind which information value occurs well-nigh oft.
The median is the data value in the middle of a sorted list of data. To discover it, you put the data in order, and and so determine which data value is in the middle of the information set.
The mean is the arithmetic boilerplate of the numbers. This is the center that most people call the boilerplate, though all iii – mean, median, and style – actually are averages.

At that place are no symbols for the mode and the median, merely the hateful is used a great deal, and statisticians gave it a symbol. There are actually two symbols, one for the population parameter and one for the sample statistic. In about cases you cannot find the population parameter, so you lot use the sample statistic to estimate the population parameter.

Definition $\PageIndex{one}$: Population Mean

The population hateful is given by

$\mu=\dfrac{\sum 10}{N}$, pronounced mu

where

$Due north$ is the size of the population.
$x$ represents a information value.
$\sum x$ means to add up all of the data values.

Definition $\PageIndex{2}$: Sample Hateful

Sample Mean:

$\overline{x}=\dfrac{\sum 10}{n}$, pronounced x bar, where

$north$ is the size of the sample.
$x$ represents a data value.
$\sum x$ means to add together up all of the data values.

The value for $\overline{x}$ is used to guess $\mu$ since $\mu$ can't be calculated in most situations.

Case $\PageIndex{1}$ finding the mean, median, and manner

Suppose a vet wants to find the average weight of cats. The weights (in pounds) of five cats are in Case $\PageIndex{1}$.

Tabular array $\PageIndex{1}$: Finding the Mean, Median, and Way
6.8	8.2	7.5	9.iv	8.2

Notice the hateful, median, and fashion of the weight of a cat.

Solution

Before starting whatsoever mathematics problem, it is always a good thought to ascertain the unknown in the problem. In this case, yous want to define the variable. The symbol for the variable is $x$.

The variable is $10 =$ weight of a cat

Hateful:

$\overline{x}=\dfrac{vi.8+viii.ii+7.5+nine.4+8.2}{5}=\dfrac{forty.1}{5}=8.02$ pounds

Median:

You need to sort the listing for both the median and style. The sorted list is in Example $\PageIndex{2}$.

Tabular array $\PageIndex{ii}$: Sorted List of Cat's Weights
half-dozen.eight	7.v	8.two	8.two	9.4

There are five data points so the middle of the list would exist the third number. (Just put a finger at each end of the listing and move them toward the center one number at a time. Where your fingers meet is the median.)

Table $\PageIndex{3}$: Sorted List of Cats' Weights with Median Marked
6.8	7.5	8.two	8.ii	9.4

The median is therefore eight.2 pounds.

Mode:

This is easiest to do from the sorted listing that is in Case $\PageIndex{2}$. Which value appears the almost number of times? The number eight.two appears twice, while all other numbers announced one time.

Mode = 8.2 pounds.

A information prepare tin can take more than one mode. If at that place is a tie betwixt ii values for the nearly number of times then both values are the style and the data is called bimodal (ii modes). If every data betoken occurs the same number of times, there is no manner. If there are more than two numbers that announced the near times, then usually there is no manner.

In Instance $\PageIndex{1}$, there were an odd number of data points. In that case, the median was merely the centre number. What happens if there is an even number of data points? What would you exercise?

Case $\PageIndex{2}$ finding the median with an even number of information points

Suppose a vet wants to find the median weight of cats. The weights (in pounds) of vi cats are in Example $\PageIndex{iv}$. Find the median.

Table $\PageIndex{4}$: Weights of Vi Cats
6.8	8.2	7.5	nine.4	viii.2	vi.three

Solution

Variable: $x =$ weight of a cat

First sort the list if it is not already sorted.

There are six numbers in the list so the number in the eye is between the 3rd and 4th number. Use your fingers starting at each end of the list in Example $\PageIndex{5}$ and move toward the center until they meet. In that location are two numbers there.

