The purpose of this assignment is to conduct a review of literature concerning a health care issue that is important to you in order to develop a proposal for implementation. 

Conduct a literature review of nine articles. The articles must be published within the last 5 years, and only one article may be a meta-analysis or systematic review. Collect articles that both support and oppose the health care issue and implementation you are proposing.

Use the “Literature Review Table” template to complete this literature review assignment, including a 250-500-word summary of the identified health care issue that is important to you and explain why.

While APA style is not required for the body of this assignment, solid academic writing is expected, and documentation of sources should be presented using APA formatting guidelines, which can be found in the APA Style Guide, located in the Student Success Center.

This assignment uses a rubric. Please review the rubric prior to beginning the assignment to become familiar with the expectations for successful completion.

In the last column we discussed the use of pooling to get a better

estimate of the standard deviation of the measurement method, es-

sentially the standard deviation of the raw data. But as the last column

implied, most of the time individual measurements are averaged and

decisions must take into account another standard deviation, the stan-

dard deviation of the mean, sometimes called the “standard error” of the

mean. It’s helpful to explore this statistic in more detail: fi rst, to under-

stand why statisticians often recommend a “sledgehammer” approach

to data collection methods; and, second, to see that there might be a

better alternative to this crude tactic. We’ll also see how to answer the

question, “How big should my sample size be?”

For the next few columns, we need to discuss in more detail the ways

statisticians do their theoretical work and the ways we use their results.

I often say that theoretical statisticians live on another planet (they don’t,

of course, but let’s say Saturn), while those of us who apply their results

live on Earth. Why do I say that? Because a lot of theoretical statistics

makes the unrealistic assumption that there is an infi nite amount of data

available to us (statisticians call it an infi nite population of data). When we

have to pay for each measurement, that’s a laughable assumption. We’re

often delighted if we have a random sample of that data, perhaps as many

as three replicate measurements from which we can calculate a mean.

That last sentence contains a telling phrase: “a random sample of that

data.” Statisticians imagine that the infi nite population of data contains

all possible values we might get when we make measurements. Statisti-

cians view our results as a random draw from that infi nite population of

possible results that have been sitting there waiting for us. If we were

to make another set of measurements on the same sample, we’d get

a different set of results. That doesn’t surprise the statisticians (and it

shouldn’t surprise us if we adopt their view)—it’s just another random

draw of all the results that are just waiting to appear.

On Saturn they talk about a mean, but they call it a “true” mean. They

don’t intend to imply that they have a pipeline to the National Institute

of Standards and Technology and thus know the absolutely correct value

for what the mean represents. When they call it a “true mean,” they’re

just saying that it’s based on the infi nite amount of data in the popula-

tion, that’s all.

Statisticians generally use Greek letters for true values—μ for a true

mean, σ for a true standard deviation, δ for a true diff erence, etc.

The technical name for these descriptors (μ, σ, δ) is parameters. You’ve

probably been casual about your use of this word, employing it to refer to,

Statistics in the Laboratory:
Standard Deviation of the Mean

say, the pH you’re varying in your experiments, or the yield you get from

those experiments, or maybe even constraints (“We have to stay within

out budgetary parameters”). You can’t be sloppy like that when you work

with a statistician: the word parameter has a very strict meaning.

Because parameters are based on an infi nite amount of data, there is no

uncertainty in their values. (We’ll see why in a minute.)

So, you’re saying to yourself, “I’m confused. And why would I even worry

about what to call these things if I don’t have that infi nite amount of data

and can’t calculate them, anyway?”

Good point. Here’s a key thing, though. Even though we’ll never know

the values of these parameters, we can still use a limited sample of data

to guess at their true values. It’s a process called estimation, so the results

are called parameter estimates, also called sample statistics.

We use a Roman letter to represent individual measurements (e.g., x1 =

3.6), and we put a “bar” above the letter when we want to indicate an

arithmetic average (a mean). For example, if x2 = 4.8, and x3 = 4.5, we would

write the mean of x1 through x3 as x
_

= 4.3. Thus,we say that the statistic x
_

is

an estimate of the parameter μ. Because there is uncertainty in the mea-

sured values that have been “drawn from the population at random,”

there is uncertainty in these parameter estimates.

Backing up a bit, how do we measure the uncertainty in measured

values? As we discussed in the last column, the estimate s of the true

standard deviation σ is given by the familiar equation:

where the Greek capital letter sigma (Σ) is the summation operator, and

its index i indicates the measurement number from 1 to n. For x1 through

x3, s = 0.6807.

