Types of data and how to present them 47:269: Research Methods I Dr. Leonard March 31, :269: Research Methods I Dr. Leonard March 31, 2010

Scientific Theory 1. Formulate theories  2. Develop testable hypotheses (operational definitions)  3. Conduct research, gather data  4. Evaluate hypotheses based on data  5. Cautiously draw conclusions 1. Formulate theories  2. Develop testable hypotheses (operational definitions)  3. Conduct research, gather data  4. Evaluate hypotheses based on data  5. Cautiously draw conclusions

Scales of Measurement / Nominal / Categories / Ordinal / Categories that can be ranked / Interval / Scores with equidistant intervals between them / Ratio / Scores with equidistant intervals and absolute zero / Nominal / Categories / Ordinal / Categories that can be ranked / Interval / Scores with equidistant intervals between them / Ratio / Scores with equidistant intervals and absolute zero

Responses are distinct Responses can be ranked Equal intervals Absolute zero NominalYESNO OrdinalYES NO IntervalYES NO RatioYES

Two major approaches to using data / Descriptive statistics / Describe or summarize data to characterize sample / Organizes responses to show trends in data / Inferential statistics / Draw inferences about population from sample (is population distinct from sample?) / Significance tests / Capture impact of random error on responses / Margin of error / Note: Statistics describe responses from a sample; parameters describe responses from a population (e.g., a census) / Descriptive statistics / Describe or summarize data to characterize sample / Organizes responses to show trends in data / Inferential statistics / Draw inferences about population from sample (is population distinct from sample?) / Significance tests / Capture impact of random error on responses / Margin of error / Note: Statistics describe responses from a sample; parameters describe responses from a population (e.g., a census)

Descriptive Statistics / N, total number of cases (responses) in a sample / Our class would be N = 33 / f, or frequency, is the number of participants who gave a particular response, x / Can also be given as percentages or proportions / Can be univariate or bivariate / How participants vary on one variable (uni-) / How participants vary on two variables (bi-) / Descriptive statistics are a good first step for analyzing any data! / They are the only statistics appropriate for nominal data / N, total number of cases (responses) in a sample / Our class would be N = 33 / f, or frequency, is the number of participants who gave a particular response, x / Can also be given as percentages or proportions / Can be univariate or bivariate / How participants vary on one variable (uni-) / How participants vary on two variables (bi-) / Descriptive statistics are a good first step for analyzing any data! / They are the only statistics appropriate for nominal data

Frequency distribution (nominal data) x (response) f (frequency) % Democrat Republican Independent Green party90.9 Totaln = 1, %

Frequency distribution (interval or ratio data) / When you need to present a wide range of scores, show responses grouped in intervals to make it easier to grasp “big picture” of data Intervalf / When you need to present a wide range of scores, show responses grouped in intervals to make it easier to grasp “big picture” of data Intervalf

/ Frequency distributions can be depicted graphically in… Bar graphs / Bars not touching because of discrete data / Nominal and ordinal data Histograms / Bars touching because of continuous data / Interval and ratio data Frequency polygons (single line) / Interval and ratio data / Frequency distributions can be depicted graphically in… Bar graphs / Bars not touching because of discrete data / Nominal and ordinal data Histograms / Bars touching because of continuous data / Interval and ratio data Frequency polygons (single line) / Interval and ratio data

What else can we do besides frequencies?  Measures of central tendency show the central or “ typical ” scores in a distribution / Mean- the average score / Median- the middle score / Mode- the most frequent score / The mean, median, and mode are related to the horizontal shape (skew) of the distribution. / In a normal distribution: Mean = Median = Mode / In a positively skewed distribution: Mode < Median < Mean / In a negatively skewed distribution: Mean < Median < Mode  Measures of central tendency show the central or “ typical ” scores in a distribution / Mean- the average score / Median- the middle score / Mode- the most frequent score / The mean, median, and mode are related to the horizontal shape (skew) of the distribution. / In a normal distribution: Mean = Median = Mode / In a positively skewed distribution: Mode < Median < Mean / In a negatively skewed distribution: Mean < Median < Mode

Which measure of central tendency??? Different measures of central tendency are appropriate depending upon the level of measurement used: NominalOrdinal Interval/Ratio    Mode Mode Mode Median Median Mean

The Mean / The most informative and elegant measure of central tendency. / The average / The fulcrum point of the distribution / The most informative and elegant measure of central tendency. / The average / The fulcrum point of the distribution