Table $\PageIndex{5}$: Sorted List of Weights of 6 Cats
6.three	6.8	vii.5	8.two	8.2	nine.4

To discover the median, simply average the two numbers.

median $=\dfrac{7.v+8.two}{ii}=7.85$ pounds

The median is 7.85 pounds.

Instance $\PageIndex{3}$ finding mean and median using technology

Suppose a vet wants to find the median weight of cats. The weights (in pounds) of six cats are in Example $\PageIndex{4}$. Find the median

Solution

Variable: $10=$ weight of a true cat

You tin do the calculations for the mean and median using the technology.

The procedure for calculating the sample mean ( $\overline{10}$ ) and the sample median (Med) on the TI-83/84 is in Figures 3.1.ane through 3.1.iv. Get-go yous need to become into the STAT menu, and then Edit. This will allow you to blazon in your data (run into Figure $\PageIndex{1}$).

Screenshot (35).png — Figure $\PageIndex{1}$: TI-83/84 Reckoner Edit Setup

Once you have the data into the estimator, you so become back to the STAT carte, movement over to CALC, so cull i-Var Stats (see Figure $\PageIndex{2}$). The calculator volition at present put i-Var Stats on the main screen. Now type in L1 (2d button and i) and so press ENTER. (Note if you lot have the newer operating system on the TI-84, then the procedure is slightly dissimilar.) If you lot press the downwardly pointer, y'all will see the rest of the output from the calculator. The results from the calculator are in Figure $\PageIndex{3}$.

Screenshot (36).png — Figure $\PageIndex{two}$: TI-83/84 Calculator CALC Carte

Screenshot (37).png — Figure $\PageIndex{3}$: TI-83/84 Calculator Input for Instance $\PageIndex{iii}$ Variable

Screenshot (38).png — Effigy $\PageIndex{four}$: TI-83/84 Figurer Results for Instance $\PageIndex{iii}$ Variable

The commands for finding the mean and median using R are as follows:

variable<-c(type in your information with commas in between)
To detect the mean, use mean(variable)
To discover the median, use median(variable)

And then for this example, the commands would be

weights<-c(6.8, 8.two, 7.5, 9.4, 8.two, 6.iii)
mean(weights)
[ane] vii.733333
median(weights)
[1] 7.85

Example $\PageIndex{four}$ affect of farthermost values on hateful and median

Suppose you have the aforementioned set of cats from Example $\PageIndex{1}$ but 1 additional cat was added to the data gear up. Example $\PageIndex{6}$ contains the six cats' weights, in pounds.

Table $\PageIndex{six}$: Weights of Six Cats
6.eight	vii.5	8.2	8.2	nine.four	22.one

Find the mean and the median.

Solution

Variable: $x=$ weight of a true cat

mean $=\overline{x}=\dfrac{6.viii+vii.5+8.2+8.2+9.4+22.1}{six}=x.37$ pounds

The data is already in order, thus the median is between 8.2 and viii.2.

median $=\dfrac{viii.2+8.ii}{2}=eight.2$ pounds

The mean is much higher than the median. Why is this? Find that when the value of 22.i was added, the mean went from eight.02 to 10.37, but the median did non change at all. This is considering the mean is afflicted by extreme values, while the median is not. The very heavy cat brought the hateful weight upwards. In this case, the median is a much meliorate measure of the center.

An outlier is a information value that is very different from the remainder of the data. It tin can exist really high or actually low. Farthermost values may exist an outlier if the extreme value is far enough from the heart. In Example $\PageIndex{4}$, the data value 22.1 pounds is an extreme value and it may be an outlier.

If there are farthermost values in the data, the median is a better measure of the eye than the mean. If in that location are no farthermost values, the mean and the median will be similar so well-nigh people utilize the mean.

The mean is not a resistant measure out because it is affected by farthermost values. The median and the mode are resistant measures because they are not affected by extreme values.