Now, let’s go to Saturn for a few minutes. On Saturn we can play with

the infi nite population of data. Let’s suppose that for the measurements

we’ve been making, μ = 4.76 (exactly) and σ = 0.30 (exactly). The estimate

of s = 0.6807 seems a bit high in comparison to σ = 0.30, but parameter

estimates can be quite variable when n is small (and to a statistician n = 3

is small), so it isn’t anything to worry about.

We won’t live long enough to look at all of the data in the infi nite popu-

lation, so let’s look at only one million pieces of data and say that’s

by Stanley N. Deming

AL

20 AMERICAN LABORATORY MARCH 2019

2121 AMERICAN LABORATORY MARCH 2019

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21

exhibit less variability than the raw data. The relationship between sx-, s,

and n is a “reciprocal square-root” function, the statistician’s “one-over-

the-square-root-of-n” effect:

Clearly, as n increases, the uncertainty in the mean decreases.

This relationship holds on Saturn, as well, and shows why on Saturn

there is no uncertainty in the mean—if n = ∞ then σ x- = 0:

This equation can be rearranged to show in general how the ratio of the

standard deviation of the mean to the standard deviation of the raw data

decreases as 1/√n :

Figure 4 illustrates this 1/√n effect. Clearly, as n increases, σx – decreases.

Doing a few replicates can reduce the uncertainty in the mean by quite

representative enough. The Gaussian distribution in Figure 1 was ob-

tained by drawing at random one million data points (statistical samples

of size n = 1) from the infinite population with μ = 4.76 and σ = 0.30.

The data have been “binned” to generate the “histogram distribution”

shown in Figure 1. The bin size is 0.04 on the horizontal axis. There are

100 bins from 3 to 7. If a sample mean had a value between 4.00 and

4.04, for example, it would be placed in bin number 26. The height of

each contiguous histogram bar represents the number of data points

that end up in that bin. Note that the mean of the one million data points

is 4.760 (to three decimal places), and the standard deviation of the one

million data points is 0.300 (to three decimal places). Figure 1 is what we

would expect to see for the individual measurements. No surprises here.

Figure 2 is a little bit diff erent. For this fi gure, we didn’t pull out only one

data point, but we pulled out two data points at a time and binned their

means. So, Figure 2 is based on two million data points, or one million

means for which n = 2. The “grand mean,” the “average of the averages”

(represented by the symbol x with two bars above it) is equal to 4.760, as

expected, but now we see that the “standard deviation of the means” sx – =

0.212, less than 0.30. Interesting.

For Figure 3 we pulled out four data points at a time and binned their

means. The grand mean is again 4.760, but sx- = 0.150, exactly half of

σ = 0.30 for the raw data. What’s going on here?

When data points are averaged, the negative deviations of some of the

data points cancel the positive deviations of other data points. Thus,

the estimated means tend to be closer to the true mean and therefore

Figure 1 – The distribution of 1,000,000 individual pieces of data (n = 1)
drawn at random from an infi nite population with μ = 4.76 and σ = 0.30.
See text

for discussion.

Figure 2 – Yellow: the distribution of 1,000,000 means, each estimated

from two pieces of data (n = 2) drawn at random from an infi nite popu-
lation with μ = 4.76 and σ = 0.30. Green in background: the underlying
distribution of raw data. See text for discussion.

2222 AMERICAN LABORATORY MARCH 2019

STATISTICS IN THE LABORATORY continued

marginal improvement in σx – decreases. Stated differently, the first few

replicates give a lot of bang for the buck; after that, it gets more and more

expensive to decrease σx -.

Many researchers want to know how big their sample size should be (a

legitimate request). Suppose a researcher asks a statistician this ques-

tion, expecting to get a simple answer: e.g., n = 3. Instead, the statistician

turns around and silently walks off in disgust. Why do statisticians be-

have this way? Because they know there is no simple answer to this

question, and they’re going to have to work with the researcher to try

to get information that the researcher might not have. Experience has

taught them that the best time to get out of a bad deal is at the begin-

ning. They don’t want to go through this excruciating process again.

The researcher might have a pooled estimate of σ for the measurement

process, but the researcher’s mean is probably going to be used to make

a decision. The question then becomes, “How uncertain can the reported

mean be and still make a good decision?” That is, how small does σx- have

to be? It’s my opinion that because of the ways companies compart-

mentalize their functions, the researcher making the measurements is

often not aware of this last piece of information. It then becomes the

statistician’s task to move across the company to discover this piece of

information so the sample size can be determined. If you know σ and σx -,

you can calculate the sample size n yourself. At this point, you don’t need

the statistician.

Here’s an example. The percentage of toluene in 500 chemical samples

of gasoline is to be estimated by making multiple gas chromatographic

measurements for each gasoline sample and using the sample mean as

an estimate of the toluene percentage. Each measurement costs $50.

