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##### Educational Research

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**Educational Research**Chapter 11 Descriptive Statistics Gay, Mills, and Airasian**Topics Discussed in this Chapter**• Preparing data for analysis • Types of descriptive statistics • Central tendency • Variation • Relative position • Relationships • Calculating descriptive statistics**Preparing Data for Analysis**• Issues • Scoring procedures • Tabulation and coding • Use of computers**Scoring Procedures**• Instructions • Standardized tests detail scoring instructions • Teacher-made tests require the delineation of scoring criteria and specific procedures • Types of items • Selected response items - easily and objectively scored • Open-ended items - difficult to score objectively with a single number as the result Objectives 1.1 & 1.2**Tabulation and Coding**• Tabulation is organizing data • Identifying all information relevant to the analysis • Separating groups and individuals within groups • Listing data in columns • Coding • Assigning names to variables • EX1 for pretest scores • SEX for gender • EX2 for posttest scores Objectives 2.1, 2.2, & 2.3**Tabulation and Coding**• Reliability • Concerns with scoring by hand and entering data • Machine scoring • Advantages • Reliable scoring, tabulation, and analysis • Disadvantages • Use of selected response items, answering on scantrons Objectives 1.4 & 1.5**Tabulation and Coding**• Coding • Assigning identification numbers to subjects • Assigning codes to the values of non-numerical or categorical variables • Gender: 1=Female and 2=Male • Subjects: 1=English, 2=Math, 3=Science, etc. • Names: 001=John Adams, 002=Sally Andrews, 003=Susan Bolton, … 256=John Zeringue Objectives 2.2 & 2.3**Computerized Analysis**• Need to learn how to calculate descriptive statistics by hand • Creates a conceptual base for understanding the nature of each statistic • Exemplifies the relationships among statistical elements of various procedures • Use of computerized software • SPSS-Windows • Other software packages Objective 2.4**Descriptive Statistics**• Purpose – to describe or summarize data in a parsimonious manner • Four types • Central tendency • Variability • Relative position • Relationships Objective 2.4**Descriptive Statistics**• Graphing data – a frequency polygon • Vertical axis represents the frequency with which a score occurs • Horizontal axis represents the scores themselves Objectives 3.1 & 3.2**Central Tendency**• Purpose – to represent the typical score attained by subjects • Three common measures • Mode • Median • Mean Objective 4.1**Central Tendency**• Mode • The most frequently occurring score • Appropriate for nominal data • Median • The score above and below which 50% of all scores lie (i.e., the mid-point) • Characteristics • Appropriate for ordinal scales • Doesn’t take into account the value of each and every score in the data Objectives 4.2, 4.3, & 4.4**Central Tendency**• Mean • The arithmetic average of all scores • Characteristics • Advantageous statistical properties • Affected by outlying scores • Most frequently used measure of central tendency • Formula Objectives 4.2, 4.3, & 4.4**Variability**• Purpose – to measure the extent to which scores are spread apart • Four measures • Range • Quartile deviation • Variance • Standard deviation Objective 5.1**Variability**• Range • The difference between the highest and lowest score in a data set • Characteristics • Unstable measure of variability • Rough, quick estimate Objectives 5.2 & 5.3**Variability**• Quartile deviation • One-half the difference between the upper and lower quartiles in a distribution • Characteristic - appropriate when the median is being used Objectives 5.2 & 5.3**Variability**• Variance • The average squared deviation of all scores around the mean • Characteristics • Many important statistical properties • Difficult to interpret due to “squared” metric • Formula Objectives 5.2 & 5.3**Variability**• Standard deviation • The square root of the variance • Characteristics • Many important statistical properties • Relationship to properties of the normal curve • Easily interpreted • Formula Objectives 5.2 & 5.3**The Normal Curve**• A bell shaped curve reflecting the distribution of many variables of interest to educators • See Figure 14.2 • See the attached slide Objective 6.1**The Normal Curve**• Characteristics • Fifty-percent of the scores fall above the mean and fifty-percent fall below the mean • The mean, median, and mode are the same values • Most participants score near the mean; the further a score is from the mean the fewer the number of participants who attained that score • Specific numbers or percentages of scores fall between ±1 SD, ±2 SD, etc. Objectives 6.1, 6.2, & 6.3**The Normal Curve**• Properties • Proportions under the curve • ±1 SD = 68% • ±1.96 SD = 95% • ±2.58 SD = 99% • Cumulative proportions and percentiles Objectives 6.3 & 6.4**Skewed Distributions**• Positive – many low scores and few high scores • Negative – few low scores and many high scores • Relationships between the mean, median, and mode • Positively skewed – mode is lowest, median is in the middle, and mean is highest • Negatively skewed – mean is lowest, median is in the middle, and mode is highest Objectives 7.1 & 7.2**Measures of Relative Position**• Purpose – indicates where a score is in relation to all other scores in the distribution • Characteristics • Clear estimates of relative positions • Possible to compare students’ performances across two or more different tests provided the scores are based on the same group Objectives 7.1 & 7.2**Measures of Relative Position**• Types • Percentile ranks – the percentage of scores that fall at or above a given score • Standard scores – a derived score based on how far a raw score is from a reference point in terms of standard deviation units • z score • T score • Stanine Objectives 9.3 & 9.4**Measures of Relative Position**• z score • The deviation of a score from the mean in standard deviation units • The basic standard score from which all other standard scores are calculated • Characteristics • Mean = 0 • Standard deviation = 1 • Positive if the score is above the mean and negative if it is below the mean • Relationship with the area under the normal curve Objective 9.5**Measures of Relative Position**• z score (continued) • Possible to calculate relative standings like the percent better than a score, the percent falling between two scores, the percent falling between the mean and a score, etc. • Formula Objective 9.5**Measures of Relative Position**• T score – a transformation of a z score where T = 10(z) + 50 • Characteristics • Mean = 50 • Standard deviation = 10 • No negative scores Objective 9.6**Measures of Relative Position**• Stanine – a transformation of a z score where the stanine = 2(z) + 5 rounded to the nearest whole number • Characteristics • Nine groups with 1 the lowest and 9 the highest • Categorical interpretation • Frequently used in norming tables Objective 9.7**Measures of Relationship**• Purpose – to provide an indication of the relationship between two variables • Characteristics of correlation coefficients • Strength or magnitude – 0 to 1 • Direction – positive (+) or negative (-) • Types of correlation coefficients – dependent on the scales of measurement of the variables • Spearman rho – ranked data • Pearson r – interval or ratio data Objectives 8.1, 8.2, & 8.3**Measures of Relationship**• Interpretation – correlation does not mean causation • Formula for Pearson r Objective 8.2**Calculating Descriptive Statistics**• Symbols used in statistical analysis • General rules for calculating by hand • Make the columns required by the formula • Label the sum of each column • Write the formula • Write the arithmetic equivalent of the problem • Solve the arithmetic problem Objectives 10.1, 10.2, 10.3, & 10.4**Calculating Descriptive Statistics**• Using SPSS Windows • Means, standard deviations, and standard scores • The DESCRIPTIVE procedures • Interpreting output • Correlations • The CORRELATION procedure • Interpreting output Objectives 10.1, 10.2, 10.3, & 10.4**Calculating Descriptive Statistics**• See the Statistical Analysis of Data module on the web site for problems related to descriptive statistics