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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

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Presenting data l.jpg

Presenting Data

Descriptive Statistics

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Nominal Level

  • No order, just a name

  • Can report

    • Mode

    • Bar Graph

    • Pie Chart

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Ordinal Level

  • Rank order only

  • Can Report

    • Mode

    • Median

    • Percentiles

    • Histograms and Pie Charts

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Interval/Ratio Level

  • Equidistant

  • Can Report

    • Mode, Median, Mean

    • Standard Deviation

    • Percentiles

    • Frequency curves, Histograms

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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

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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

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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)

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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

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Normal Curve

  • To get a statistical summary, including an imposed normal curve in Minitab:

  • Select: Stat > Basic Statistics > Display Descriptive Statistics > Graph > Graphical Summary

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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

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  • 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

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  • 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

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Statistical Hypotheses

  • Statistical Hypotheses are statements about population parameters.

  • Hypotheses are not necessarily true.

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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.

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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.

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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%.

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Statistical Significance

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

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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.

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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.

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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.