Previous experience has indicated that individual measurements have

a standard deviation of 0.10% toluene (this is σ, the method standard

deviation). However, the client requires a standard deviation of 0.025%

toluene (this will be σx-). How big should your sample size be?

You can almost calculate n in your head. If the ratio of σ x- to σ is

0.025%/0.10% = 1/4, then √n = 4 and n = 16. You must make 16 replicate

measurements on each of the 500 chemical samples for a total of 8,000

measurements. But this will cost $400,000. Your client is going to balk at

this. They’ll ask, “Isn’t there a cheaper way to get the results we need?”

Of course there’s a cheaper way. To get there, let’s look at an assumption

statisticians usually make when they solve sample size questions like

this. They assume σ is what it is, and it can’t be changed. They then apply

the 1/√n sledgehammer to come up with a sample size, as we did above.

But statisticians are often wrong about their assumption, and σ can be

changed. Suppose we bought a better chromatograph that gave mea-

surements with σ = 0.025% toluene instead of 0.10% toluene. With that

new chromatograph, the calculation of sample size would be n = 1. Only

500 measurements would be needed, and the cost running the samples

would be only $25,000.

Figure 5 illustrates the idea. Suppose you start out making 16 measure-

ments per sample ($800/sample) using the old chromatograph and

you suddenly realize you could save money if you bought a better

a bit. For example, when n = 4, σx – is decreased by a factor of 2. But to

decrease σx – by another factor of 2, the number of experiments must be

quadrupled to 16. Clearly, as the number of replicates is increased, the

Figure 4 – Illustration of the “one-over-the-square-root-of-n” effect. The

ratio σ x – / σ decreases as 1/√n.

Figure 3 – Yellow: the distribution of 1,000,000 means, each estimated

from four pieces of data (n = 4) drawn at random from an infinite popu-
lation with μ = 4.76 and σ = 0.30. Green in background: the underlying
distribution of raw data. See text for discussion.

2323 AMERICAN LABORATORY MARCH 2019

AL

samples 101 through 220 (the yellow rectangle labeled RECOVER). After

that, it’s pure SAVINGS, spending only $50 per sample rather than $800

per sample (the green rectangle). The total cost of the project (red area)

will be $190,000 ($90,000 for the chromatograph and $100,000 for the

measurements). This is a lot better than the $400,000 it was going to

cost. (The total cost would have been only $115,000 if you’d realized the

benefits of a better chromatograph at the beginning of the project.)

Don’t try to do with statistics what you can do cheaper with an improved

measurement method. The 1/√n sledgehammer isn’t always the best

way to solve sample size problems.

In conclusion: a) σx- is important for most decision-making, b) you can

make σx- as small as you want by using a large enough sample size,

c) you can calculate your sample size yourself, and d) sometimes it’s less

expensive to make σx- small just by using a better measurement method

with a smaller σ.

I n t h e n e x t m o d u l e w e ’ l l s e e h o w σ x- c a n b e u s e d t o c a l c u l a t e a

confidence interval for the mean.

Stanley N. Deming, Ph.D., is an analytical chemist masquerading as a stat-

istician at Statistical Designs, El Paso, Texas, U.S.A.; e-mail: standeming@

statisticaldesigns.com; www.statisticaldesigns.com

chromatograph. By the time you’ve finished your 100th sample (1600

measurements up to this point, an integrated COST of $80,000), you’ve

put together the funding (the upper yellow rectangle in the figure,

$90,000) and the new chromatograph you’re purchased has just arrived.

Starting with sample 101 you use the new chromatograph and start sav-

ing 15 measurements × $50 per measurement = $750 per sample, which

you can use to recover the $90,000 cost of the new chromatograph from

Figure 5 – An illustration of financial considerations when deciding

whether or not to use a more precise measurement method. See text

for discussion.

