Measures of Central Tendency
When we are representing a sample quantitatively, we rely on expressions that will summarize the data for our audience. These expressions or measures are called typical values.
If we are trying to summarize and analyze strictly quantitative (interval or ratio data) we may start by creating a histogram and draw a curve over the top, connecting the center of each of the bars.? When we do this, we create a curve that represents the distribution of the scores. Sometimes our measure of central tendency describes the center of this curve; sometimes it describes the most likely score to occur; sometimes our measure is the balance point, a mathematically symmetrical place (though it may not appear to you that way when you look at the graph.)
THE BALANCE POINT – THE “AVERAGE” OR ARITHMETICAL MEAN
The most commonly used measure of central tendency (I am sure you are familiar with it) and the one that people usually “mean” when they say average is the MEAN (as it were.) I too will refer to the mean as the “average” for the remainder of this course.
You can calculate a mean easily by hand for very small samples, or with the aid of a calculator or spreadsheet for large distributions:
Here is a small sample that can be done by hand:
2, 3, 4, 5, 6
Step 1: Count the number of observations (scores) in the sample. We call this the “N”?
N = 5
Step 2: Add the scores together?
2 + 3 + 4 + 5 + 6 = 20
Step 3: Divide by “N”?
20/5 = 4
The mean of this small distribution is 4.
Do this one by hand or with a calculator =
2, 7, 3, 9, 3, 4, 7
N = ?
Sum of the scores = ?
Sum divided by “N” = ?
I will trust that you can figure it out, but feel free to email me for confirmation of the right answer if you like.
Generally, when we are working with large collections of quantitative data, we use the mean to describe the typical score. However, there are circumstances when we might want to consider a different summary measure. The mean is very (mathematically) sensitive to extreme scores or “outliers.” The degree of distortion that a uncharacteristically high or low score can cause in an otherwise smooth or somewhat “normal” distribution can lead to a misinterpretation of the significance of the data. So it is good to be familiar with the other two measures of central tendency, even though you may rarely (if ever) use them.
THE CENTER OF THE CURVE – THE MEDIAN
If we line all our scores up in rank order, we can pick a middle value.?
When we have an odd number of scores, the middle value is right there:
1, 3, 5, 7, 9 – it’s obvious that the middle value is 5.
5 is the median.
When we have an even number of scores, the middle value is “hidden”:
1, 3, 5, 7, 9, 11 – it just takes a little simple middle school arithmetic to find it.
Locate the middle two scores (in this case, that’s 5 and 7)
Add?them together; 5 + 7 = 12
Divide the resulting sum by 2; 12/2 = 6
6 is the MEDIAN of this small distribution.
When is it most appropriate to use the median?
When our distribution of scores is very uneven, especially with many scores bunched at one end and just a few stragglers or outliers at the other end. The outliers would likely distort our ability to see what a typical score in this distribution is.?
Let’s say that we did a study of the effects of a particular herbal extract on the longevity of a sample of mice who were given the extract every day of their lives from birth. We had 8 mice and this is how long they lived in days:
499 502 510 523 524 530 539 815
If we merely calculated the arithmetic mean, the effect of the age of our one rodent Methuselah might cause that summary measure to suggest to our clinical audience that the herbal extract helped our little rodent subjects to actually live longer. Instead, it is more likely that his extended lifespan was a fluke, and the reporting the median would give a more honest and accurate accounting of the research results.
Just compare -?
The median age in this sample is 523 + 524 divided by 2 or 523.5 days old.
The mean age is 555.25
If the average age of the mice we have been raising in our lab is 525 days, the second result would be more misleading to anyone interested in the effects of the extract.
THE MOST COMMON SCORE – THE MODE
I mentioned in the earlier modules that we sometimes work with qualitative data called “nominal” data. An example of such data in the health sciences would be diagnostic categories:
Diabetes = 22
Asthma = 40
Psoriasis = 13
In this sample, asthma is the most common diagnosis that we encounter. We might use a numbering system of our own devising, or an established one like the ICD-9. But an any rate, whether we report it by name, by our own numerical label, or someone else’s, if we report that asthma was the most typical diagnosis in this sample, we are reporting the MODE.
We may use the mode with plain old quantitative data too. Let’s say we weighed our mice at death (the ones who took the herbal extract described above.) These are the weights we obtained (in ounces):
.5, .5, .5, .5. .47, .5, .51, .49
Although we could do a simple mean calculation, it would also be appropriate to use the mode and say that the average weight of one of our experimental mice was 1/2 ounce.
