5. Univariate Data
Good to start at the univariate level
Univariate: one variable at a time
– Investigate the responses
– Assess usability for the rest of the analysis
6. Frequency Table
Shows how often each response was
given by the respondents
Most useful with nominal or ordinal
– Interval/ratio has too many categories
In Minitab, Select: Stat>Tables>Tally
7. Charts and Graphs
Use a bar graph or pie chart if the variable
has a limited number of discrete values
– Nominal or ordinal measures
Histograms and frequency curves are best for
interval/ratio measures
In Minitab, Select: Graph > (and then type)
8. Normal Curve
The normal curve is critical to assessing
normality which is an underlying assumption
in inferential statistical procedures
– And in reporting of results
Kurtosis: related to the bell-shape
Skewness: symmetry of the curve
– If more scores are bunched together on the left
side, positive skew (right)
– If most scores are bunched together on the right
side, negative skew
9. Normal Curve
To get a statistical summary, including
an imposed normal curve in Minitab:
Select: Stat > Basic Statistics > Display
Descriptive Statistics > Graph >
Graphical Summary
10. Measures of Central Tendency
Mode: most frequently selected
– Bimodal = two modes
– If more than two modes, either multiple
modes or no mode
Median: halfway point
– Not always an actual response
Mean: arithmetic mean
11. Percentiles
The median is the 50 percentile
A percentile tells you the percentage of
responses that fall above and below a
particular point
Interquartile range = 75th percentile –
25th percentile
– Not affected by outliers as the range is
12. Z-scores
Standard deviations provide an estimate
of variability
If scores follow a ‘normal curve’, you
can comparing any two scores by
standardizing them
– Translate scores into z-scores
– (Value – mean) / standard deviation
14. In statistics, we test one hypothesis against
another…
The hypothesis that we want to prove is
called the alternative hypothesis, Ha.
Another hypothesis is formed which
contradicts Ha.
–This hypothesis is called the null
hypothesis, Ho. Ho contains an
equality statement.
15. Errors
Truth
Ho is true Ho is false
Reject Ho Type I Error OK
Decision
Fail to
Reject Ho
OK Type II
Error
16. P-value
The choice of is subjective.
The smaller is, the smaller the
critical region. Thus, the harder it is to
Reject Ho.
The p-value of a hypothesis test is the
smallest value of such that Ho would
have been rejected.
17. Interval Estimates
Statisticians prefer interval estimates.
Something depends on amount of
variability in data and how certain we want
to be that we are correct.
The degree of certainty that we are correct
is known as the level of confidence.
– Common levels are 90%, 95%, and 99%.
X Something
19. P-value
The probability, computed assuming that Ho is
true, that the test statistic would take a value
as extreme or more extreme than that actually
observed is called the p-value of the test.
The smaller the p-value, the stronger the
evidence against Ho provided by the data.
If the p-value is as small or smaller than alpha,
we say that the data are statistically significant
at level alpha.
20. Power
The probability that a fixed level alpha
significance test will reject Ho when a
particular alternative value of the
parameter is true is called the power of the
test to detect that alternative.
One way to increase power is to increase
sample size.
21. Use and Abuse
P-values are more informative than the results of
a fixed level alpha test.
Beware of placing too much weight on traditional
values of alpha.
Very small effects can be highly significant,
especially when a test is based on a large
sample.
Lack of significance does not imply that Ho is
true, especially when the test has low power.
Significance tests are not always valid.
