Understanding Data Analysis: Descriptive Statistics and Hypothesis Testing, Slides of Research Methodology

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Presenting Data

Descriptive Statistics

Nominal Level

• No order, just a name
• Can report
  • Mode
  • Bar Graph
  • Pie Chart

Ordinal Level

• Rank order only
• Can Report
  • Mode
  • Median
  • Percentiles
  • Histograms and Pie Charts

Interval/Ratio Level

• Equidistant
• Can Report
  • Mode, Median, Mean
  • Standard Deviation
  • Percentiles
  • Frequency curves, Histograms

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

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

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

• 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

Normal Curve

• To get a statistical summary, including an imposed normal curve in Minitab:
• 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 – 25 th 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.
• Hypotheses are not necessarily true.

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.

Errors

Truth Ho is true Ho is false Decision Reject Ho Type I Error OK Fail to Reject Ho

OK Type II Error

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.

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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

Statistical Significance

• Statistically significant: if the probability of obtaining a statistic by chance is less than the set alpha level (usually 5%)

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.

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.

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.