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Presenting Data
Descriptive Statistics
Nominal Level

No order, just a name
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Can report
– Mode
– Bar Graph
– Pie Chart
Ordinal Level
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Rank order only
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Can Report
– Mode
– Median
– Percentiles
– Histograms and Pie Charts
Interval/Ratio Level
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Equidistant
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Can Report
– Mode, Median, Mean
– Standard Deviation
– Percentiles
– Frequency curves, Histograms
Univariate Data
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Good to start at the univariate level
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Univariate: one variable at a time
– Investigate the responses
– Assess usability for the rest of the analysis
Frequency Table
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Shows how often each response was
given by the respondents
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Most useful with nominal or ordinal
– Interval/ratio has too many categories

In Minitab, Select: Stat>Tables>Tally
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)
Normal Curve
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The normal curve is critical to assessing
normality which is an underlying assumption
in inferential statistical procedures
– And in reporting of results
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Kurtosis: related to the bell-shape
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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
Normal Curve
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To get a statistical summary, including
an imposed normal curve in Minitab:
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Select: Stat > Basic Statistics > Display
Descriptive Statistics > Graph >
Graphical Summary
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
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
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
Statistical Hypotheses

Statistical Hypotheses are statements
about population parameters.
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Hypotheses are not necessarily true.
In statistics, we test one hypothesis against
another…
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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.
Errors
Decision
Reject Ho
Fail to
Reject Ho
Truth
Ho is true
Ho is false
Type I Error
OK
OK
Type II
Error
P-value


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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.
Interval Estimates


Statisticians prefer interval estimates.
X  Something

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%.
Statistical Significance

Statistically significant: if the probability
of obtaining a statistic by chance is less
than the set alpha level (usually 5%)
P-value
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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.
Power
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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.
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.