Equally a consumer you need to be aware that people choose the measure of centre that best supports their claim. When you read an commodity in the paper and it talks about the "boilerplate" it usually means the mean but sometimes it refers to the median. Some articles will utilise the word "median" instead of "boilerplate" to exist more specific. If you demand to make an important conclusion and the data says "average", information technology would be wise to enquire if the "boilerplate" is the mean or the median before you decide.

Every bit an example, suppose that a company wants to utilise the mean salary as the average bacon for the visitor. This is because the high salaries of the administration will pull the hateful higher. The company tin can say that the employees are paid well because the average is high. However, the employees want to use the median since it discounts the farthermost values of the administration and will give a lower value of the average. This will make the salaries seem lower and that a raise is in order.

Why apply the hateful instead of the median? The reason is because when multiple samples are taken from the aforementioned population, the sample means tend to be more consequent than other measures of the center. The sample mean is the more reliable measure of center.

To sympathise how the different measures of centre related to skewed or symmetric distributions, run into Figure $\PageIndex{5}$. As you tin can see sometimes the mean is smaller than the median and mode, sometimes the mean is larger than the median and mode, and sometimes they are the same values.

Screenshot (39).png — Figure $\PageIndex{5}$: Hateful, Median, Mode as Related to a Distribution

One last type of average is a weighted average. Weighted averages are used quite often in real life. Some teachers use them in calculating your grade in the course, or your grade on a projection. Some employers utilise them in employee evaluations. The idea is that some activities are more important than others. Equally an example, a fulltime teacher at a customs college may be evaluated on their service to the college, their service to the community, whether their paperwork is turned in on time, and their pedagogy. Even so, instruction is much more of import than whether their paperwork is turned in on time. When the evaluation is completed, more weight needs to exist given to the teaching and less to the paperwork. This is a weighted boilerplate.

Definition $\PageIndex{iii}$

Weighted Boilerplate

$\dfrac{\sum x w}{\sum w}$ where $due west$ is the weight of the data value, $ten$.

Example $\PageIndex{5}$ weighted average

In your biological science class, your final grade is based on several things: a lab score, scores on 2 major tests, and your score on the concluding exam. In that location are 100 points available for each score. The lab score is worth 15% of the form, the 2 exams are worth 25% of the course each, and the final exam is worth 35% of the course. Suppose you earned scores of 95 on the labs, 83 and 76 on the ii exams, and 84 on the terminal exam. Compute your weighted boilerplate for the course.

Solution

Variable: $x=$ score

The weighted average is $\dfrac{\sum x westward}{\sum westward}=\dfrac{\text { sum of the scores times their weights }}{\text { sum of all the weights }}$

weighted boilerplate $=\dfrac{95(0.15)+83(0.25)+76(0.25)+84(0.35)}{0.15+0.25+0.25+0.35}=\dfrac{83.four}{1.00}=83.four \%$

A weighted average can exist institute using technology.

The procedure for calculating the weighted average on the TI-83/84 is in Figures three.1.vi through iii.i.9. First you lot demand to go into the STAT card, and then Edit. This will allow you to blazon in the scores into L1 and the weights into L2 (see Effigy $\PageIndex{6}$).

Screenshot (40).png — Effigy $\PageIndex{six}$: TI-3/84 Calculator Edit Setup

In one case you have the data into the calculator, you then get dorsum to the STAT menu, move over to CALC, and and so choose 1-Var Stats (run into Figure $\PageIndex{7}$). The estimator will at present put i-Var Stats on the main screen. Now type in L1 (2nd button and 1), then a comma (button to a higher place the 7 button), and so L2 (2nd button and 2) and then press ENTER. (Note if you have the newer operating system on the TI-84, then the procedure is slightly different.) The results from the calculator are in Figure $\PageIndex{9}$. The $\overline{x}$ is the weighted average.

