Using Statistics to Answer Questions
Lecture, Chapter 9
What are Statistics?
nStatistics is the branch of mathematics that involves the collection, analysis, and interpretation of data and aids researchers in making decisions during research.
nDescriptive statistics involves procedures to summarize a set of data to better understand it.
nInferential statistics involves procedures to analyze data after an experiment is completed; used to determine whether the IV has had a significant effect on the DV.
nMeasurement is the assignment of symbols to events according to a set of rules
nA scale of measurement is a set of measurement rules.
nNominal scale is a simple classification system in which events are recorded in categories.
nOrdinal scale permits events to be rank ordered.
nInterval scale permits rank ordering with the assumption of equal intervals between adjacent events.
nRatio scale permits rank ordering with the assumptions of equal intervals between adjacent events and a true zero point.
Another Explanation (Kaplan & Saccuzzo, 2005)
nMeasurement is the application of rules for assigning numbers to objects.
nProperties of measurement scales:
Magnitude – the property of “moreness;” when a particular instance has more,
less, or equal amounts of the given quantity than does another instance.
Equal intervals – when the difference between two points at any place on the scale
has the same meaning as the difference between two other points.
Absolute 0 – occurs when nothing of the property being measured exists.
Measures of Central Tendency
nThe mode is the number or event that occurs most frequently in a distribution.
nThe median is the number or score that divides the distribution into equal halves.
nThe mean is the arithmetic average, found by adding all scores and dividing by the number of scores.
nThe type of information you are seeking and the scale of measurement you are using determines the measure of central tendency you should use.
nWhat measure of central tendency is most useful for our Satisfaction with Life Scale (sample=9)?
nPie chart – graphical representation of the percentage allocated to each alternative as a slice of a circular pie.
nBar graph – a graph in which the frequency for each category of a qualitative variable is represented as a vertical column. The columns of a bar graph do not touch.
nHistogram – a graph in which the frequency for each category of a quantitative variable is represented as a vertical column which touches the adjacent column.
nFrequency polygon – a graph constructed by placing a dot in the center of each bar of a histogram and then connecting the dots.
nLine graph – frequently used to display the relationship between IV and DV found in experimental results.
Ordinate – vertical or y axis of a graph
Abscissa – horizontal or x axis of a graph
Measures of Variability
nVariability is the extent to which scores spread around the mean.
nThe range is the distance from the lowest score to the highest score in a distribution.
nVariance is a single number that represents the total amount of variation in a distribution.
nStandard deviation is the square root of the variance; related to the normal curve.
nNormal distribution is a symmetrical, bell-shaped distribution having half the scores above the mean and half the scores below the mean.
nThe normal distribution or normal curve is a hypothetical probability distribution that is frequently used in social science research.
nAs sample size increases, the distribution of sample means approximates normal curve.
Characteristics of the Normal Curve
nSymmetric-each side is an exact representation of the other
nTotal area under curve = 100%
nMean = 0 (for z scores)
Review of Hypotheses
nPreviously, we learned that hypotheses should involve synthetic statements, in that they can be either true or false.
nIn experimental research, you have both null and experimental hypotheses which state both possibilities in the synthetic statement.
nThe null hypothesis states that there is not a statistically significant relationship or difference.
nThe experimental hypothesis states that there is a statistically significant relationship or difference.
nSignificance level indicates to what degree experimental results (correlations or differences between groups) would occur by chance.
nUsually, we use .05 level of significance, which means that if the result occurs 5 or fewer times by chance in 100, then we assume that the difference we found did not occur by chance.
If result occurs by chance more than 5 times in 100, we say that it is not
significant and therefore accept the null.
If result occurs by chance 5 or fewer times in 100, we say that it is significant and
therefore reject the null.
nDegrees of freedom (df) is a mathematical term indicating the ability of a number in a specified set to assume any value; the higher the df, the greater the likelihood of yielding statistically significant results.
nSo far, we have looked at statistics describing one variable in one group of participants.
nIn correlation research, we are still using only one group, but we are measuring the degree of association of two variables within that group.
nWe, therefore, have two sets of scores for each participant.
nThe null hypothesis would be as follows:
There is not a statistically significant correlation between Life Satisfaction scores
and GPA in experimental psychology students.
nThe experimental hypothesis would be as follows:
There is a statistically significant correlation between Life Satisfaction scores and
GPA in experimental psychology students.
nCorrelation coefficient is a single number representing the degree to which the two variables (Life Satisfaction score and GPA) are related and ranges from -1 to +1.
Perfect negative correlation (-1) indicates that as one variable increases, the other
No correlation (0) indicates that as one variable increases, the other stays the
Perfect positive correlation (+1) indicates that as one variable increases, the other
nPearson Product-Moment correlation coefficient is used when both variables are interval or ratio scales of measurement and appear linear; the most widely used coefficient.
Other Correlation Coefficients
nIn inferential statistics, we have at least two groups and are measuring at least one variable.
nA t Test is used to evaluate the means of two groups to determine whether or not the difference between them occurred by chance.
Null – There is not a statistically significant difference in Life Satisfaction scores
between the two groups.
Experimental – There is a statistically significant difference in Life Satisfaction
scores between the two groups.
nThe null hypothesis states that all differences between groups are due to chance, not because of the IV.
One vs. Two Tail Tests
nIn a directional hypothesis, you are placing your chance factor of 5% entirely on one side of the distribution.
nIn a nondirectional hypothesis, you are dividing your chance factor in half and placing the 2.5% halves on both sides of the distribution.
nSince critical value for the one-tailed test is lower, it is easier to obtain statistically significant results with a directional hypothesis. Calculations between one and two tailed tests are the same, but different columns (with different critical values) should be consulted in the t table.
nIf your hypothesis is non-directional, you will need to use a two-tailed t-test.
nIf your hypothesis is directional, you will need to use a one-tailed t-test.
Type I and Type II Errors
nSince we determine that results are significant when the occurrence of the result by chance are 5 times or less in 100, we still have a probability of 1 in 20 that the results did occur by chance.
nType I error occurs when the experimental hypothesis is accepted and actually the null hypothesis is true.
nType II error occurs when the null hypothesis is accepted and actually the experimental hypothesis is true.
nEffect size is the magnitude or size of the experimental treatment.
nEffect size differs from statistical significance in that a significance test indicates the likelihood that the IV truly had an effect; effect size indicates how much or little the IV affected the DV.
nCohen’s d is often used (.2 to .5 = small effect; .5 to .8 = medium effect; .8+ = large effect)
nPearson product-moment correlation can be obtained, then squared (r˛) to see how much of the variance is accounted for by the variable.