The Median / The middle most score in a distribution. / The scale value below which and above which 50% of the distribution falls / Not the fulcrum: The halfway point / The middle most score in a distribution. / The scale value below which and above which 50% of the distribution falls / Not the fulcrum: The halfway point

The Median / If N is odd, then median is the center score / If N is even, then median is the average of the two centermost score / If N is odd, then median is the center score / If N is even, then median is the average of the two centermost score

The Median / If the median occurs at a value where there are tied scores, use the tied score as the median

The Mode / The most frequent score in the distribution

One more thing…  These measures of central tendency vary in their sampling stability = match between the sample mean (e.g., x) and the population mean ( μ ). Mode Median Mean Note: Roman (r, s, x) characters are used for sample statistics while Greek ( , ,  ) characters are used for population statistics.  These measures of central tendency vary in their sampling stability = match between the sample mean (e.g., x) and the population mean ( μ ). Mode Median Mean Note: Roman (r, s, x) characters are used for sample statistics while Greek ( , ,  ) characters are used for population statistics. Least sampling stability Most sampling stability

Review of central tendency / Which one is the only appropriate measure for nominal data? / The mode / How do you find the median when there is an odd number of scores? / Simply locate the score in the middle / …when there is an even number of scores? / Average the two middle scores / Which measure is most sensitive to extreme scores and why? / The mean because it takes all scores into account and can be swayed by positive or negative skew / Which measure has the most sampling stability and why? / The mean because it is the most accurate representation of the overall sample / Which one is the only appropriate measure for nominal data? / The mode / How do you find the median when there is an odd number of scores? / Simply locate the score in the middle / …when there is an even number of scores? / Average the two middle scores / Which measure is most sensitive to extreme scores and why? / The mean because it takes all scores into account and can be swayed by positive or negative skew / Which measure has the most sampling stability and why? / The mean because it is the most accurate representation of the overall sample

Application of central tendency / In 2006, the median home price in Boston was $386,300. (San Francisco was $518,400; Washington D.C was $258,700). / How do you interpret these numbers? / Why are housing prices framed in terms of the median rather than the mean or the mode? / In 2006, the median home price in Boston was $386,300. (San Francisco was $518,400; Washington D.C was $258,700). / How do you interpret these numbers? / Why are housing prices framed in terms of the median rather than the mean or the mode?

Measures of variability / Measures of central tendency …indicate the typical scores in a distribution …are related to skew (horizontal) / Measures of variability …show the dispersion of scores in a distribution …are related to kurtosis (vertical) / Measures of central tendency …indicate the typical scores in a distribution …are related to skew (horizontal) / Measures of variability …show the dispersion of scores in a distribution …are related to kurtosis (vertical)

Measures of variability / Range - the difference between the highest and lowest score / Variance - the total variation (distance) from the mean of all the scores / Standard deviation - the average variation (distance) from the mean of all the scores / Range - the difference between the highest and lowest score / Variance - the total variation (distance) from the mean of all the scores / Standard deviation - the average variation (distance) from the mean of all the scores

Measures of variability Range = Highest Score – Lowest Score Most sensitive to extreme scores! Range = Highest Score – Lowest Score Most sensitive to extreme scores!

Measures of variability / Again, variance is the overall distance from the mean of all scores (requires squaring the distance of each score from the mean) / Not as useful as the standard deviation -- the average distance scores fall from the mean / Again, variance is the overall distance from the mean of all scores (requires squaring the distance of each score from the mean) / Not as useful as the standard deviation -- the average distance scores fall from the mean

Measures of variability / Standard deviation, like the mean, is the most informative and elegant measure of variability. / The average distance of scores from the mean score -- deviation is distance! / Also like the mean, standard deviation has the most sampling stability / Standard deviation, like the mean, is the most informative and elegant measure of variability. / The average distance of scores from the mean score -- deviation is distance! / Also like the mean, standard deviation has the most sampling stability

How would these standard deviations differ? Mean = 6 Mean = 7.9 Range = 8 Range = 10 6

Standard deviation and shape of distribution Mean = 15 Std. Dev. = Mean = 15 Std. Dev. = 0.9 Mean = 15

Properties of Normal Distributions All normal distributions are single peaked, symmetric, and bell-shaped Normal distributions can have different values for mean and standard deviation but… All normal distributions follow the rule 68.3% of data within 1 standard deviation of the mean 95.4% of data within 2 standard deviations of the mean 99.7% of data within 3 standard deviations of the mean

 99.7% % % % % Mean