Please spend some time at the following websites to become more familiar with the measures of central tendency, how to use them, and how to interpret them most accurately:
DIG Stats. Retrieved?August 1, 2011 from http://www.cvgs.k12.va.us/DIGSTATS/
Using Data & Statistics (2006). Retrieved?August 1, 2011 from http://www.mathleague.com/help/data/data.htm
Measures of dispersion
We won’t get the whole picture accurate picture of what a distribution of scores looks like if we only notice where they are “clumping.” We have to look at how they are spread out and the typical span in which we find most of our observations. This information is crucial because it tells us what the expected difference from the average is, and lets us know when we can be confident that the difference between a particular observation (or set of observations) and the average is unusual, possibly “significant.”
The least complicated to calculate (but also least informative) measure of spread or dispersion is the RANGE. You can find the range by subtracting the lowest score in your data set from the highest.
2, 3, 4, 5, 6
What’s the highest score?? 6
What’s the lowest score? 2
6 – 2 = 4
4 is the range for this data set.
You try it with this one:
6, 8, 3, 10, 12
What’s the highest score? What’s the lowest score? Subtract them and you get….(I think you can handle this.)
Obviously, we don’t learn much about a distribution from this summary measure. We can’t really see the difference between this –
1, 2, 4, 4, 4, 4, 4, 12
1, 3, 9, 9, 9, 9, 9, 12
very different distributions that have exactly the same range!
There are variations on the range that are used and are easy to calculate and somewhat more informative. Called the Interquartile Range and Semi-interquartile Range, they rely on some easy to calculate measures of position to narrow down the scope so we can begin to get a picture of where most of the scores are falling and thus begin to blend the central tendency with the typical pattern of dispersion for a more accurate picture of the distribution we are examining. If you’d like to know more about these visit –
Basic Statistical Concepts (2011). Retrieved August 1, 2011 from http://www.statsoft.com/textbook/elementary-concepts-in-statistics/
MOST VITAL TO YOUR UNDERSTANDING WHAT YOU READ IN A RESEARCH ARTICLE IS THE CONCEPT OF THE STANDARD DEVIATION.
The standard deviation defines an area below and above the mean about which it is expected that a majority of the scores will fall. Once we get outside that area, in either direction (+/-), we are approaching the zone in which it is rare to find an observation in our distribution, and quite possible that the reason we would find a score there is of interest to us in accomplishing our research goals.
(Click here for a brief PowerPoint demonstration of how to calculate the: STANDARD DEVIATION)
Please visit the following links to learn more about means, standard deviations and the STANDARD NORMAL DISTRIBUTION.
Summarizing & Presenting Data. Retrieved?August 1, 2011 from http://surfstat.anu.edu.au/surfstat-home/chap1ex.html
Normal Distribution. Retrieved?August 1, 2011 from http://davidmlane.com/hyperstat/A6929.html
The Normal Distribution. Retrieved?August 1, 2011 from http://www-stat.stanford.edu/~naras/jsm/NormalDensity/NormalDensity.html
Trochim, W.K. (2006). Research Methods Knowledge Base. Retrieved August 1, 2011 from http://www.socialresearchmethods.net/kb/contents.php
In the previous module, you found resources for your area of research interest. For this third segment of the SLP, please?find 2 more articles from professional, scholarly?journals.?They must be no more than 5 years old. Search ProQuest to find them.?Save the articles, as you will use them in future modules.
Before you choose the articles, review what has to be included in your paper so you are sure to choose articles with the appropriate information.
In?a 2-page paper,
1. Write an introduction to the topic and remind me of?your research?hypothesis. If I have given you feedback to improve it, use the improved version.
2. Submit an annotated bibliography for each of the 2?articles. Be sure to include the following:
Describe the source’s use of?descriptive statistical measures you learned about in this module.?
Describe the statistics used and how the researcher interprets them in relation to his/her study.
Discuss how the findings in that article compare or contrast with the hypothesis you developed for your SLP area of research. In other words, does the article support help your hypothesis? Or contradict it? Please explain.
Use headings and subheadings to guide the flow of your paper. If you need a reminder about writing an annotated bibliography, review the link provided in SLP 1.?
ASSIGNMENT EXPECTATIONS:? Please read before completing assignments.??
Paper?should be 2 pages in length (double-spaced).
Please use major sections corresponding to the major points of the assignment, and where appropriate use sub-sections (with headings).
Remember to write in a scientific manner (try to avoid using the first person except when describing a relevant personal experience).
Quoted material should not exceed 10% of the total paper (since the focus of these assignments is on independent thinking and critical analysis).? Use your own words and build on the ideas of others.???
When material is copied verbatim from external sources, it MUST be properly cited.? This means that material copied verbatim must be enclosed in quotes and the reference should be cited either within the text or with a footnote.
Credible professional sources are used (for example, government agencies, nonprofit organizations, academic institutions, scholarly journals). Wikipedia is not acceptable.
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