Screenshot (41).png — Figure $\PageIndex{7}$: TI-83/84 Calculator CALC Carte

Screenshot (42).png — Figure $\PageIndex{8}$: TI-83/84 Reckoner Input for Weighted Average

Screenshot (43).png — Effigy $\PageIndex{nine}$: TI-83/84 Calculator Results for Weighted Average

The commands for finding the mean and median using R are every bit follows:

10<-c(type in your data with commas in between)
w<-c(blazon in your weights with commas in between
weighted.mean(ten,w)

So for this instance, the commands would be

x<-c(95, 83, 76, 84)
w<-c(.xv, .25, .25, .35)
weighted.mean(x,w)
[1] 83.4

Case $\PageIndex{six}$ weighted average

The faculty evaluation procedure at John Jingle Academy rates a kinesthesia fellow member on the post-obit activities: teaching, publishing, commission service, community service, and submitting paperwork in a timely manner. The process involves reviewing student evaluations, peer evaluations, and supervisor evaluation for each teacher and awarding him/her a score on a scale from ane to ten (with x beingness the best). The weights for each activity are xx for teaching, 18 for publishing, 6 for committee service, 4 for community service, and two for paperwork.

Ane faculty member had the following ratings: 8 for teaching, 9 for publishing, 2 for committee piece of work, 1 for community service, and 8 for paperwork. Compute the weighted average of the evaluation.
Another faculty member had ratings of 6 for teaching, 8 for publishing, 9 for committee work, ten for customs service, and 10 for paperwork. Compute the weighted average of the evaluation.
Which kinesthesia fellow member had the higher boilerplate evaluation?

Solution

a. Variable: $x=$ rating

The weighted average is $\dfrac{\sum x west}{\sum w}=\dfrac{\text { sum of the scores times their weights }}{\text { sum of all the weights }}$

evaluation $=\dfrac{8(20)+9(eighteen)+2(vi)+ane(four)+8(ii)}{xx+18+6+4+2}=\dfrac{354}{50}=7.08$

b. evaluation $=\dfrac{half dozen(xx)+8(18)+9(6)+10(iv)+10(2)}{20+xviii+half-dozen+4+2}=\dfrac{378}{l}=7.56$

c. The second faculty member has a college boilerplate evaluation.

Yous can find a weighted boilerplate using technology. The last thing to mention is which boilerplate is used on which type of data.

Fashion can exist constitute on nominal, ordinal, interval, and ratio data, since the mode is simply the information value that occurs most often. You are but counting the data values. Median tin be plant on ordinal, interval, and ratio information, since you need to put the data in order. As long as there is order to the data you can observe the median. Mean can be found on interval and ratio information, since you must have numbers to add together.

Homework

Do $\PageIndex{1}$

Cholesterol levels were collected from patients 2 days later they had a middle assail (Ryan, Joiner & Ryan, Jr, 1985) and are in Instance $\PageIndex{7}$. Observe the mean, median, and way.

Table $\PageIndex{7}$: Cholesterol Levels
270	236	210	142	280	272	160
220	226	242	186	266	206	318
294	282	234	224	276	282	360
310	280	278	288	288	244	236

The lengths (in kilometers) of rivers on the South Island of New Zealand that flow to the Pacific Ocean are listed in Case $\PageIndex{8}$ (Lee, 1994). Find the mean, median, and style.

Tabular array $\PageIndex{8}$: Lengths of Rivers (km) Flowing to Pacific Ocean
River	Length (km)	River	Length (km)
Clarence	209	Clutha	322
Conway	48	Taieri	288
Waiau	169	Shag	72
Hurunui	138	Kakanui	64
Waipara	64	Rangitata	121
Ashley	97	Ophi	80
Waimakariri	161	Pareora	56
Selwyn	95	Waihao	64
Rakaia	145	Waitaki	209
Ashburton	90

The lengths (in kilometers) of rivers on the South Island of New Zealand that menstruum to the Tasman Sea are listed in Example $\PageIndex{9}$ (Lee, 1994). Observe the mean, median, and mode.