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1046080.44

Rubic_Print_

Format

20.0%

Format 10.0%

5.0%

5.0%

Course Code	Class Code	Assignment Title	Total Points
HCA-540	HCA-540-O500	Health Care Research Literature Review	125.0
Criteria	Percentage	1: Unsatisfactory (0.00%)	2: Less Than Satisfactory (80.00%)	3: Satisfactory (88.00%)	4: Good (92.00%)	5: Excellent (100.00%)	Comments	Points Earned
Content	70.0%
Literature Review Table	40.0%	The literature review table is not complete.	The literature review table, including all required criteria, is incomplete or incorrect.	The literature review table, including all required criteria, is mostly complete with only minor errors.	The literature review table, including all required criteria, is complete and correct.	The literature review table, including all required criteria, is complete and includes substantial relevant supporting detail.
Summary of Health Care Issue		20.0%	Summary of the identified health care issue and relative significance to the student is not included.	Summary of the identified health care issue and relative significance to the student is incomplete or incorrect.	Summary of the identified health care issue and relative significance to the student is included but lacks explanation and relevant supporting details.	Summary of the identified health care issue and relative significance to the student is complete and includes explanation and relevant supporting details.	Summary of the identified health care issue and relative significance to the student is thorough and includes substantial explanation and relevant supporting details.
References		10.0%	N/A	References are not included.	Minimum required number and type of relevant references are not included or do not both support and oppose the argument.	Minimum required number and type of relevant references are included and adequately support and oppose the argument.	More than the minimum number and type of relevant references are included and substantially support and oppose the argument.
Organization and Effectiveness
Thesis Development and Purpose	7.0%	Paper lacks any discernible overall purpose or organizing claim.	Thesis is insufficiently developed or vague. Purpose is not clear.	Thesis is apparent and appropriate to purpose.	Thesis is clear and forecasts the development of the paper. Thesis is descriptive and reflective of the arguments and appropriate to the purpose.	Thesis is comprehensive and contains the essence of the paper. Thesis statement makes the purpose of the paper clear.
Argument Logic and Construction	8.0%	Statement of purpose is not justified by the conclusion. The conclusion does not support the claim made. Argument is incoherent and uses noncredible sources.	Sufficient justification of claims is lacking. Argument lacks consistent unity. There are obvious flaws in the logic. Some sources have questionable credibility.	Argument is orderly, but may have a few inconsistencies. The argument presents minimal justification of claims. Argument logically, but not thoroughly, supports the purpose. Sources used are credible. Introduction and conclusion bracket the thesis.	Argument shows logical progressions. Techniques of argumentation are evident. There is a smooth progression of claims from introduction to conclusion. Most sources are authoritative.	Clear and convincing argument that presents a persuasive claim in a distinctive and compelling manner. All sources are authoritative.
Mechanics of Writing (includes spelling, punctuation, grammar, language use)			5.0%	Surface errors are pervasive enough that they impede communication of meaning. Inappropriate word choice or sentence construction is used.	Frequent and repetitive mechanical errors distract the reader. Inconsistencies in language choice (register) or word choice are present. Sentence structure is correct but not varied.	Some mechanical errors or typos are present, but they are not overly distracting to the reader. Correct and varied sentence structure and audience-appropriate language are employed.	Prose is largely free of mechanical errors, although a few may be present. The writer uses a variety of effective sentence structures and figures of speech.	Writer is clearly in command of standard, written, academic English.
Paper Format (Use of appropriate style for the major and assignment)	Template is not used appropriately or documentation format is rarely followed correctly.	Template is used, but some elements are missing or mistaken; lack of control with formatting is apparent.	Template is used, and formatting is correct, although some minor errors may be present.	Template is fully used; There are virtually no errors in formatting style.	All format elements are correct.
Documentation of Sources (citations, footnotes, references, bibliography, etc., as appropriate to assignment and style)	Sources are not documented.	Documentation of sources is inconsistent or incorrect, as appropriate to assignment and style, with numerous formatting errors.	Sources are documented, as appropriate to assignment and style, although some formatting errors may be present.	Sources are documented, as appropriate to assignment and style, and format is mostly correct.	Sources are completely and correctly documented, as appropriate to assignment and style, and format is free of error.
Total Weightage	100%

LiteratureReview Table

Student Name:

Summary of Health Care Issue (

2

50-500 words):

		Criteria	Article 1	Article 2	Article 3
		APA-Formatted Article Citation with DOI
		Article Permalink
		How Does the Article Relate to your proposal?
		Quantitative, Qualitative (Explain how you know.)
		Purpose Statement
		Setting (Indicate where the study took place.)
		Sample
		Method
		Outcomes/Results/ Key Findings of Study
		Limitations of the Study
		Future Recommendations of the Researcher

Criteria

APA-Formatted Article Citation with DOI

Article Permalink

How Does the Article Relate to your proposal?

Quantitative, Qualitative (Explain how you know.)

Purpose Statement

Setting

(Indicate where the study took place.)

Sample

Method

Outcomes/Results/ Key Findings of Study

Limitations of the Study

Future Recommendations of the Researcher

Article 4	Article 5	Article 6

Criteria

APA-Formatted Article Citation with DOI

Article Permalink

How Does the Article Relate to your proposal?

Quantitative, Qualitative (Explain how you know.)

Purpose Statement

Setting

(Indicate where the study took place.)

Sample

Method

Outcomes/Results/ Key Findings of Study

Limitations of the Study

Future Recommendations of the Researcher

Article 7

Article 8

Article 9

2