Educational Research Descriptive Statistics Chapter th edition Chapter th edition Gay 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

Tabulation and Coding Tabulation is organizing data Identifying all relevant information to the analysis Separating groups and individuals within groups Listing data in columns

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.

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

Descriptive Statistics Purpose – to describe or summarize data in a parsimonious manner Four types Central tendency Variability Relative position Relationships

Descriptive Statistics Graphing data – a frequency polygon Vertical axis represents the frequency with which a score occurs Horizontal axis represents the scores themselves

Central Tendency Purpose – to represent the typical score attained by subjects Three common measures Mode Median Mean

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

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

Variability Purpose – to measure the extent to which scores are spread apart Four measures Range Quartile deviation Variance Standard deviation

Variability Range The difference between the highest and lowest score in a data set Characteristics Unstable measure of variability Rough, quick estimate

Variability Quartile deviation One-half the difference between the upper and lower quartiles in a distribution Characteristic - appropriate when the median is being used

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

Variability Standard deviation The square root of the variance Characteristics Many important statistical properties Relationship to properties of the normal curve Easily interpreted Formula

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

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.

The Normal Curve Properties Proportions under the curve  1 SD  68%  1.96 SD  95%  2.58 SD  99% Cumulative proportions and percentiles

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

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

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

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

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

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

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

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 correlations coefficients – dependent on the scales of measurement of the variables Spearman Rho – ranked data Pearson r – interval or ratio data

Measures of Relationship Interpretation – correlation does not mean causation Formula for Pearson r

Calculating Descriptive Statistics Symbols used in statistical analysis General rules form 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

Calculating Descriptive Statistics Using SPSS Windows Means, standard deviations, and standard scores The DESCRIPTIVES procedures Interpreting output Correlations The CORRELATION procedure Interpreting output

Formula for the Mean

Formula for Variance

Formula for Standard Deviation

Formula for Pearson Correlation

Formula for Z-Score