Table $\PageIndex{9}$: Lengths of Rivers (km) Flowing to Tasman Sea
River	Length (km)	River	Length (km)
Hollyford	76	Waimea	48
Cascade	64	Motueka	108
Arawhata	68	Takaka	72
Haast	64	Aorere	72
Karangarua	37	Heaphy	35
Cook	32	Karamea	eighty
Waiho	32	Mokihinui	56
Whataroa	51	Buller	177
Wanganui	56	Grey	121
Waitaha	40	Taramakau	80
Hokitika	64	Arahura	56

Eyeglassmatic manufactures eyeglasses for their retailers. They research to encounter how many defective lenses they fabricated during the time period of January i to March 31. Case $\PageIndex{10}$ contains the defect and the number of defects. Find the mean, median, and mode.

Tabular array $\PageIndex{ten}$: Number of Defective Lenses
Defect Type	Number of Defects
Scratch	5865
Correct shaped - small	4613
Flaked	1992
Wrong axis	1838
Chamfer wrong	1596
Crazing, cracks	1546
Incorrect shape	1485
Wrong PD	1398
Spots and bubbles	1371
Incorrect height	1130
Right shape - big	1105
Lost in lab	976
Spots/chimera - intern	976

Impress-O-Matic printing company's employees take salaries that are contained in Example $\PageIndex{eleven}$.

Employee	Salary ($)
CEO	272,500
Driver	58,456
CD74	100,702
CD65	57,380
Embellisher	73,877
Folder	65,270
GTO	74,235
Handwork	52,718
Horizon	76,029
ITEK	64,553
Mgmt	108,448
Platens	69,573
Polar	75,526
Pre Printing Manager	108,448
Pre Press Manager/ IT	98,837
Pre Press/ Graphic Artist	75,311
Designer	90,090
Sales	109,739
Administration	66,346

Table $\PageIndex{11}$: Salaries of Print-O-Matic Printing Company Employees
a. Detect the hateful and median.
b. Find the hateful and median with the CEO'south salary removed.
c. What happened to the hateful and median when the CEO's salary was removed? Why?
d. If y'all were the CEO, who is answering concerns from the union that employees are underpaid, which average of the consummate data set would you prefer? Why?
e. If you were a platen worker, who believes that the employees demand a enhance, which boilerplate would you prefer? Why?

Impress-O-Matic press company spends specific amounts on fixed costs every month. The costs of those fixed costs are in Example $\PageIndex{12}$.

Monthly charges	Monthly toll ($)
Bank charges	482
Cleaning	2208
Calculator expensive	2471
Lease payments	2656
Postage	2117
Uniforms	2600

Table $\PageIndex{12}$: Fixed Costs for Print-O-Matic Press Visitor
a. Detect the hateful and median.
b. Detect the hateful and median with the bank charger removed.
c. What happened to the hateful and median when the bank charger was removed? Why?
d. If it is your job to oversee the fixed costs, which average using te consummate data set would yous prefer to utilise when submitting a report to administration to show that costs are low? Why?
e. If information technology is your job to find places in the upkeep to reduce costs, which average using the complete data set would you lot prefer to use when submitting a report to administration to bear witness that stock-still costs need to exist reduced? Why?

State which blazon of measurement scale each represents, and then which center measures can be use for the variable?
1. Y'all collect data on people'southward likelihood (very likely, likely, neutral, unlikely, very unlikely) to vote for a candidate.
2. You lot collect information on the diameter at chest height of trees in the Coconino National Forest.
3. You collect data on the year wineries were started.
4. You collect the drink types that people in Sydney, Australia drinkable.
State which type of measurement scale each represents, then which center measures can be apply for the variable?
1. You collect information on the height of plants using a new fertilizer.
2. You collect data on the cars that people drive in Campbelltown, Australia.
3. You lot collect data on the temperature at different locations in Antarctica.
4. You collect data on the first, 2nd, and third winner in a beer competition.
Looking at Graph iii.i.1, land if the graph is skewed left, skewed correct, or symmetric and and then land which is larger, the mean or the median?

