Hi jabeda!!
RANGE:
The range is the distance between the highest and lowest score.
Numerically, the range equals the highest score minus the lowest
score.
In this case:
Range = highest score - lowest score = 6 - 0 = 6 hs
-------------------------------------------------------
VARIANCE:
The variance is an index of variability used to characterize the
dispersion among the measures in a given population. Numerically, the
variance equals the average of the several squared deviations from the
mean:
Variance = sum((Xi-M)^2) /N for i = 1 to N ;
where each Xi is one of the N different scores, M is the mean of the
scores and N is the number of scores.
In this problem:
M = (2 + 5 + 6 + 1 + 4 + 0 + 3) / 7 = 21/7 = 3
Variance = [(2-3)^2 + (5-3)^2 + (6-3)^2 + (1-3)^2 + (4-3)^2 + (0-3)^2
+ (3-3)^2] / 7 = (1 + 4 + 9 + 4 + 1 + 9 + 0)/7 = 28/7 = 4
Variance = 4
-----------------------------------------------------
STANDARD DEVIATION:
Standard deviation is a measure of how widely values are dispersed
from the mean value in a set of data. That is, the standard deviation
is the average squared deviation (variance) returned to standardized
format. The standard deviation is the square root of the variance.
Unlike the variance, which is a somewhat abstract measure of
variability, the standard deviation can be readily conceptualized as a
distance along the scale of measurement.
Standard Deviation = sqrt(Variance) = sqrt(4) = 2 hs
-------------------------------------------------------
Co-efficient of Variation:
The coefficient of variation is a measure of relative dispersion. It
is used to compare the variations ( dispersion ) of two different
series. Numerically is the ratio between S.D. and the mean expressed
as a percentage:
C.V. = (S.D. / M) x 100
In this case:
C.V. = (2hs / 3 hs) x 100 = 0.667 x 100 = 66.7%
------------------------------------------------
First of all you may want to know why my results are different from
yours; the answer comes from the ambiguous definition of Variance and
Standard Deviation. I have used the formula for the calculation of the
Variance considering the entire population. You have used the
"unbiased estimator" for the Variance considering a SAMPLE SUBSET OF
OBSERVATIONS OR SCORES:
Variance = sum((Xi-M)^2) /(N-1) for i = 1 to N ;
where each Xi is one of the N different scores, M is the mean of the
scores and N is the number of scores.
And from this Variance you will obtain the S.D. and the C.V.
For this problem:
Variance = 28/6 = 4.667
S.D. = sqrt(4.667) = 2.16 hs
C.V. = 2.16 / 3 = 0.72 = 72%
For a further explanation about this "unbiased estimators" see the
following pages at QuickMBA.com:
"Standard Deviation and Variance":
http://www.quickmba.com/stats/standard-deviation/
See also "Variance and standard deviation Parts 1 & 2" at HyperStat
Online:
Part 1:
http://www.ruf.rice.edu/~lane/hyperstat/A16252.html
Part 2: At this page you have a discussion about why the S.D. "...is
the most commonly used measure of spread...The standard deviation has
proven to be an extremely useful measure of spread in part because it
is mathematically tractable. Many formulas in inferential statistics
use the standard deviation...":
http://davidmlane.com/hyperstat/A40397.html
After reading several pages and articles related to this topic, I
conclude that the best measure of variability is the Standard
Deviation, some of the advantages of this estimator are:
- It is based on all observations.
- It is useful for further algebraic treatment.
- It is not affected by sampling fluctuations.
- It is less erratic.
Some examples of it use are in business:
"Business interpretation of standard deviation:
- The “risk” of an investment is usually the standard deviation of
some measure of return on the investment
- stock risk: standard deviation of stock price
- bond fund risk: standard deviation of annual rate of return"
http://www.stat.ncsu.edu/~st350_info/reiland/Variab2/sld002.htm
For example at the page "Standard Deviation" of the Chemistry Learning
Center (CLC) at Virginia Tech we can read the following statements:
"...The most common measure of the error in an experimental quantity
is the standard deviation of a set of data...
...Statistically, for a large set of measurements about 68% will lie
within one standard deviation of the average value, 95% will lie
within 2 and 99.7% will lie within 3. For example, if you have a set
of data reported as 1.00 +- 0.1, 68% of the data will lie within the
range 0.9 to 1.1, 95% within the range 0.8 to 1.2, and 99.7% within
the range 0.7 to 1.3.
Standard deviations are typically reported to only one significant
figure."
Please visit this page to see more info related to the two different
formulas for S.D. (using N or N-1):
http://learn.chem.vt.edu/tutorials/error/stddev.html
See also the following articles about how the S.D. is used:
"A note on standard deviation" by Harley Weston, Department of
Mathematics and Statistics, University of Regina :
http://mathcentral.uregina.ca/rr/database/RR.09.95/weston2.html
"What is the Standard Deviation?" at The Financial Forecast Center™:
http://www.neatideas.com/stdev.htm
"How Standard Deviation works?" by David Harrell, a staff writer for
Morningstar.com:
http://news.morningstar.com/news/ms/Investing101/riskybusinesstwo.html
"Standard Deviation" from the Stats page of the Children's Mercy
Hospital :
http://www.cmh.edu/stats/definitions/stdev.htm
Search strategy:
Variance population sample
Standard deviation definition
Standard deviation interpretation
I hope this helps you, but if you find something unclear in this
answer, please post a request for an answer clarification before rate
it.
Best Regards.
livioflores-ga |
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