Key Statistical Concepts
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Descriptive Statistics
Frequency Distribution: "a set of scores arranged in order of magnitude along the x-axis
and the frequency of each score is represented along the y-axis" (Coolidge, 2000, p. 48).
Types:
- frequency histogram: similar to bar graphs but has no spaces between the bars
- frequency polygon: points are connected with straight lines.
Types based on the general shape of the distribution
- normal distribution: bell-curve
- skewed distribution: negatively and positively (Students in Colorado remember positive skew as the right
side of the snowy mountain that is appropriate for skiing. So they ski to the right->skew to the right)
A "job-aid" is available in the Salkind chapter on how to construct a frequency distribution graph.
General procedures and advices (Coolidge, 2000):
- group data into intervals (5 to 10)
- define the size of the interval widths based on understandable units
- Make sure the intervals do not overlap.
Activities
Review Salkind Chapter on "frequency distribution" and create a
paper-pencil distributions for one of
the following sample data.
m06_fakespring.xls
m06_fakefall.xls - see
what quantitative data look like
Inferential Statistics
- Used to make inferences about populations based on the behavior of a sample.
- Concerned with how likely it is that a result based on a sample or samples are the
same as results that might be obtained from an entire population.
Hypothesis
Hypothesis (chapter 7 of Salkind; continuation of Module 2): "An if-then statement of conjecture that
relates variables to one another." A good hypothesis translates a problem statement or a research question into
a form that can be tested through statistical techniques. Some of the statistics books state that inferential
statistics is about hypothesis testing.
Null Hypothesis (Ho) (what is not true)
At the starting point of a research, hypothesis is normally stated as "null hypothesis," in the absence
of any other information or a priori (before the fact) knowledge. "There is no significant difference between learning
outcomes and format of training." In theory, all hypothesis-testing should start with a null hypothesis when there is
no other evidence to support a non-null (alternative) hypothesis. It is like a default position, safe and conservative
(Coolidge, 1998).
Research (Alternative) Hypothesis (Ha) (what is true)
A definite statement of the relationship between variables. Each null hypothesis corresponds to one or more
research hypotheses.
"There is significant difference between learning outcomes and format of training - in online or classroom
setting." - nondirectional
"Learning outcomes as indicated by test scores are significantly higher in online setting than in classroom
setting." - directional
Research hypothesis is less conservative, because it is more sensitive to differences than null hypothesis.
That means, it is more likely to show a test as significant.
Statistically Significant and Level of Significance
Statistically significant means that differences are due to systematic influence and not due to chance or random errors.
As in the following example, if the fear-of-fat score differences of Australian and Indian students are statistically
significant (p<0.05), that means the difference is due to cultural difference. However, the world is not perfect and
chance has effects on many things. A level of significance is associated with every statistical tests.
If the findings are significant at 0.05, the translation is that there is 1 chance in 20 that any differences found
were not due to the systematic influence. So the level of significance is the risk associated with not being 100% confident
that the difference is due to cultural difference. In other words, it is an estimate of the probability that we are wrong
when we say there is difference or no difference between the two samples (alpha level).
The alpha level (the level of risk or uncertainty) should be set based on the nature of the tests. 0.1 (being 90%
confident) is used for exploratory tests that allows for larger chance factors; 0.05 (being 95% confident) is used for
many educational tests; 0.01 (being 99% confident) is used for science that needs a higher accuracy and tolerates low
chance factors.
Type I and II Error, alpha, and p level
Another way to understand alpha and p level is from the Type I and Type II errors. Type I error occurs when a researcher
rejects null hypothesis, when it is actually true. This is considered to be a serious error because it can mislead people
to believe the effects of some treatment. For example, a new drug does not work when the researcher claims that it works.
Type II error occurs when a researcher retains null hypothesis when it is actually false. For example a new drug works
when the researcher concludes that it does not.
p level (or alpha) is also the probability of committing a Type One Error. p<.05 means a Type one error should be
less than 5 chances out of 100.
Note: In statistics, we use "significant" and "nonsignificant" to report the findings, but not
"insignificant." In significant is a value judgment not a statistical concept.
An "off-topic" but interesting example: cultural difference and fear of fat.
The mean Fear-of-Fat score of Australian students is 100; the one of Indian students is 125. By eyeballing, we see
there is a difference. But is this difference significantly different? In other words, is the difference due to chance
(e.g., sampling error) or due to the cultural difference of the two groups of students?
We answer the question through hypothesis-testing. "There is no significant difference between the two groups of
students in fear of fat." We set a level of significance (known as "alpha") and then ran independent t-test. When a=0.1
(10%), there is a 10% probability that we are wrong when we say the results are due to chance. a=0.05 (5%); a=0.01 (1%).
In other words, significance level is the risk associated with not being 100% confident that your results are due to the
intervention.
When interpreting the statistical results, we compare p (probability of the differences are due to chance) with a
(level of risks). When p<a, we reject the null hypothesis and conclude that there is significant difference between the
fear-of-fat scores. This procedure applies to the testing of research hypothesis.
Confidence Interval
Because a sample normally doesn't perfectly represent the population, inferential statistics identifies how likely
(90%, 95%, or 99%) the sample results represent the results that would occur in the population.
By generating hypotheses, we make probability statements that the results we see in samples would also be found
in population. Our confidence in the probability statements are at 90%, 95%, or 99%. This is known as confidence interval.
Confidence interval corresponds to alpha. When a=0.05, we are 95% confident about our probability statement.
Degree of Freedom (df)
"The value that is different for different statistical tests and approximates the sample size of number of
individual cells" (Salkind, 2000, p.367).
"A complicated statistical term which in some statistical tests is roughly correlated with the total number
of participants or observations but always slightly less. The df is actually based in the estimation of the standard
deviation and indicates the number of numbers that are free to vary in estimation theory." (Coolidge, 2000, p. 153).
- Correlation: df=N-2 (N: no of participants)
- t test: df=N-1 (t for one); and N1-1 + N2-1 (tea for two).
The value of DF should always be specified in statistical reports. DF is referred to when comparing a test
statistic (t value in t test; f ratio in ANOVA) to a critical value at the corresponding alpha level (e.g., a=.05).
The test is significant if the test statistic is larger than the critical value. As t-table and f-table indicate,
critical values vary with degree of freedom.
Note: If any statistical software is used to run a test (rather than hand-calculation with formulas), one can
make conclusions only by the p level obtained. T and f tables are used when the results are obtained by hand calculation.
Resources
m06_glossary.doc -
Glossary of Key Terms used in the Research Essays