Graph 3.i.one : Skewed or Symmetric Graph
Looking at Graph iii.1.2, state if the graph is skewed left, skewed right, or symmetric and then state which is larger, the mean or the median?

Graph 3.1.two : Skewed or Symmetric Graph
An employee at Coconino Community College (CCC) is evaluated based on goal setting and accomplishments toward the goals, job effectiveness, competencies, and CCC core values. Suppose for a specific employee, goal 1 has a weight of 30%, goal 2 has a weight of 20%, job effectiveness has a weight of 25%, competency one has a goal of 4%, competency 2 has a goal has a weight of 3%, competency iii has a weight of 3%, competency four has a weight of three%, competency 5 has a weight of 2%, and cadre values has a weight of x%. Suppose the employee has scores of 3.0 for goal 1, three.0 for goal 2, two.0 for chore effectiveness, iii.0 for competency 1, ii.0 for competency 2, 2.0 for competency 3, iii.0 for competency 4, 4.0 for competency 5, and iii.0 for core values. Discover the weighted average score for this employee. If an employee has a score less than ii.5, they must take a Performance Enhancement Programme written. Does this employee need a plan?
An employee at Coconino Community College (CCC) is evaluated based on goal setting and accomplishments toward goals, job effectiveness, competencies, CCC cadre values. Suppose for a specific employee, goal i has a weight of twenty%, goal 2 has a weight of twenty%, goal 3 has a weight of ten%, chore effectiveness has a weight of 25%, competency 1 has a goal of 4%, competency 2 has a goal has a weight of 3%, competency iii has a weight of iii%, competency 4 has a weight of 5%, and core values has a weight of x%. Suppose the employee has scores of 2.0 for goal 1, ii.0 for goal 2, iv.0 for goal three, iii.0 for job effectiveness, ii.0 for competency i, 3.0 for competency ii, 2.0 for competency 3, 3.0 for competency four, and 4.0 for core values. Discover the weighted average score for this employee. If an employee that has a score less than 2.v, they must have a Performance Enhancement Plan written. Does this employee demand a plan?
A statistics class has the following activities and weights for determining a grade in the course: test 1 worth 15% of the grade, test ii worth fifteen% of the course, examination 3 worth 15% of the grade, homework worth ten% of the grade, semester project worth 20% of the grade, and the terminal examination worth 25% of the form. If a educatee receives an 85 on test ane, a 76 on exam 2, an 83 on test 3, a 74 on the homework, a 65 on the project, and a 79 on the final, what grade did the student earn in the class?
A statistics form has the following activities and weights for determining a grade in the course: test 1 worth xv% of the grade, exam ii worth xv% of the grade, examination 3 worth xv% of the grade, homework worth 10% of the grade, semester project worth xx% of the grade, and the final test worth 25% of the course. If a student receives a 92 on test 1, an 85 on test 2, a 95 on test 3, a 92 on the homework, a 55 on the project, and an 83 on the concluding, what grade did the student earn in the course?

Answer

1. mean = 253.93, median = 268, mode = none

iii. hateful = 67.68 km, median = 64 km, mode = 56 and 64 km

5. a. mean = $89,370.42, median = $75,311, b. mean = $79,196.56, median = $74,773, c. See solutions, d. See solutions, e. Encounter solutions

seven. a. ordinal- median and fashion, b. ratio – all three, c. interval – all iii, d. nominal – style

ix. Skewed correct, mean higher

eleven. 2.71

13. 76.75

Are The Measures Of Center The Best Statistics To Use With Theseã¢â‚¬â€¹ Data?,

Source: https://stats.libretexts.org/Bookshelves/Introductory_Statistics/Book:_Statistics_Using_Technology_%28Kozak%29/03:_Examining_the_Evidence_Using_Graphs_and_Statistics/3.01:_Measures_of_Center

Posted by: hadlockemenceapery2002.blogspot.com